Discharge Characteristics of Triangular-notch Thin-plate Weirs

Discharge

Characteristics

of

Triangular-notch

Thin-plate

Weirs

GEOLOGICAL

SURVEY

WATER-SUPPLY

PAPER

16I7-B

Discharge

Characteristics

of

Triangular-notch

Thin-plate

Weirs

By

JOHN

SHEN

STUDIES

OF

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

GEOLOGICAL

SURVEY

WATER-SUPPLY

PAPER

1617-B

A

comprehensive

study

of

the

discharge

characteristics

of

triangular-notch

thin-plate

weirs

UNITED

STATES

GOVERNMENT

PRINTING

OFFICE,

WASHINGTON

:

1981

UNITED

STATES

DEPARTMENT

OF

THE INTERIOR

JAMES

G.

WATT,

Secretary

GEOLOGICAL

SURVEY

Doyle

G.

Frederick,

Acting

Director

Library

of

Congress

Cataloging

in

Publication

Data

Shen,

John.

Discharge

characteristics

of

triangular-notch

thin-plate

weirs.

Geological

Survey

water-supply

paper

;

1617-B

Bibliography:

p.

Supt.

of

Docs,

no.:

I

19.13:1617-B

1.

Wiers.

2.

Stream

measurements.

I.

Title.

II.

Series:

United

States.

Geological

Survey.

Water-supply

paper;

1617-B.

TC175.S5625

551.48'3'0287

80-607045

For

sale

by

Superintendent

of

Documents,

U.S.

Government

Printing

Office

Washington,

D.C.

20402

PREFACE

An

initial

draft

of

the

report

on

the

discharge

characteristics

of

tri-

angular-notch,

thin-plate

weirs

was

prepared

in

1960.

In

draft

form

it

was

the

basis

for

the

development

of

recommended

flow-measurement

standards. In

1974,

the

discharge

equation

and

the

coefficients

of

discharge

proposed

in

the

report

were

adopted

as

part

of

an

interna-

tional

standard.

The

equation

and

coefficients

have

been

quoted

exten-

sively

in

several

foreign

national

standards,

books,

and

reports.

In

its

present

revised

form,

the

report

provides

background

and

verification

for

the

triangular-notch

weir

as

a convenient,

reliable,

and

inexpensive

device

for

measuring

small

flows

in

open

channels.

in

CONTENTS

Page

Preface

________________________________________

III

Symbols_______________________________________

VII

Abstract

_______________________________________

B1

Introduction

_____________________________________

1

Description

of

the

weir

______________________________

2

Basic

equation

of

discharge

____________________________

3

Dimensional

analysis

________________________________

4

Significance

of

the

ratios

in

equation

3

______________________

5

Coefficient

of

discharge

for

one liquid

_____________________

5

Modified

equations

for

one

liquid

_________________________

6

Review

of

the

literature

________________________________

7

Thomson's

experiments

______________________________

7

Barr's

experiments

_________________________________

8

Barr-Strickland

formula

____________________________

8

Cone's

experiments

_________________________________

9

Yarnall's

experiments

_______________________________

10

Greve's

experiments

______________________________

10

Lenz's

experiments

_______________________________

11

Experiments

of

Numachi,

Kurokawa,

and

Hutizawa

_____________

13

Other

experiments

_______________________________

13

Summary

____________________________________

15

Analysis

of

data

___________________________________

16

Evaluation

of

k

h

__________________________________________________

16

An

alternative

for

k

h

______________________________________________

19

Evaluation

of

C

e

_________________________________________________

22

Influence

of

h/P

and

PIB

___________________________________________

30

Comparison

of

formulas

______________________________

30

Basis

of

comparison

______________________________

30

90°-notch

weir

___________________________________

32

Other

notch

angles

_______________________________

32

Conclusions

_____________________________________

32

References

_____________________________________

39

Supplemental information

_____________________________

41

Requirements

for

precise

measurements

___________________

42

Specifications

for

installation

_______________________

42

Restrictions

on

the

geometric

parameters

________________

42

Restrictions

on

h

and

P

_________________________________________

42

Effect

of

neglecting

k

h

_________________________________________

42

Effect

of

approach-channel

conditions

___________________

43

Measurement

of

6

_____________________________________________

44

Measurement

of

h

_____________________________________________

44

Determination

of

gage

zero

________________________

44

VI

CONTENTS

ILLUSTRATIONS

Page

FIGURE

1.

The

triangular-notch,

thin-plate

weir

_________________

B3

2.

Evaluation

of

k

h

illustrated

with

Numachi,

Kurokawa,

and

Hutizawa

data

(v

=

90°)

_____________________________

17

3.

Evaluation

of

k

h

illustrated

with

Lenz

data

_____________

18

4-11.

Values

of:

4.

k

h

as

a

function

of

v

__________________

20

5.

k

as

a

function

of

v

_____________________-

21

6.

C

e

,

notch

angles

approximately

120°

__________

23

7.

C

e

,

notch

angles

approximately

90°

_________

24

8.

C

e

,

notch

angles

approximately

60°

________

25

9.

C

e

,

notch

angles

approximately

45°

______

26

10.

C

e

,

notch

angles

approximately

27°

_______

27

11.

C

e

,

notch

angles

of

10°

and

13°

_______________

28

12.

Coefficient

of

discharge

C

e

as

a

function

of

v

___________

29

13.

Coefficient

of

discharge

C

e

as

a

function

of

h/P

and

P/B

(v

=

90

°)

___

31

14-19.

Comparisons

of

formulas

for:

14.

90°-notch

weir

________________________

33

15.

Notch

angles

of

approximately

120°

______

34

16.

Notch

angles

of

approximately

60°

_____

35

17.

Notch

angles

of

approximately

45°

__________

36

18.

Notch

angles

of

approximately

27°

______

37

19.

Notch

angles

of

approximately

10°

_________

38

20.

Effect

of

using

h

instead

of

h

e

in

equation

6

____________

43

TABLES

TABLE

1.

Scope

of

Greve's

experiments

______________

Bll

2.

Scope

of

Lenz's

experiments

with

water

____________

12

3.

Values

ofN

and

a

for

equation

16

(Lenz)

______

13

CONVERSION

FACTORS

The

inch-pound

system

of

units

is

used

throughout

this

report.

The

following

factors

may

be

used

to

convert

inch-pound

units

to

International

System

of

Units

(SI):

Multiply

inch-pound

units

By

ft

(feet)

in

(inch)

ft

2

(square

feet)

Length

0.30480

25.4

Area

0.09290

To

obtain

SI

units

m

(meters)

mm

(millimeters)

m

2

(square

meters)

CONTENTS

VII

Multiply

inch-pound

units

By

To

obtain

SI

units

Volume

ft

3

(cubic

feet)

0.02832

m

3

(cubic

meters)

Temperature

°F

(degrees

Fahrenheit)

5/9

(°F-32)

°C

(degrees

Celsius)

SYMBOLS

a

exponent

in

equation

16

b

width

of

rectangular

notch

B

width

of

approach

channel

C

coefficient

of

discharge

C

e

coefficient

of

discharge

(based

on

effective

head)

E

relative

error

g

acceleration

due

to

gravity

h

piezometric

head

referred

to

vertex

of

notch

h

e

effective

head

j

exponent

in

equation

9

h

adjustment

quantity

h

head-adjustment

quantity

L

a length

M

coefficient

in

equation

12

m

exponent

in

equation

12

N

coefficient

in

equation

16

n

exponent

in

equation

12

P

height

of

weir

notch

above

bottom

of

channel

Q

discharge

R

Reynolds

number

S

slope

of

sides

of

notch

W

Weber

number

7

specific

weight

6

included

angle

between

sides

of

notch

H

dynamic

viscosity

p

density

a

surface tension

STUDIES

OF

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

DISCHARGE

CHARACTERISTICS

OF

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

By

JOHN

SHEN

ABSTRACT

The

triangular-notch,

thin-plate

weir

is

a

convenient,

inexpensive,

and

relatively

precise

flow-measuring

instrument.

It

is

frequently

used

to

measure

the

flow

of

water

in

laboratories

and

in

small,

natural

streams.

This

report

includes

an

extensive

review

of

the

literature

and

presents

a

comprehensive

analysis

of

the

discharge

characteristics

of

triangular-notch

weirs.

Previously

published

data

are

analyzed

in

the

light

of

the

effective-head

concept,

and

a

new

discharge formula

is

proposed.

Coefficients

are

recom-

mended

and

requirements

for

precise

measurements

are

described.

Limits

of

applicabili-

ty

are

discussed.

INTRODUCTION

The

triangular-notch,thin-plate

weir

is

used

widely

for

measuring

the

flow

of

liquids

in

flumes

and

open

channels.

Simple

in

design

and

easily

made

from

readily

available

materials,

it

is

inexpensive,

convenient

to

use,

and

easy

to

maintain.

In

permanent

or

portable

form

it

is

frequent-

ly

used

to

measure

the

flow

of

water

in

laboratories

and

in

small,

natural

streams.

When

several

forms

of

weirs

or

flumes

might

be

used,

the

triangular-notch

weir

is

often

preferred

because

of

its

greater

ac-

curacy

at

low

flows

or

its

lesser

sensitivity

to

approach-channel

geometry

and

velocity

distribution.

Within

the

range

of

conditions

for

which

verification

data

are

adequate, and

with

reasonable

care

in

its

construction, installation,

and

use,

the

triangular-notch,

thin-plate

weir

is

a

relatively

precise

instrument.

The

triangular-notch

weir

has

been

the

subject

of

considerable

ex-

perimental

research

and

published

discussion.

Unfortunately,

however,

most

of

the

laboratory

investigations

have

been

restricted

to

a

narrow

range

of

notch

angles

and

channel

geometries.

The

90°-notch

weir

has

been

most

extensively

studied.

With

few

exceptions,

water

has

been

the

liquid

used

in

laboratory

tests.

A

large

number

of

empirical

discharge

formulas

have

been

proposed

for

triangular-notch

weirs.

Most

of

these

were

designed

to

fit

a

par-

Bl

B2

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

ticular

set

of

experimental

data.

None

provides

a

comprehensive

solu-

tion,

even

for

a

single

liquid.

Deficiencies

in

these

formulas

are

con-

cealed

with

numerous

limits

of applicability

which

greatly

restrict

their

usefulness.

This

report

presents

a

comprehensive

analysis

of

the

discharge

characteristics

of

triangular-notch,

thin-plated

weirs.

Previously

published

data

from

various

sources

have

been

reanalyzed,

and

some

of

the

traditional

discharge

formulas

are

compared

with

an

original

solu-

tion.

The

report

is

based

on

one

of

a

series

of

studies

of

weirs

and

spillways

which

was

undertaken

by

the

U.S.

Geological

Survey

under

the

direction

of

C.

E.

Kindsvater.

Previously

published

reports

in

the

series

are

concerned

with

broad-crested

weirs

(Tracy,

1957),

rec-

tangular,

thin-plate

weirs

(Kindsvater

and

Carter,

1959),

and

embankment-shaped

weirs

(Kindsvater,

1964).

Portions

of

this

report

are

adapted

from

unpublished

reports

and

drafts

of

weir

standards

prepared

by

C.

E.

Kindsvater.

DESCRIPTION

OF

THE

WEIR

The

weir

which

is

the

subject

of

this

study

is

a

symmetrical,

V-shaped

notch

in

a

vertical

thin

plate.

The

line

which

bisects

the

angle

of

the

notch

is

vertical

and

equidistant

from

the

sides

of

the

approach

chan-

nel.

The

weir

plate

is

smooth,

plane,

and

perpendicular

to

the

sides

as

well

as

the

bottom

of

the

approach

channel.

Figure

1

shows

a

triangular-notch,

thin-plate

weir

installed

at

the

end

of

a

rectangular

channel.

The

crest

surfaces

of

the

weir

notch

are

plane

surfaces

which

form

sharp,

right-angle

corners

at

their

intersection

with

the

upstream

face

of

the

weir

plate.

The

width

of

the

crest

surface

varies,

but

it

is

general-

ly

between

1/32

and

1/16

inch.

If

the

weir

plate

is

thicker

than

1/16

inch,

the

downstream

edges

of

the

notch

are

chamfered

to

make

an

angle

of

not

less

than

45°

with

the

surface

of

the

crest.

Ideally,

the

channel

upstream

from

the

weir

is

straight,

smooth,

horizontal,

rec-

tangular,

and

of

sufficient

length

to

develop

the

normal

(uniform

flow)

turbulence

and

velocity

distribution

for

all

discharges.

Usually,

however,

it

is

less

than

ideal,

and

baffles

or

screens

are

provided

in

order

to

simulate

a

normal

velocity

distribution.

Channel

and

tailwater

conditions

downstream

from the

weir

are

such

as

to

permit

a

free,

fully

ventilated

flow

from

the

notch.

Provisions

for

ventilation

ensure

that

pressure

on

the

nappe

surfaces

is

atmospheric.

The

tailwater

is

low

enough

that

it

does

not

interfere

with

the

ventilation

of

the

nappe

or

free

flow

from the

notch.

The

head

on

the

weir

is

the

measured

vertical

distance

from

the

water

surface

to

the

vertex

of

the

notch.

The

head-measuring

section

is

located

a

sufficient

distance

upstream

from

the

weir

to

avoid

the

region

of

surface

draw-down,

and

it

is

sufficiently

close

to

the

weir

that

the

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B3

Upstream

face

of

weir

plate

^45°

FIGURE

l.-The

triangular-notch,

thin-plate

weir.

energy

loss

between

the

measuring

section

and

the

weir

is

negligible.

A

distance

of

4

to

5

h

is

recommended.

BASIC

EQUATION

OF

DISCHARGE

The

traditional

equation

of

discharge

for

triangular-notch

weirs

is

derived

on

the

basis

of

an

assumed

analogy

between

the

weir

and

the

B4

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

orifice.

In

the

derivation,

an

approximate

velocity

equation

is

in-

tegrated

over

assumed

area

limits

of

the

nappe

in

the

plane

of

the

weir.

The

result

is

an

equation

which

is

useful

mainly

because

it

is

dimen-

sionally

correct

and

because

it

is

used

almost

universally

as

the

basis

for

the

analysis

of

experimental

data.

It

is

used

herein

as

the

basic

equation

of

discharge.

In

its

traditional

form,

the

basic

discharge

equation

is

Q=CV20

tan

*60

(1)

lo

£

in

which

Q

is

the

volume

rate

of

flow

or

discharge

in

cubic

feet

per

sec-

ond;

C is

the

nondimensional

coefficient

of

discharge;

g

is

the

accelera-

tion

due

to

gravity

in

feet

per

second

per

second;

0

is

the

angle

included

between

the

sides

of

the

notch,

usually

measured

in

degrees;

and

h

is

the

potentiometric

head

or

height

of

the

upstream

liquid

surface

measured

with

respect

to

the

vertex

of

the

notch,

in

feet

(in

metric

units,

discharge

would

be

stated

in

cubic

meters

per

second,

head

in

meters,

and

g

in

meters

per

second

per

second;

C,

being

nondimen-

sional,

would

remain

unchanged

in

numerical

value).

Whereas

C is

assumed

to

be

a

coefficient

of

contraction

in

the

tradi-

tional

derivation,

in

actuality

it

is

an

experimentally

determined

coeffi-

cient

which

is

dependent

upon

all

the

variables

needed

to

describe

the

channel,

the

weir,

and

the

discharging

liquid.

Thus,

in

the

absence

of

a

rigorous

theoretical

solution,

dimensional

relations

must

be

used to

guide

the

analysis

of

experimental

data

on

triangular-notch

weirs.

DIMENSIONAL

ANALYSIS

The

principal

variables

needed

to

define

the

discharge

characteristics

of

a

triangular-notch,

thin-plate

weir

in

a

rectangular

channel

(fig.

1)

are:

Q,

the

discharge;

B,

the

width

of

the

approach

channel;

P,

the

height

of

the

notch

vertex

with

respect

to

the

floor

of

the

approach

channel;

h,

the

head

on

the

weir,

referred

to

the

vertex

of

the

notch;

6,

the

angle

included

between

the

sides of

the

notch;

p,

the

density

of

the

liquid;

n,

the

viscosity

of

the

liquid;

a,

the

surface

tension

of

the

liquid;

and

7,

the

specific

weight

of

the

liquid.

If

the

discharge

is

selected

as

the

dependent

variable,

a

complete

statement

of

the

discharge

function

is

Q-f<jB,P,h>9

t

p

9

p

t

o,y).

(2)

From

this

equation

a

nondimensional

discharge

ratio

can

be

expressed

as

a

function

of

five

nondimensional

ratios,

g

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B5

in

which

g=ylp.

The

dependent

ratio

in

equation

3

is

proportional

to

the

coefficient

of

discharge.

The

first

three

ratios

on

the right-hand

side

describe

the

geometry

of

the

weir,

approach

channel,

and

flow

pattern;

the

last

two

ratios

are

the

Reynolds

number

(R)

and

the

Weber number

(W).

SIGNIFICANCE

OF

THE

RATIOS

IN

EQUATION

3

The

first

independent

ratio

in

equation

3,

h/P,

is

a

measure

of

the

depth-contraction

characteristic.

Because

velocities

in

the

channel

upstream

from

the

weir

are

proportional

to

the

head,

the

h/P

ratio,

in

combination

with

h/B

and

0,

is

also

a

measure

of

the

relative

magnitude

of

the

velocity

in

the

approach

channel.

The

hIB

ratio,

in

combination

with

6,

is

a

measure

of

the

width-

contraction

characteristic

(resembling

bIB

for

rectangular-notch

weirs;

see

Kindsvater

and

Carter,

1959).

A

more

convenient

ratio

to

serve

the

same

purpose

is

P/B,

which

is

the

quotient

of

h/B

divided

by h/P.

Thus,

as

a

width-contraction ratio,

P/B

depends

on

h/P

as

well

as

0.

The

notch

angle,

0,

is

a

nondimensional

measure

of

the

shape

of

the

weir

notch.

It

is

also

a

measure

of

the

cross-sectional

shape

of

the

discharging

liquid

stream

in

the

plane

of

the

weir.

In

contrast

with

the

discharge

from

most

other

forms

of

weirs,

the

shape

of

the

stream

from

a

triangular-notch

weir

is

geometrically

similar

at

all

values

of

head,

except

as

it

is

influenced

by

h/P

and

hIB.

The

Reynolds

number,

R,

is

a

measure

of

the

relative

influence

of

fluid

viscosity.

It

is

usually

expressed

as

R

=

VLp//t,

in

which

V

is

a

typical

velocity,

L

is

a

significant length,

p

is

the

mass

density,

and

/A

is

the

dynamic

viscosity

of

the

liquid.

In

this

instance

the

head,

h,

is

the

dominant

length

parameter.

As

the

velocity

is

also

a

function

of

h,

the

influence

represented

by

the

Reynolds

number

can

be

expressed

in

terms

of

h,

p,

and

p.

The

Weber

number,

W, is

a

measure

of

the

relative

influence

of

sur-

face

tension.

It

is

usually

expressed

as

W

=

VVx/v/o/p

in

which

a

is

the

coefficient

of

surface

tension

of

the

liquid.

As

in

the

Reynolds

number,

h

is

the

most

significant

length

parameter

for

use

in

the

Weber

number.

Because

V

is

also

a

function

of

h,

the

influence

represented

by

the

Weber

number

can

be

expressed

in

terms

of

h,

a,

and

p.

COEFFICIENT

OF

DISCHARGE

FOR

ONE

LIQUID

In

equation

3

the

dependent

variable

is

proportional

to

the

coefficient

of

discharge.

The

independent

variables

include

three

geometric

ratios

and

two

fluid-property

ratios.

As

explained

in

the

preceding

section,

one

of

the

geometric

ratios,

hIB,

can

be

replaced

with

P/B,

which

is

more

convenient

because

it

is

a

constant

for

a

given

weir

installation.

B6

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

For

one

liquid

over

a

limited

temperature

range,

/A,

a,

and

p

can

be

assumed

to

be

constants.

Under

these

circumstances,

the

effects

of

viscosity

and

surface

tension

are

the

absolute

magnitude

of

h

alone. Thus,

both

R

and

W

in

equation

3

can

be

replaced

with

the

quan-

tity

h.

It

follows

that,

for

one

liquid

and

a

limited

range

of

temperatures,

(4)

In

this

report

the

only

liquid

of

interest

is

water

at

normal

atmospheric

temperatures.

MODIFIED

EQUATIONS

FOR

ONE

LIQUID

The

separate

quantity

h

in

equation

4

represents

the

combined

and

inseparable

effects

of

viscosity

and surface

tension

on

the

coefficient

of

discharge.

The

principal

effects of

viscosity

are

those

which

are

associated

with

flow

pattern

modifications

due

to

boundary

resistance

and

separation.

The

principal

effects

of

surface

tension

are

associated

with

the

curvature

of

the

nappe

and

"clinging"

on

the

crest

surfaces

of

the

notch.

A

detailed

discussion

of

these

effects

is

contained

in

the

paper

by

Kindsvater

and

Carter

(1959)

on

rectangular,

thin-plate

weirs.

The

following

pertinent

conclusions

are

drawn

from

that

paper:

1.

The

combined

effects

of

viscosity

and

surface

tension

on

a

notch

weir

are

commensurate

with fictitious

increases

in

head

and

notch

width.

2.

The

coefficient

of

discharge

can

be

expressed

in

terms

of

the

geometric

ratios

alone

if

the

effects

of

viscosity

and

surface

tension

are

accounted

for

by

an

adjustment

of

the

measured

values

of

head

and

notch

width

(for

triangular-notch

weirs,

for

which

notch

width

is

a

function

of

h,

only

the

measured

values

of

head

need

be

adjusted.)

3.

The

adjustment

of

measured

values

of

h

can

be

accomplished

very

simply,

as

indicated

by

the

equation

h

e

=

h+kh,

(5)

in

which

h

e

is

the

effective

(adjusted)

head,

h

is

the

measured

head,

and

kfr

is

a

quantity

which

must

be

determined

by

experiment.

1

4.

Use

of

the

effective

head,

h

e

,

in

place

of

h,

results

in

the

following

modifications

of

equations

1

and

4

for

one

liquid:

1

Subsequent

to

the

initial

exposition of

the

"effective-head

concept"

by

Kindsvater

in

1956,

it

was

applied

successful-

ly

to

several

kinds

of

weirs

by

others

(Carter,

1956;

Schlag,

1962a;

Schlag,

1962b;

John

Shen,

written

commun.,

1962;

Burgess

and

White,

1966,

among

others).

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B7

W

(6)

and

Equation

6,

in

combination

with

experimentally

derived

values

of

C

e

and

kh,

is

proposed

as

a

new,

comprehensive

discharge formula

for

tri-

angular-notch,

thin-plate

weirs.

2

Equation

7

defines

C

e

as

a

function

of

geometry

alone;

i.e.,

independent

of

fluid-property

effects

represented

by

h.

The

adequacy

and

validity

of

these

equations

must

be

examined

in

the light

of

experimental

data.

The

data

used

for

this

purpose

have

been

abstracted

from

the

literature.

REVIEW

OF

THE

LITERATURE

THOMSON'S

EXPERIMENTS

James

Thomson,

Professor

of

Civil

Engineering

at

Queen's

College,

Belfast,

Ireland,

was

among

the

earliest

experimental

investigators

of

the

triangular-notch

weir

(Thomson,

1858,

1861).

Thomson's

ex-

periments

were

made

at

a

pond

in

an

open

field

near

Belfast.

His

pur-

pose

in

making

the

tests

was

to

acquire

experimental

evidence

to

sup-

port

his

proposal

that

triangular-notch

weirs

be

used

instead

of

rec-

tangular

weirs

when

the

range

of

discharges

to

be

measured

included

very

small

flows.

Professor

Thomson's

experimental

equipment

was

crude

by

modern

standards,

and

the

range

of

conditions

covered

by

his

experiments

was

small.

Nevertheless,

some

of

his conclusions

are

pertinent

here.

Two

notch

angles,

90°

and

127°,

were

investigated.

For

the

90°

notch,

Thomson

recommended

a

constant

value

of

C

(eq

1)

=

0.593.

For

the

127°

notch,

he

recommended

C-

0.617.

Experiments

made

with

the

vertex

of

the

notch

at

the

level

of

the

channel

(P=0),

with

the

channel

wide

enough

to

provide

negligible

approach

velocities,

showed

no

ap-

preciable

influence

which

could

be

attributed

directly

to

the

value

of

P.

With

reference

to the

127°

notch,

Thomson

noted

an

instability

which

he

associated

with

the

nappe's

tendency

to

cling

to

the

upper

portions

of

the

wetted

crest

surfaces.

This

led

him

to

recommend

for

larger

discharges

that

two

or

more

90°

notches

(located

side-by-side)

be

used

instead

of

a

single,

large-angle

notch.

2

Equation

6

is

described

as

the

Kindsvater-Shen

equation

in

the

international

standard

(International

Standards

Organization,

1975)

which

was

based

in

part

on

the

1960

draft

of

this

report.

B8

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

BARR'S

EXPERIMENTS

One

of

the

most

extensive

among

the

early

investigations

began

as

an

effort

to

check

Thomson's

conclusions

(Barr,

1910).

James

Barr,

then

a

Carnegie

Research

Scholar

at

Glasgow

University,

Scotland,

performed

his

experiments

in

the

James

Watt

Engineering

Laboratories.

Many

of

the

traditional

formulas

for

90°-notch

weirs

are

based

on

the

results

of

Barr's

experiments.

Because

his

instrumenta-

tion

and

technique

were

good,

his

data

are

among

those

selected

for

analysis

in

this

report.

Barr's

experiments

were

performed

primarily

with

three

different

90°

notches.

One

weir

was

made

of

brass

and

two of

iron

plate.

The

upstream

edges

of

the

crest

surfaces

were

square

and

sharp,

and

the

width

of

the

crest

surfaces

varied

from

1/16

inch

to

1/12

inch.

A

54°

notch

was

also

tested.

The

weirs

were

placed

at

the

end

of

a

channel

which

was

4

feet

wide

and

22

feet

long.

The

vertex

of

the

notches

was

about

2

feet

above

the

floor

of

the

approach

channel

for

most

of

Barr's

tests.

To

investigate

the

effect

of

channel

width

and

weir

height,

Barr

used

false

walls

and

floors

in

the

approach

channel.

The

false

walls

extended

3

feet

upstream,

and

the

floor

extended

3.5

feet

upstream

from

the

weir

plate.

Tests

were

also

made

to

investigate

the

effects

of

roughness

and

projections

on

the

upstream

face

of

the

weir.

For

this

purpose,

roughness

was

produced with

a

mixture

of

varnish

and

emery

particles.

Projections

consisted

of

narrow

metal

strips

which

were

bolted

to

the

weir

bulkhead.

On

the

basis

of

his

experiments,

Barr

disagreed

with

Thomson's

con-

clusion

that

the

coefficient

of

discharge

was

a

constant.

His

results

in-

dicated

that

the

coefficient

varied

with

the

head.

He

concluded

that

the

coefficient

was

increased

by

roughness

and

projections

on

the

upstream

face

of

the

weir.

He

also concluded

that

the

coefficient

was

independent

of

channel

width

if

the

width

was

at

least

eight

times

the

head.

For

channel

widths

less

than

Sh,

the

coefficient

increased

as

the

width

decreased.

He

also

found

that

the

coefficient

decreased

slightly

with

decreasing

values

of

P

when

P

was

less

than

Kh,

where

K

is

a

coefficient

equal

in

value

to

the

head

in

inches.

BARR-STRICKLAND

FORMULA

A

widely

quoted

formula

for

90°-notch

weirs

was

derived

by

T.

P.

Strickland

(1910)

on

the

basis

of

Barr's

experiments.

As

an

equation

for

C,

the

Barr-Strickland

formula

is

C=

0.566+^^-

(8)

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B9

CONE'S

EXPERIMENTS

Subsequent

to

the

work

of

Thomson

and

Barr,

the

triangular-notch

weir

was

widely

used

and

extensively

investigated

in

the

United

States

as

well

as

Europe.

One

of

the

earliest

investigators

in

this

country

was

V.

M.

Cone

(1916),

whose

work

on

rectangular,

trapezoidal,

circular,

and

triangular

thin-plate

weirs

was

performed

in

the

laboratory

of

the

Colorado

Agricultural

Experiment

Station

at

Fort

Collins,

Colorado,

under

the

sponsorship

of

the

U.S.

Department

of

Agriculture.

The

channel

used

for

Cone's

experiments

was

20

feet

long,

10

feet

wide,

and

6

feet

deep.

The

vertex

of

the

weir

notch

was

4.5

feet

above

the

channel

floor.

Cone

tested

five

triangular-notch

weirs,

with

notch

angles

equal

to

120°,

90°, 60°,

30°,

and

28°4'.

All

weirs

were

made

of

V4-inch

brass

plate,

with

sharp

upstream

edges

and

crest-surface

widths

of

1/16

inch

(except

the

120°

notch,

for

which

the

crest

surface

was

1/32-inch

wide).

A

total

of

98

tests

was

made,

covering

a

range

of

heads

from

0.20

to

1.35

feet.

Cone,

like

Thomson,

observed

that

notch

angles

of

120°

or

larger

were

impractical

because

of

the

tendency

for

the

nappe

to

cling

to

the

upper

portion

of

the

wetted

crest

surfaces.

On

the

basis

of

his

ex-

periments,

he

proposed

a

general

formula

for

all

notch

angles

between

28°

and

109°.

As

an

equation

for

C,

Cone's

formula

is

r

0.576

0.00584

,

Q

.

u

=

+

'

(y

)

in

which

S

is

the

slope

of

the

sides

of

the

notch,

expressed

decimally,

and

50.75

Cone

also

made

a

limited

number

of

tests

on

the

90°

notch

with

the

floor

of

the approach

channel

at

the

vertex

of

the

notch.

He

concluded

that

the

coefficient

of

discharge

was

slightly

greater

for

P=0

than

for

P=4.5

feet.

Thus,

the

results

of

tests

made

by

Thomson,

Barr,

and

Cone

give

conflicting

indications

of

the

influence

of

P.

With

regard

to

crest-surface

width,

Cone

concluded

that

"thin

edge"

and

"sharp

edge"

need

not

imply

that

the

weir

notches

are

knife-edged.

He

observed

that

the

allowable

width

depends

on

the

head.

Ex-

periments

indicated

that

"edges

V4-inch

thick

showed

that

while

water

would

adhere

to

the

notch

edges

with

a

head

of

0.15

foot,

there

was

no

adherence

with

heads

of

0.2

foot

and

over."

He

recommended

that

head

measurements

"be

made

either

at

a

distance

of

at

least

4/&

upstream

from

the

notch

or

at

a

distance

of

at

least

2h

sidewise

from

the

end

of

the

crest

of

the

notch"

(Cone,

1916, p.

1089-1090).

BIO

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

YARNALL'S

EXPERIMENTS

D.

R.

Yarnall's

principal

work

on

triangular-notch

weirs

was

based

on

research

performed

at

the

University

of

Pennsylvania

(Yarnall,

1912,

1926).

His

tests

were

made

in

a

channel

6

feet

wide

and

9

feet

long,

with

only

4

feet

8

inches

between

baffles

and

weir

plate.

The

ver-

tical

distance

from

the

vertex

of

the

notch

to

the

floor

of

the

approach

channel

was

3.6

feet.

The

notch

angles

tested

by

Yarnall

were

90°,

53°8',

27°,

and

13°8',

and the

width

of

the

weir-crest

surfaces

was

1/32

inch.

A

total

of

61

tests

were

made

on

the

four

weirs,

with

a

range

of

heads

from

0.35

to

1.26

feet.

A

notable

feature

of

Yarnall's

experiments

was

the

short

channel,

which

he

had

equipped

with

simple

but

effective

baffles

in

order

to

pro-

duce

a

uniform

velocity

distribution

upstream

from

the

weir.

Such

demonstrations

of

the

practicality

of

short

weir

"boxes"

led

to

their

widespread

use

as

portable

flow-meters

in

hydraulics

laboratories,

in-

dustrial

plants,

and

field

applications.

GREVE'S

EXPERIMENTS

The

most extensive

investigation

of

different

notch

angles

is

represented

by

the

work

of

Professor

F.

W.

Greve

at

Purdue

Universi-

ty

(Greve,

1932).

A

total

of

16

different

weirs,

with

notch

angles

vary-

ing

from

25°03'

to

118°11',

were

investigated.

All

of

the

weirs

were

tested

in

a

channel

which

was

5

feet

wide

and

16

feet

long

from

baffles

to

weir

plate.

The

distance

from

notch

vertex

to

channel

floor

varied

from

4.5

to

5

feet.

The

weir

plates

were

cut

from

3

/s-inch

soft

steel

plates,

with

the

downstream

edges

of

the

notches

beveled

to form

a

crest

which

was

sharp

on

the

upstream

edge

and

1/32-inch

wide.

The

report

stated

that

the

plates

were

coated

with

asphalt

paint,

but

not

specifically

that

the

paint

was

applied to

the

surfaces

in

the

immediate

vicinity

of

the

crest.

Thus,

there

is

some

uncertainty regarding

the

smoothness of

the upstream

face

and

the

sharpness

and

width

of

the

crest

during

the

tests.

Table

1

shows

a

summary

of

the

scope

of

Greve's

experiments.

In

summarizing

the

results

of

his

tests,

Greve

presented

a

com-

prehensive

discharge

formula

for

all

notch

angles

between

20°

and

120°.

His

formula

for

C

is

C=

(11)

(tan

|)0.004^0.03

Referring

to

supplementary

tests

on

a

10°

notch,

Greve

observed

that

the

small

notch

exhibited

characteristics

of

"excessive

clinging"

and

gave

"results

which

were

not

in

harmony"

with

his

general

formula.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

Bll

TABLE

1.-Scope

ofGreve's experiments

Series

W-3.3

W-3.4

______

W-3.0

W-3.1

W-3.5

W-3.6

W-3.6A

_____

W-3.7

W-3.8

_

W-3.9

______

W-3.10

______

W-3.11

_______

W-3.12

W-3.13

W-3.14

W-3.15

Notch

angle

25°03'

36°53'

40°00'

44°

24'

45°23'

53°

55'

53°55'

59°07'

69°38'

81°

52'

94°39'

98°47.5'

102°2(y

110°00'

118°H'

Number

of

tests

37

30

27

46

27

13

32

17

31

43

33

46

22

31

36

30

Range

of

head

(ft)

From

0.167

.146

.282

.201

.153

.251

.241

.190

.217

.243

.342

.219

.318

.257

.217

.251

To

0.991

1.090

.854

1.025

1.178

1.128

1.211

1.199

1.254

1.216

1.199

1.111

.988

1.047

.945

.725

Range

of

temperature

(°F)

From

62

63

69

65

71

72

71

77

71

74

69

72

74

80

To

69

71

73

79

67

73

72

84

75

77

74

76

81

82

A

final

statement,

based

on

attempts

to

correlate

his

results

with

the

results

of

others,

is

noteworthy:

"it

is

unwise

to

formulate

a

general

ex-

pression

for

discharge

from

data

compiled

from

different

sources

unless

such

expression

incorporates

certain

factors

which

correlate

variation

in

the

physical

characteristics

of

the

testing

plants"

(Greve,

1932,

p.

33).

LENZ'S

EXPERIMENTS

The

effect

of

viscosity

and

surface

tension

were

objectives

of

the

ex-

periments

made

by

Professor

Arno

T.

Lenz

at

the

University

of

Wisconsin

(Lenz,

1943).

Using

water

and

two

different

oils

at

temperatures

ranging

from

51°

to

102°

F,

Lenz's

tests

covered

a

range

of

viscosities

from

1

to

150

times

that

of

water,

surface

tensions

from

1

to

0.41

times

that

of

water,

and densities

from

1

to

0.85

times

that

of

water.

Six

different

weirs,

with

notch

angles

ranging

from

10°

to

90°,

were

investigated.

The

tests

were

made

in

a

steel

tank

3.5

feet

wide

and

7

feet

9

inches

long

from

baffles

to

weir

plate.

The

vertex

of

the

notch

was

3

feet

above

the

floor

of

the

tank.

The

weir

plates

were

made

of

Vs-inch

brass,

sharp-edged

on

the

upstream

side

and

beveled

on

the

downstream

side

to

form

a

sharp-edged

crest

1/32-inch

wide.

Table

2

shows

a

summary

of

the

scope

of

Lenz's

tests

which,

for

the

purpose

of

this

report,

includes

only

those

experiments

using

water.

B12

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

TABLE

2.

-Scope

ofLenz's

experiments

with

water

Series

90

D_

_

__

60D_

_

__

45D

__

_ _

28D

_

20

D_

_

__

20

D

_

__

10

D

__ _ _

Notch

angle

89°53.2

;

59°51.0'

44°40.0'

27°55.0'

l$°55.ff

20°

2.5'

10°20.1'

Number

of

tests

14

33

18

24

19

33

Range

of

head

(ft)

From

0.23

.16

.20

.26

.18

.24

To

0.50

.60

.55

.70

.88

Range of

temperatures

From

61

54

56

51

54

52

To

63

62

60

59

67

65

54

On

the

basis

of

his

experiments,

Lenz

derived

a

general

formula

for

the

coefficient

of discharge

M

C=0.560+

'

(12)

in

which

R=

N

/g^3

/Gi/p)

is

the

Reynolds

number;

W=(pgh

2

)/a,

is

the

Weber

number;

and

M,

n,

and

m

are

empirically

determined

functions

of

the

notch

angle,

M=0.475+

°'

225

>

(13)

-

\0.80

/

0

\0-09

(14)

n~

0.165

(tan

-^

,

and

«

0.170

,

1(

.v

m-

(15)

-

'

0.035

That

the

formula

is

not

truly

general

is

indicated

by

the

fact

that

strict

limits

of

applicability

were

placed

on

R,

W,

C,

and

6.

For

water

at

one

temperature

(70°

F),

limits

on

R

and

W

are

re-

moved,

and

C

can

be

computed

with

the

equation

C=0.560+

>

(16)

h

a

in

which

AT

and

a,

like

M,

n,

and

m,

are

functions

of

0

alone.

In

equation

16,

R

and

W

are

functions

of

the

quantity

h,

because,

for

one

liquid

at

a

constant temperature,

the

fluid

properties

are

constant

(See

the

discus-

sion

leading

to

eq

4).

Lenz

gives

experimentally

determined

values

of

AT

and

a

in

table

3.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B13

TABLE

3.

-

Values

o/N

and

a/or

equation

16

(Lenz)

Notch

angle,

9

Constant

N

__

a

_

90°

.

0.0159

.588

60°

0.0203

.582

45°

0.0238

.579

28°04'

0.0315

.575

20°

0.0390

.573

10°

0.0624

.569

Basing

his

comparison

on

the

tests

of

Barr,

Yarnall,

Greve,

Ho

and

Wu

(Ho

and

Wu,

1931),

Lenz

concluded

that

equation

16

would

give

results

accurate

to

within

1

percent

for

all

notch

angles

between

28°

and

90°.

He

observed

that

increasing

the

temperature

of

water

from

40°

F

to

165°

F

decreases

the

computed

value

of

C

about

1

percent.

EXPERIMENTS

OF

NUMACHI,

KUROKAWA,

AND

HUTIZAWA

The

most extensive

investigation

of

a

single

notch

angle

and

a

single

liquid

is

represented

by

the

experiments

performed

by

Numachi,

Kurokawa,

and

Hutizawa

(1940)

at

Tohoku

Imperial University

in

Japan.

These

tests,

all

of

which

were

made

with

water

and a

90°

triangular-notch

weir,

covered

a

range

of

values

of

h

from

0.16

to

0.85

feet,

P

from

0.31

to

2.16

feet,

and

B

from

1.44

to

3.87

feet.

The

channel

in

which

the

tests

were

made

was

26

feet

long

and

3.87

feet

wide.

False

walls

and

a

false

floor,

extending

15

feet

upstream

from

the

weir,

were

used

to

produce

30

different

values

of

PIE.

The

weir

was

made

of

bronze,

and

the

weir

crest

surface

was

5/64

inches

wide.

Subsequently,

12

additional

series

of

tests

were

made

by

Numachi,

Kurokawa,

and

Hutizawa

(1943).

Made

with

the

same

weir,

these

sup-

plementary

tests

covered

a

lower

range

of

values

of

P,

from

0.01

to

0.164

feet.

Values

Of

B

varied

from

0.985

to

2.10

feet.

On

the

basis

of

their

data,

Numachi

and

Hutizawa

proposed

a

general

equation

for

the

coefficient

C,

(17)

0=0.574^(0.055.^)

£-

(M-

Within

the

range

of

conditions

covered

by

the

tests,

equation

17

gives

results

which

are

within

1

percent

of

the

experimental

data.

OTHER

EXPERIMENTS

Among

the

many

published

accounts

of

experiments

on

triangular-

notch

weirs,

some

are

of

interest

because

they

were

especially

con-

cerned

with

certain

details

of

the

discharge

characteristics.

For

exam-

ple,

Switzer

(1915),

Cozzens

(1915),

and

Smith

(1934,

1935)

made

tests

B14

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

to

determine

the

influence

of

temperature

on

the

discharge

of

water.

On

the

basis

of

tests

on

90°

and

54°

notches,

with

temperatures

rang-

ing

from

39°

F

to

165°

F,

Switzer

concluded

that

the

effect

on

the

coefficient

of

discharge

was

less

than

2

percent

and

"inappreciable

com-

pared

with

other

factors

present".

Cozzens,

on

the

other

hand,

presented

a curve

to

be

used

to

make

a

small

correction

over

a

range

of

temperatures

from

60°

F

to

220°

F.

Smith

developed

an

empirical

for-

mula

which

incorporated

the

kinematic

viscosity

in

a

correction

coeffi-

cient.

Professor

H.

W.

King

(1916,

1954)

made

tests

on

90°,

60°,

and

22.5°

notches

at

the

University

of

Michigan.

His

weirs

were

made

of

steel,

with

a

sharp

upstream

crest

edge

and

a

crest-surface

width

of

Vs

inch.

The

vertex

of

the

notch

was

2

feet

above

the

floor

of

the

experimental

channel.

On

the

basis

of

his

experiments,

Professor

King

developed

the

following

formulas:

(90°

notch)

C--'

(18)

(60°

notch)

C=

0.595

ft

0

-

01

,

(19)

(22.5°

notch)

C=9L,

(20

)

An

investigation

performed

at

Cornell

University

by

Ho

and

Wu

(1931)

is

notable

for

the

large

range

of

heads

(from

0.129

to

3.43)

covered

by

the

tests.

All

of

these

tests

were

made

in

a

channel

6

feet

wide,

12

feet

deep,

and

30

feet

long

from

baffles

to

weir

plate. Weirs

tested

included

notch

angles

of

90°, 60°, 37°,

28°,

and

19°.

The

ver-

tices

of

notches

were

a

minimum

distance

of

8

feet

above

the

channel

floor.

The

weir

plates

were

made

of

galvanized

sheet

metal

or

brass,

with

sharp,

square,

upstream

edges.

Lenz

(1943)

observed

that

his

general

formula

for

water

(eq

16)

was

confirmed

within

1

percent

by

the

Ho

and

Wu

data.

From

1931

to

1934,

four

investigations

were

made

at

Princeton

University

by

senior

students

under

the

direction

of

Professor

L.

F.

Moody.

Most

pertinent

of

these

is

the

investigation

by

Allerton

(1932),

which

involved

tests

with

notch

angles

of

90°, 60°,

54°,

and

30°.

Of

particular

interest

is

the

fact

that

Allerton

analyzed

his

data

on

the

basis

of

a

formula

which

involved

a

head

adjustment

similar

to

that

used

in

equation

5.

The

formula

was

attributed

to

Moody,

but

no

men-

tion

of

the

formula

is

made

in

reports

of

two

subsequent

investigations

made

at

Princeton.

Investigations

which

placed

emphasis

on

the

influence

of

viscosity

were

made by

Mawson

(1927),

Thornton

(1929),

and

Smith

(1934,

1935).

Because

they

ignored

the

equal

or

larger

influence

of

surface

tension,

however,

the

results

of

the

studies

were

inconclusive.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B15

Hertzler

(1938)

was

especially

concerned

with

120°

notches.

On

the

basis

of

his

tests

on

six

different

weirs,

all

with

6

=

120°,

he

proposed

the

formula

SUMMARY

Published

accounts

of

experimental investigations

agree

on

very

few

details

of

the

discharge

characteristics

of

triangular-notch

weirs.

Some

of

the

disagreements

are

doubtless

the

result

of

differences

in

equip-

ment

and

technique.

More

significantly,

however,

conclusions

drawn

from

the

experiments

are

believed

to

differ

because

of

certain

inade-

quacies

in

traditional

methods

of

analyzing

the

data.

It

is

reasonable

to

assume

that

experimental

equipment

was

ade-

quate

and

that

the

work

was

done

with

skill

and

care.

Nevertheless,

there

are

indications

that

some

of

the

weirs

were

neither

sharp-edged,

thin-crested,

nor

smooth-faced.

Crude

equipment

and

technique

in

other

instances

resulted

in

systematic

errors

the

measure-

ment

of

Q,

h,

and

6.

The

effect

of

these

deviations

cannot

be

determined

from

the

information

available.

There

is

some

agreement

that

weirs

with

notch

angles

less

than

about

20°

or

greater

than

about

100°

exhibit

characteristics

of

in-

stability

which

are

intermittent

clinging

of

the

nappe

to

the

surface

of

the

crest.

Similar

behavior

was

observed

at

extremely

low

heads

on

all

notch

angles.

Most

investigators

found

that

C

varied with

notch

angle.

This

is

a

clear

indication

of

a

shortcoming

in

the

traditional,

so-called

theoretical

equation

of

discharge

(eq

1).

Only

one

of

the

many

investigations

(by

F.

Numachi

and

others)

included

a

systematic study

of

the

influence

of

weir

placement

and

channel

geometry.

However,

this

study

was

made

with

only

one

notch

angle

and

one

liquid.

Several

investigators

demonstrated

the

practicality

of

using

short

approach

channels

if

they

are

effectively

equipped

with

baffles.

No

comprehensive

studies

have

been

made

of

the

influence

of

velocity

distribution

and turbulence

in

the

approach

channel.

Few

investigators

attempted

to

isolate

and

evaluate

the

influence

of

the

physical

properties

of

the

liquid,

although

several

attempted

to

determine

the

effect

of

temperature

on

the

discharge

of

one

liquid

(water).

Their

conclusions

are

remarkable

for

their

disagreement.

Lenz

made considerable

progress

in

isolating

the

separate

effects

of

viscosity

and

surface

tension,

but

the

range

of

conditions

covered

in

the

in-

vestigation

was

quite

limited.

For

this reason,

his

proposed

general

for-

mulas

are

restricted

in

applicability.

They

are

also

inconvenient

to

use.

B16

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

Most

investigators

working with

water

at

normal

temperatures

reported

a

correlation

between

C

and

h.

Empirical

formulas

were

evolved

to account for

this

correlation,

but

no

explicit

recognition

was

given

to

the

fact

that

the

h

correlation

is

evidence

of

the

combined

effects

of

viscosity

and surface

tension.

Thus,

the

"scale

effect"

implica-

tions

of

the

phenomenon

were

ignored.

It

is

believed

that

the

effective-

head

concept

(eq

5)

provides

a

reasonable

and

practical

method

of

ac-

counting

for

the

effects

the

absolute

magnitude

of

h.

No

additional

experiments

were

made

for

this

investigation.

Instead,

experimental

data

obtained

from

some

of

the

sources

reviewed

above

are

used

in

the

following

analysis.

ANALYSIS

OF

DATA

EVALUATION

OF

k

h

The

coefficient

C

e

in

equations

6

and

7

is

described

as

a

function

of

geometric

ratios

(hIP,

PIE,

and

6)

when

h

e

,

the

"effective

head,"

is

deter-

mined

from

equation

5,

h

e

=h+kh.

(5)

In

this

equation,

h

is

the

measured

head

on

the

notch and

k^

is

an

ex-

perimentally

determined quantity

that

accounts

for

the

combined

effects

of

viscosity

and

surface

tension.

Comparison

of

equations

4

and

7

shows

that

the

effect

of

the

effective-head

adjustment

is

to

relieve

the

coefficient

of

discharge

of

dependency

on

h.

This

criterion

is

used

to

determine

the

value

of

kh

which

is

applicable

to

a

specific

set

of

ex-

perimental

data.

For

one

liquid

over

a

small

temperature

range,

and

for

one

notch

angle,

k^

is

assumed

to

be

a

constant.

The

validity

of

the

several

basic

assumptions

the

effective

head

concept

has

been

demonstrated

in

a

of

rectangular

weirs.

Nevertheless,

the practicality

of

the

concept

as

applied

specifically

to

triangular-notch

weirs

must

be

established

on

the

basis

of

experimental

data

for

triangular-notch

weirs.

The

results

of

ex-

periments

by

Barr,

Cone,

Yarnall,

Greve, Lenz,

and

Numachi,

Kurokawa,

and

Hutizawa

are

selected

for

this

purpose.

Because

the

flow

of

water

at

ordinary

atmospheric

temperatures

is

of

principal

in-

terest

to

the

Geological

Survey,

and

because

data

on

other

liquids

are

limited

in

many

ways,

the

data

used

are

only

those

which

involve

water

in

the

temperature

range

from

40°

F

to

85°

F.

In

the

validation

process,

the

quantity

k^

is

evaluated

from

ex-

perimental

data

by

a

trial

procedure.

For

a

series

of

tests

in

which

h

is

the

principal

variable,

k^

is

determined

by

successive

approximations

as

the

quantity

which

will

remove

the

correlation

between

C

and

h.

The

DISCHARGE

OVER TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B17

U.DJ

0.62

0.61

0.60

0.59

0.58

0.57

0.63

0.62

0.61

0.60

0.59

0.58

0.57

n

RR

\

A

*;

^

^^

£--

a»

A

L

^

A

f

i

r*-

~~^

C as

a

A-

B Ce

as

A

A A

|

A

^

4

~x

functk

A

x

a

func

L_

_

x

>n

of

/j

-x

-x.

~x

Q

/P

zir

^

eM^i-c

X

p=

-A

-

X

-A

>o^

EXPLANATION

Symbol

Head,

in

feet

A

O.K

h

<0.3

X

0.3<

h

<0.5

0

0.5<

/?

<0.7

Notes:

fi=3.87

ft,

*/7=0.003

ft,

P=0.64-2.16

ft

-x

x

tion

o:

.^x^

/7/P

T38^

6^>^<

X

r-o-

^T

0.06

0.08

0.10

0.12

0.14

0.16

0.18 0.20

0.22

0.24 0.26

0.28

0.30

0.32

VALUE

OF

h/P

FIGURE

2.-Evaluation

of

k

h

illustrated

with

Numachi,

Kurokawa,

and

Hutizawa

data

(0

=

90°).

procedure

is

illustrated

in

figure

2

by

its

application

to

some

of

the

data

reported

by

Numachi,

Kurokawa,

and

Hutizawa.

3

These

data

were

selected

because

they

were

obtained

with

different

values

of

P.

Thus,

the

influence

of

h

can

be

isolated from

the

influence

on

hIP.

The

tests

shown

in

figure

2

are

for

a constant, large

value

of

B

and

5

different

values

of

P,

all

with

a

90°-notch

weir.

Figure

2A

shows

C

plotted

as

a

function

of

hIP.

Ranges

of

values

of

h

are

indicated

by

the

symbols

shown

in

the

legend.

The

scatter

in

values

of

C

is

systematically

The

procedure

is

illustrated

and

discussed

in

greater

detail

in

the

paper

by

Kindsvater

and

Carter

(1959,

p.

783-787,

figs.

3,

4,

5,

6)

on

rectangular weirs.

B18

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

to

the

magnitude

of

h

as

shown

by

the

dashed

curves.

There

is

no

clear-

ly

defined

correlation

with

P.

Figure

25

shows

C

e

plotted

as

a

function

of

hIP.

The

value

of

k^

which

was

used

in

the

computations

of

C

e

for

this

figure

was

0.003

foot.

Evidence

of

the

lack

of

correlation

between

C

e

and

h

is

shown

by

the

fact

that

one

curve

drawn

through

all

the

points

is

horizontal,

and

the

scatter

is

greatly

reduced.

Thus,

the

validity

of

the

effective-head

concept

is

demonstrated,

and

the

value

k^=

0.003

foot

is

shown

to

be

appropriate

for

the

tests

by

Numachi

and others.

Another

demonstration

of

the

validation

procedure

is

shown

in

figure

3,

which

is

based

on

the

results

of

Lenz's

tests

on

weirs

with

three

0.64

0.63

0.62

0.61

0.60

0.59

0.58

0.57

0.63

0.62

0.61

0.60

0.59

0.58

0.57

0.56

-*-&-*

EXPLANATION

Symbol

0

(degrees)

k^

feet

0

89°53'

0.003

X

59°5V

0.004

A

44°40'

0.005

Notes:

P=3.0

ft,

P/B=0.86

B Ce

as

a

function

of

h/P

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20 0.22

0.24

0.26

VALUE

OF

h/P

FIGURE

3.-Evaluation

of

k

h

illustrated

with

Lenz

data.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B19

different

notch

angles.

Values

of

the

notch

angles

are

shown

in

the

legend.

In

the

top

part

of

figure

3,

C

is

plotted

as

a

function

of

hIP,

but

because

P

is

constant

for

each

weir,

the

separation

and

curvature

of

the

lines

drawn

through

the plotted

points

show

the

influence

h

and

6.

The

lower

part

of

figure

3

shows

C

e

plotted

as

a

function

of

hIP

for

the

Lenz

data.

In

this

instance,

different

values

of k^

were

derived

for

each value of

6,

as

shown

in

the

legend.

Here

again

the

effectiveness

of

kh

in

eliminating

h

as

an

independent

variable

is

demonstrated

by

the

fact

that

curves

drawn

through

the

points

are

essentially

horizontal.

Coincidentally,

a

single

curve

fits

the

three

sets

of

points

reasonably

well,

showing

that

6

has

a

minimal

effect

on

C

e

for

notch

angles

be-

tween

45°

and

90°.

Values

of

PIB

and

h/P

represented

by

the

tests

shown

in

figures

2

and

3

are

in

the

range

in

which

these

variables

also

have

negligible

influence

on

C

e

.

This

fact

is

demonstrated

subsequently.

Values

of

k^

for

values

of

6

between

10°

and

120°

were

determined

from

the

tests

made

by

Barr,

Cone,

Yarnall,

Greve,

Lenz,

and

Numachi

and

others.

The

results

are

shown

on

figure

4.

Values

of

k^

for

those

few

tests

which

were

made

for

very

small

or

very

large

notch

angles

are

not

well

defined.

Deviations

from

the

smooth

curve

drawn

through

the

points

can

be

attributed

to

differences

in

physical

characteristics

of

weirs

and

test

channels.

AN

ALTERNATIVE

FOR

k

h

The

quantity

k^

is

added

to

the

measured

head

as

an

adjustment

for

the

combined

effects

of

viscosity

and

surface

tension.

The

procedure

is

justified

with

the

observation

that,

in

general, the

effects

of

the

two

fluid

properties

are

commensurate

with

a

fictitious

increase

in

head

and

notch

width.

Because

the

width

is

a

function of

the

head

for

triangular-

notch

weirs,

a

quantity

added

to

the

head

is

simultaneously

an

adjust-

ment

of

the

width.

However,

because

the

width

is

a

function

of

6

as

well

as

h,

it

is

not

unreasonable

that

figure

4

shows

k^

to

be

a

function

of

6.

As

an

alternative

to

k^,

a

factor

k

is

proposed,

where

k

is

measured

perpendicularly

to the

sides

of

the

notch.

Thus,

fc-Afcsin-!--

(22)

z

Values

of

k

were

computed

for

the

data

shown

in

figure

4

to

determine

whether

k

might

be

constant

for

all

notch

angles.

The

results

are

shown

in

figure

5.

The

figure

shows

that

a

value

of

k=

0.002

foot

could

be

used

for

values

of

0

between

40°

and

90°.

Somewhat

smaller

values

of

k

are

indicated

for

values

of

6

less

than

40°.

The

quantity

k

is

not

well

defined

for

values

of

6

greater

than

90°.

w

CO

o

0.024

50

60

70

80

VALUE

OF

0,

IN

DEGREES

FIGURE

4.

-Value

of

k

h

as

a

function

of

0.

90

100

110

120

s

o

Sd

so

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B21

»

To

<o

E

.-

c

>

N

CO

1=

fc

P

C

3

1-

-s

D

O

8

w

.

in

^

HI

C4H

DC

°

O

C!

ill

O

-«

UJ

*0

i

3

<

13

l

U3

H

05

O

133d

Nl

'*

dO

3mVA

B22

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

It

is

apparent

that

the

effective

head

in

the

modified

basic

equation

of

discharge

(eq

6)

can

be

evaluated

in

terms

of

either

k

or

k^.

If

k^

is

used,

values

of k^

are

read

from

figure

4.

If

k

is

used,

values

of

k

can

be

read

from

figure

5,

or,

in

the

range

from

40°

to

90°,

h

e

can

be

computed

as

(23)

sn

Which

of

these

alternatives

is

the

more

convenient

is

conjectural.

Hereafter

in

this

report,

h

e

is

computed

with

values

of k^ from

figure

4.

EVALUATION

OF

C

e

Figures

6

to

11,

inclusive, show

values

of

C

e

computed

from

results

of

the

tests

by

Barr,

Cone,

Yarnall,

Greve,

Lenz,

and

Numachi,

Kurokawa,

and

Hutizawa.

In

the

figures,

C

e

is

plotted

as

a

function

of

h/P.

The

data

cover

a

long

range

of

notch

angles

from

120°

to

10°.

In-

cluded

on

each figure

is

a

legend

which

identifies

the

investigators

and

gives

corresponding

values

of

6,

P/B, and

k^

Values

of

kh

used

in

com-

puting

C

e

for

each

set

of

tests

are

the

values

plotted

in

figure

4.

The

average

values

of

C

e

were

determined

visually

by

fitting

a

line

to

the

plotted

points.

The

average

value

and

the

limits

representing

1

percent

deviations

from

the

average

are

shown

as

horizontal

lines

in

each

figure.

The

experimental

data

plotted

in

figures

6

through

11

show

a

range

of

values

of

h/P

from

0.03

to

0.35

and

a

range

of

values

of

PIE

from

0.

15

to

1.5.

In

this

range,

which

includes

the

most

common

limits

for

prac-

tical

applications,

it

is

apparent

that

C

e

is

virtually

independent

of

h/P

and

PIE.

Thus,

within

this

range

C

e

is

remarkably

well

defined

as

a

function

of

6

alone.

Scatter

in

the

test

points

at

smaller

values

of

h/P

is

attributed

to

the

influence

of

intermittent

nappe-clinging

at

small

values

of

h.

The

influence

of

PIE

at

higher

values

of

hIP

is

discussed

in

the next

section.

Figure

12

shows

the

relationship

between

6

and

the

average

values

of

C

e

determined

from

figures

6

through

11.

The

dashed

portions

of

the

summary

curves for

C

e

(fig.

12)

and

kh

(fig.

4)

indicate

that

there

is

some

uncertainty regarding

the

practicality

of

weirs

with

notch

angles

less

than

20°

or

greater

than

100°.

0.62

0.61

o

u_

0.60

O

HI

D

t

0.59

0.58

X

o

0 0

o

x

o-

X

o o o

o

O

<B>

O|

0

O

1

8

1

O

^

1

0

o

0

EXPLANATION

Symbol

Name

6

PIB

AT,

(ft)

degrees

X

Cone's

120°

0.45

0.003

formula

O

Greve

118°11'

0.95

0.005

X X

+

1

-1

Pet.

.588

(avg)

Pet.

0.05

0.10

0.15

0.20

VALUE

OF

hIP

FIGURE

6.-Value

of

C

e

,

notch

angle

approximately

120°.

0.25

0.30

CO

O

0.35

a

o

K

f

w

to

oo

W

to

LL

O

0.62

0.61

0.60

0.59

0.58

0.57

o

0

o

X

o

8

°

0

o o

-

Y^

X

7

O

0

1

<0>07O

Q

OI

O

--X

-O*B>-

O-A

-0

1L

-

f

!

*

2

Q

AD

a

EXPLANATION

Symbol

Name

0

PIB

*y,(ft)

degrees

V

Barr

90°00'

0.50

0.003

x

Cone

90°00'

0.45

0.0035

n

Yarnall

90°00'

0.60

0.003

0

Greve

94°39'

0.95 0,003

+

Numachi

90°00'

0.16-1.04

0.003

and

others

A

Lenz

89°53'

0.86

0.003

4

v

o

+

*

;^,r!7.t

f

-

r

4

^r

7^

Q

-fD

I

+1

Pet.

-

0.578

(avg)

-1

Pet.

0.05

0.10

0.15

0.25

0.20

VALUE

OF

hIP

FIGURE

7.-Value

of

C

e

,

notch

angle

approximately

90

c

0.30

0.35

0.40

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B25

Is

**

o

Oq

in

^

0

O

Q3

-

_

<D

0

z-g

§

^

**>

CL

UJ

E

*

(0

£

z

c

"o

n

E

X

to

o

o o

000

o o o

S

in

co

o

oo

o

o o

oo

r^

J-

o o

in

o o o

CO

O> O>

in in in

^

(0 0)

C

>

N

^

(33

c

>

(D

^

n

o

<l

c

.

i

+

L

1

S

!

1

i

^

I

1

«

1

!

-

i

I

O

^.

1

<

\

1

!

-

1

a

a?

i

§

§F

1

x

1

<

\

1

<«

*

X

1

1*^1

c

1

r

**

i

<

i

X

44

1

*

O

|

O

o

1

?

.

J

£

^-

a

x

a

s

100

°

a

ffj

t

O

-

i

<i

i

-ft

1

a

-

'

Y/\

4

°Q

0-

1

in

CO

o

CO

0

in

CM

o

CM

O

in

o

2

o

in

o

o"

a.

£

<

'I

Ll-

Q,

O

Pu

UJ

(D

jo

amvA

w

to

0.60

0.59

0.58

0.57

O

o

<n>

o

X

0

O

'

X

"

'

"

X

K

'X

"X

X

3

Symbol

Name

O

Greve

X

Lenz

O

0

<

X

<

x x

X

EXPLANATION

6

degrees

45°23'

44°40'

0

©

£

°

O

PIB

k

h

(ft)

0.95

0.005

0.86

0.005

-

+

1

Pet.

-

-0.580

(avg)

-1

Pet.

0.05 0.10

0.15

VALUE

OF

h/P

FIGURE

9.-Value

of

C

e

,

notch

angle

approximately

45°.

0.20

0.25

0.30

O

3

O

^

1

O

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B27

If

£

°£

coo

*

0

QQ

LO

c

^

d

o

w

z

£

*

c

2

8*

g

S=

I-

§

S

E

5

w

<N

c\

z

*

^

a.

<D

t

x

1

S

£

m

J°

o

«

z

o

>

"5

E

x

c

>

w

to

;

o o

0

d d

LO

CD

O>

CO

d

>

b

LO

)

CO

LO

?^

CM

CN

i

<D

3

i_

0)

-

0

-J

1

0

<

c

'

2

i

T-

e

j_

«

o

°oo

o

^*

e

o

0

5

X

«

a-

X

*"

«3«

a

X

O

OC

o

<

fc

^

x

o

oO

o

1

3

-

!

5

1

E

1

8

c

i

n

Q-

|

D

t-

'

1

X

a

1

.

1

a

|

i

1

X

1

3

1

X

|

4

^

i

1

^ 5^

I

X**

1

D

1

*>

!

i

1-1

i

I

i

)

0

1

a

1

O

|

1

-

x

o o

oj

o

0

o

CO

0

LO

CM

O

CM

O

LO

0

o

LO

0

o

i

£

.2

O

"ho

ui

i

3

_e

O

*^

o

a>

13

CO

o

6

a

o do

0.62

0.61

0.60

0.59

0.58

0.

O

0

°

0

^o

0

cb

0

o

0

EX

PLAN

AT

8

Symbol

Name

degr

Q

Yarnall

13°C

©

Lenz

10°;

o

a

O

o

<o>

o

a>

o

a

ON

ees

P/B

k

h

(ft)

)8'

0.60

0.022

>0'

0.86 0.017

+

1

Pet.

o

a

-1

Pet.

Q

05

0.10 0.15

0.20

0.25 0.30

0.597

(avg)

0.35

Q.i

VALUE

OF

hIP

FIGURE

11.-Value

of

C

e

,

notch

angles

of

10°

and

13°.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B29

T3

s

O

I

cj

T I

a

M

g

B30

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

INFLUENCE

OF

h/P

AND

P/B

A

few

tests

involving

different

values

of

P

and

B

were

made

by

Barr,

and

some

tests

withP=0

were

reported

by

Thomson

and

Cone.

Among

the

experimental

data

available

to

the

writer,

however,

the

only

data

covering an

extensive

range

of

values

of

hJP

and

PIE

were

those

ob-

tained

from

the

experiments

on

90°

notches

made

by

Numachi,

Kurokawa,

and

Hutizawa.

Values

of

C

e

were

computed

from

the

results

of

the

Numachi-

Kurokawa-Hutizawa

tests,

using

kh=

0.003

foot.

Smooth

curves

inter-

polated

from

the

plotted

data

were cross-plotted

to

obtain

the

family

of

curves

showing

C

e

as

a

function

of

hIP

and

PIE

on

figure

13.

The

results

are

a

comprehensive

solution

for

90°

-notch

weirs

that,

because

of

the

degree

of

extrapolation

from

the

data,

must

be

regarded

as

unsubstan-

tiated.

The

h/P

limit

lines

at

the

lower

left

corner

of

figure

13

illustrate

the

limited

hIP

range

of

the

Numachi-Kurokawa-Hutizawa

tests,

and

the

C

e

values

from

those

tests,

plotted

in

figure

7,

are

all

within

1

per-

cent

of

the

average

value

(0.578).

COMPARISON

OF

FORMULAS

BASIS

OF COMPARISON

Derived

from

tests

in

which

hIP

and

PIE

were

of

negligible

significance,

the

traditional

formulas

for

triangular-notch

weirs

are

generally

of

the

form

C=/«U),

(24)

in

which

an

empirically

derived

term

involving

h

accounts

for

the

com-

bined

effects

of

viscosity

and

surface

tension

for

water

at

ordinary

temperatures.

Unfortunately,

the

significance

of

h

was

not

understood

by

the

originators

of

most

of

these

formulas.

Consequently,

the

more

conservative

investigators

specified

a

range

of applicability

which

was

based

on

the

limits

of

their

test

conditions.

In

order

to

compare

the

proposed

new

formula

(eq

6)

with

the

classic

formulas,

values

of

C

were

computed

from

the

classic

formulas

and

from

the

equation,

_h+kh

2

-

obtained

from

the

solution

of

equations

1

and

6.

Values

of

C

e

and

kh

were

obtained

from

figures

10

and

4,

respectively.

The

comparisons

are

shown

in

figures

14-19,

inclusive,

in

which

C is

shown

as

a

function

of

h

for

selected

values

of

6.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B31

&

I

o

I

8

S

5

6

c>

o

CO

d d

dO

3niVA

B32

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

gO'-NOTCH

WEIR

The

most

frequently

used

triangular-notch

weir

is

one

with

a

notch

angle

of

90°

.

Some

of

the

classic

formulas

are

intended

for

use

with

this

weir

alone.

Figure

14

shows

a

comparison

of

values

of

C

computed

from

the

well-known

formulas

of

Thomson,

Barr-Strickland

(eq

8),

Cone

(eq

9,

10),

Greve

(eq

11),

Lenz

(eq

16),

and

King

(eq

18)

and

the

formula

determined

from

equation

25

in

combination

with

figures

12

and

4.

The

Thomson

"formula"

is

actually

just

C=

0.593,

because

Thom-

son

did

not

propose

a correlation

with

h.

It

is

emphasized

that

figure

14

is

a

comparison

of

formulas,

whereas

figure

7

is

a

comparison

of

the

data

from

which

the

formulas

were

derived.

Thus,

the disagreement

shown

in

figure

14,

especially

at

small

values

of

h,

is

to

a

large

degree

evidence

of

inadequacies

of

the

for-

mulas

as

representations

of

the

data

on

which

they

were

based.

Figure

7

shows

that

the

source

data

for

all

the

formulas

agree

remarkably

well

in

defining

C

e

for

use

in

equation

6.

OTHER

NOTCH

ANGLES

Figures

15

through

19

show

comparisons

of

well-known

formulas

for

weirs

with

120°,

60°,

45°,

27°,

and

10°

notch

angles.

The

solid

line

in

each figure was

computed

from

equation

25,

with

values

of

C

e

and

k^

from

figures

12

and

4,

respectively.

The

corresponding

comparisons

of

experimental

data

are

shown

in

figures

6

and

8

through

11.

It

is

demonstrated

again

that

the

solution

derived

from

the

effective-head

concept

is

substantiated

by

the

experimental

data,

whereas

the

classic

formulas

show

justifiable

differences

at

small

values

of

h.

CONCLUSIONS

The

triangular-notch,

thin-plate

weir

is

potentially

an

accurate,

con-

venient

means

of

measuring

the

flow

of

liquids.

Classic

discharge

for-

mulas

are

inadequate,

however,

because

they

fail

to

recognize

the

full

significance

of

the

pertinent

variables.

Although

the

triangular-notch

weir

is

not

subject

to

a

completely

analytical

solution,

available

ex-

perimental

data

are

sufficient

to

substantiate

a practical

solution

which

is

based

on

dimensional

analysis

and

experimental

coefficients.

Limita-

tions

of

the

data

restrict

the

use

of

the

solution

to

water.

The

equation

of

discharge

proposed

in

this

report

is

^

5/2

(6)

z

in

which

h

e

,

the

effective

head,

is

defined

by

the

equation

(5)

and

values

of

k^

are

shown

in

figure

4.

DISCHARGE

OVER

TRIANQULAR-NOTCH

THIN-PLATE

WEIRS

B33

do

amvA

0.64

0.63

EXPLANATION

Thomson

formula

Cone

formula

(eqs

9,

10)

Greve

formula

(eq

11)

Hertzler

formula

(eq

21)

Equation

25

0.56

1.0 1.2

1.4

VALUE

OF

h,

IN

FEET

FIGURE

15.

-Comparison

of formulas

for

notch

angles

of

approximately

120°.

0.64

0.63

0.62

EXPLANATION

Cone

formula

(eqs

9,

10)

King

formula

(eq

19)

Greve

formula

(eq

11)

Lenz

formula

(eq

16)

Equation

25

0,57

1.6

1.0 1.2

1.4

VALUE

OF

h,

IN

FEET

FIGURE

16.-Comparison

of

formulas

for

notch

angles

of

approximately

60

1.8

2.0

2.2

2.4

CO

Cn

0.64

0.63

EXPLANATION

Cone

formula

(eqs

9,

10)

-

Greve

formula

(eq

11)

-

Lenz

formula

(eq

16)

Equation

25

0.57

1.0

1.2

1.4

VALUE

OF

h,

IN

FEET

FIGURE

17.

-Comparison

of

formulas

for

notch

angles

of

approximately 45°.

0.65

0.64

EXPLANATION

Cone

formula

(eqs

9,

10)

- -

Greve

formula

(eq

11)

Lenz

formula

(eq

16)

Equation

25

0.2

0.4

0.6 0.8

1.0

1.2

1.4

1.6 1.8

2.0

2.2

2.4

VALUE

OF

h,

IN

FEET

0.58

0.57

FIGURE

18.

-Comparison

of

formulas

for

notch

angles

of

approximately

27°.

to

CO

W

co

0.68

0.67

-

Lenz

formula

(eq

16)

Equation

25

0.61

0.60

0.2

0.4

1.0

1.2

.1.4

VALUE

OF

h,

IN

FEET

1.6

1.8

2.0

2.2

2.4

FIGURE

19.-Comparison

of

formulas

for

notch

angles

of

approximately

10°.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B39

In

general,

the

coefficient

of

discharge,

C

e

,

is

an

experimentally

determined

function

of

the

geometric

parameters

h/P,

PIB,

and

6.

However,

within

the

range

of

the

most

common

practical

limits

of

h/P

and

PIB,

C

e

is

a

function

of

6

alone

(fig.

12).

In

this

range,

values

of

C

e

computed

from

the

results

of

experiments

by

Barr,

Cone,

Yarnall,

Greve,

Lenz,

and

Numachi,

Kurokawa,

and

Hutizawa

show

remarkable

agreement.

For

0=

90°

the

available

data

are

sufficient

to

define

C

e

over

a

wider

range

of

values

of

hIP

and

PIB

(fig.

13).

REFERENCES

Allerton,

R.

W.,

1932,

Flow

of

water

over

triangular

weirs:

Princeton

University,

Thesis

(mech.

eng.),

56

p.

29

figs.

Barr,

James,

1910,

Experiments

upon

the

flow

of

water

over

triangular

notches:

Engi-

neering

(London),

v.

89,

(April

8

and

15),

p. 435

ff.

Burgess,

J.

S.,

and

White,

W.

R.,

1966,

The

triangular-profile

(Crump)

weir-two-dimen-

sional

study

of

discharge

characteristics,

Hydraulics

Research

Station

(Wallingford,

U.K.),

HRS

Report

INT

52.

Carter,

R.

W.,

1956,

A

comprehensive

discharge

equation

for

rectangular-notch

weirs,

Master's

thesis,

Georgia

Institute

of

Technology.

Cone,

V.

M.,

1916,

Flow

through

weir

notches

with

thin

edges

and

full

contractions:

U.S.

Department

of

Agriculture,

Journal

of

Agricultural

Research,

v.

V,

no.

23

(March

6),

p.

1051.

Cozzens,

H.

A.,

Jr.,

1915,

Flow

over

V-notch

weirs:

Power,

v.

42,

no.

21

(November

23),

p.

714.

Greve,

F.

W.,

1932,

Flow

of

water

through

circular,

parabolic,

and

triangular

vertical-

notch

weirs:

(Purdue

Univ.,

Eng.

Bull.,

v.

XVI,

no.

2,

Research

Series

No.

40

(March),

84

p.,

15

figs.

Hertzler,

R.

A.,

1938,

Determination

of

a

formula

for

the

120°

V-notch

weir:

Civil

Eng.

(November),

p.

756.

Ho,

Chitty,

and

Wu,

S.

L.,

1931,

The

flow

of

water

over

sharp

crested

weir

notches-

rectangular,

trapezoidal,

and

triangular:

Cornell

Univ.,

Master's

thesis,

155

p.,

78

figs.

International

Standards

Organization,

1975,

Liquid

flow

measurement

in

open

channels

using

thin-plate

weirs

and

venturi

flumes

(ISO

1438-1975).

Kindsvater,

C.

E.,

and

Carter,

R.

W.,

1959,

Discharge

characteristics

of

rectangular

thin-plate

weirs:

Am.

Soc.,

Civil

Engineers

Trans.,

v.

124,

p.

772.

Kindsvater,

Carl

E.,

1964,

Discharge

characteristics

of

embankment-shaped

weirs:

U.S.

Geol.

Survey

Water-Supply

Paper

1617-A,

114

p.,

57

figs.

King,

H.

W.,

1916,

Flow

of

water

over

right-angled

V-notch

weir:

Univ., Michigan

Technic,

v.

29, no.

3

(October),

p.

189.

1954,

Handbook

of

Hydraulics

(Revised

by

Ernest

F.

Brater):

New

York,

McGraw-Hill

Book

Co.,

567

p.,

165

figs.

Lenz,

A.

T.,

1943,

Viscosity

and

surface

tension

effects

on

V-notch

weir

coefficients:

Am.

Soc.

Civil

Engineers

Trans.,

v.

108,

p.

759.

Mawson,

Humbert,

1927,

Applications

of

the

principles

of

dimensional

and

dynamical

similarity

to

the

flow

of

liquids

through

orifices,

notches

and

weirs:

Inst.

Mech.

Engineers,

Proc.,

v.

1,

p.

1033.

Numachi,

F.,

Kurokawa,

T.,

and

Hutizawa,

S.,

1940,

Uber

den

Uberfallbeiwert

eines

rechtwinkelig-dreieckigen

Messwehrs:

Soc.

Mech.

Engineeers

(Japan)

Trans.,

v.

6,

no.

22,

(February),

p.

110.

B40

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

1941,

Uber

den

Uberfallbeiwert

eines

rechtwinkelig-dreieckigen

Messwehrs:

Tohoku

Imperial

Univ.,

(Japan)

Tech.

reports,

v.

13,

no.

2,

p.

350.

-1943,

Uber

den

Uberfallbeiwert

eines

rechtwinkelig-dreieckigen

Messwehrs,

2.

Mitteilung:

Tohoku

Imperial

Univ.,

(Japan)

Tech.

reports,

v.

13,

no.

3,

p.

473.

Rehbock,

Th.,

1929,

discussion

of

paper

by

Ernest

W.

Schoder

and

Kenneth

B.

Turner,

Precise

weir

measurements:

Am.

Soc.

Civil

Engineers

Trans.,

v.

93,

p. 1148.

Schlag,

Albert,

1962,

Note

sur

la

mesure des

debits

pour

deversoir

triangulare,

La

Tri-

bune

de

Cebedeau

(Liege,

Belgium),

v.

15,

no.

218,

p. 22.

1962b,

Formule

de

debit

de

deversoirs

dont

le

seuil

est

constitue

par

une

s£rie

de

dents

situees

dans

un

plan

horizontal,

La Tribune

du Cebedeau

(Liege,

Belgium),

v.

15,

no.

221,

p.

180.

Smith,

E.

S.,

Jr.,

1934,

The

V-notch

weir

for

hot

water:

Am.

Soc.

Mech.

Engineers

Trans.,

v.

56,

p.

787.

1935,

The

V-notch

weir

for

hot

water:

Am.

Soc.

Mech.

Engineers

Trans.,

v.

57,

p.

249.

Strickland,

T.

P.,

1910,

Mr.

James

Barr's

experiments

upon

the

flow

of

water

over

tri-

angular

notches:

Engineering

(London),

v.

90,

(October

28),

p.

598.

Switzer,

F.

G.,

1915,

Tests

of

the

effect

of

temperature

on

weir

coefficients:

Engineering

News,

v.

73, no.

13,

(April

1),

p.

636.

Thomson,

James,

1858,

On

experiments

on

the

measurement

of

water

by

triangular

notches

in

weir

boards:

Brit.

Assoc.

Adv.

Sci.,

Annual

report,

p.

181.

1861,

On

experiments

on

the

measurement

of

water

by

triangular

notches

in

weir

boards:

Brit.

Assoc.

Adv.

Sci.,

Annual

report,

p.

151.

Thornton,

B.

M.,

1929,

Measurement

of

fluid

flow

with

triangular

notches:

Power

Engineer

(August),

London,

v.

24,

p.

314.

Tracy,

H.

J.,

1957,

Discharge

characteristics

of

broad-crested

weirs:

U.S.

Geological

Survey

Circ.

397,

15

p.,

11

figs.

Yarnall,

D.

R.,

1912,

the

V-notch

weir

method

of

measurement:

Am.

Soc.

Mech.

Engineers

Trans.,

v.

34,

p.

1055.

1926,

Accuracy of

the

V-notch

weir

method

of

measurement:

Am.

Soc.

Mech.

Engineers

Trans.,

v.

48,

p.

939.

SUPPLEMENTAL

INFORMATION

B42

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

REQUIREMENTS

FOR

PRECISE

MEASUREMENTS

SPECIFICATIONS

FOR

THE

INSTALLATION

Use

of

the

triangular-notch,

thin-plate

weir

for

precise

measure-

ments

is

restricted

to

steady,

free,

and

fully

ventilated

flows.

The

recommended

coefficients

are

applicable

for

water

only,

in

the

approx-

imate

range

of

temperatures

from

40°

F

to

85°

F.

General

specifications

for

a

satisfactory

weir

installation

are

de-

scribed

on

page

B2.

Additional

restrictions

related to

the

actual

measurements

are

discussed

below.

RESTRICTIONS

ON

THE

GEOMETRIC PARAMETERS

Values

of

v

less

than

20°

or

greater

than

100°

are

not

recommended

for

precise

measurements.

Experimental

data

for

weirs outside

this

range are

limited.

Furthermore,

the

data

available

indicate

that

weirs

with

very

small

and

very

large

notch

angles

exhibit

characteristics

of

instability

which

result

from

the

fact

that

the

nappe

intermittently

clings

to

the

upper

surfaces

of

the

crest.

Practical

restrictions

on

hIP

and

PIE

are

suggested

by

the

observa-

tion

that

head-measurement

difficulties

and

errors

result

from

surges

and

waves

which

occur

in

the

approach

channel when

the

velocity

of

ap-

proach

is

large

in

comparison

with

the

depth

of

flow.

The

available

data

are

not

adequate

to

establish

the

limiting

values

of

hIP

and

PIB

which

are

associated

with

this

condition.

Therefore,

the

user

himself

must

determine

practical

limits

which

are

consistent

with

the

required

ac-

curacy

for

head

measurements.

These

limits

are

separate

from

restric-

tions

on

the

use

of

figure

12,

which

is

applicable

only

when

the

effects

of

hIP

and

PIB

are

negligible.

RESTRICTIONS

ON

h

AND

P

Restrictions

on

the

magnitude

of

h

are

the

unpredictable,

unsteady

clinging

phenomena

which

occur

at

small

heads.

To

ensure

a

freely

discharging,

stable

nappe,

a

minimum

value

of

h=Q.2

foot

is

recommended.

There

is

disagreement

in

the

literature

regarding

an

independent

correlation

between

P

and

the

coefficient

of discharge.

It

is

tentatively

recommended

that

P

be

limited

to

values

greater

than

0.3

foot.

EFFECT

OF

NEGLECTING

k

It

is

apparent

from

equations

5

and

6

that

the

relative

influence

of

kh

decreases

as

h

increases

in

magnitude.

Consequently,

the

error

which

results

from

using

h

instead

of

h

e

in

equation

6

is

appreciable

only

for

small

values

of

h.

The

relative

error

in

the

discharge

is

plotted

as

a

function

of

h

in

figure

20.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B43

0.4

0.8

1.2

VALUE

OF

h,

IN

FEET

1.6

2.0

FIGURE

20.

-Effect

of

using

h

instead

of

h

e

in

equation

6.

EFFECT

OF

APPROACH-CHANNEL

CONDITIONS

Specifications

for

a satisfactory

weir

installation

include

the

require-

ment

that

turbulence

and

velocity

in

the

channel

upstream

from

the

weir

be

such

as

to

simulate

normal

(uniform

flow)

conditions

in

a

smooth,

horizontal,

rectangular

channel.

If

the

effective

notch

area

is

very

small

in

comparison

with

the

area

of

the

approach

channel,

ap-

proach

velocities

are

negligible,

and

the

shape

of

the

channel

is

in-

significant.

Short

channels

are

satisfactory

if

they

are

provided

with

baffles

capable

of

producing

a

nearly

normal

turbulence

and

velocity

distribution.

When

in

doubt,

the

velocity

distribution

should

be

checked

with

a

velocity

traverse

located

at

or

near

the

headwater-gage

section.

The

experimental

data

available

are

inadequate

to

evaluate

the

influence

of

velocity

distribution

over

a

full

range

of

values

of

v,

hIP,

and

PIB.

However,

the

remarkable

consistency

and

agreement

shown

by

the

data

plotted

in

figures

6

through

11

indicate

a

corresponding

lack

of

correlation

with

the different

approach-channel

conditions

which

are

represented

by

those

data.

Therefore,

in

the

range

of

values

of

h/P

from

0.03

to

0.35

and

PIB from

0.1

to

1.5,

the

influence

of

approach-channel

velocity

distribution

is

believed

to

be negligible.

B44

FLOW

OF

WATER

OVER

WEIRS

AND

DAMS

MEASUREMENT

OF

v

Precise

discharge

computations

require

that

v

be

measured

accurate-

ly.

One

of

several

satisfactory

methods

of

measuring

v

is

described:

(1)

two

true

disks

of

different,

micrometered

diameters

are

placed

in

the

notch

with

their

edges

tangent

to

the

sides of

the

notch;

(2)

the

vertical

distance

between

the

centers

(or

two

corresponding

edges)

of

the

two

disks

is

measured

with

a

micrometer

caliper;

(3)

the

angle

v

is

twice

the

angle

whose

sine

is

equal

to

the

difference

between

the

radii

of

the

disks

divided

by

the

distance

between

the

centers

of

the

disks.

Other

equally

satisfactory

methods

of

measuring

v

are

described

in

the

listed

references.

MEASUREMENT

OF

h

The

head

is

measured

with

a

hook

gage,

point

gage,

or

precise

ma-

nometer.

Hook

and

point

gages

preferably

are

used

in

a

stilling

well.

However,

if

approach

velocities

and

surface

disturbances

in

the

ap-

proach

channel

are

negligible,

the

gages

can

be

mounted

over

the

head-

water

surfaces.

Stilling

wells

and

manometers

are

connected to

piezometers

located

in

the

floor

or

walls

of

the

channel.

To

prevent

er-

rors

due

to

excessive

head

losses

or

surface

drawdown,

the

piezometnc

section

is

located

upstream

from

the

weir

a

distance

equal

to

from

four

to

five

times

the

maximum

head.

Gages

are

mounted

as

close

to

the

weir

as

possible

in

order

to

avoid

errors

in

the

gage

zero

due

to

deflec-

tion

or

movement

under

different

water

loads.

DETERMINATION

OF

GAGE

ZERO

Accuracy

of

head

measurements

is

critically

dependent

upon

the

determination

of

the

gage

zero,

which

is

defined

as

the

gage

reading

corresponding

to

the

level

of

the

vertex

of

the

notch.

Numerous

accept-

able

methods

of

determining

the

gage

zero

are

in

use.

One

such

method

is

described:

(1)

still

water

in

the

channel

upstream

from

the

weir

is

drawn

to

a

level below

the

vertex

of

the

notch;

(2)

a

temporary,

precise

hook

gage

is

mounted

over

the

approach

channel,

with

its

point

a

short

distance

upstream

from

the

weir

plate;

(3)

a

true

cylinder

of

known

(micrometered)

diameter

is

placed

with

its

axis

horizontal,

with

one

end

resting

in

the

notch

and

the

other

end

balanced

on

the

point

of

the

tem-

porary

hook

gage;

(4)

a

machinist's

level

is

placed

on

the

top

of

the

cylinder

and the

hook

gage

is

adjusted

to

make

the

cylinder

precisely

horizontal;

(5)

the

reading

of

the

temporary

gage

is

recorded;

and

(6)

the

temporary

hook

gage

is

lowered

to

the

water

surface

in

the

ap-

proach

channel

and

the reading

is

recorded.

DISCHARGE

OVER

TRIANGULAR-NOTCH

THIN-PLATE

WEIRS

B45

The

permanent

headwater

gage

is

adjusted

to

read

the

level

in

the

stilling

well,

and this

reading

is

recorded;

(7)

the

distance

from

the

bot-

tom

of

the

cylinder

to

the

vertex

of

the

notch

is

computed

with

the

known

values

of

v

and

the

diameter

of

the

cylinder;

(8)

the

vertical

distance from

the

vertex

of

the

notch

to the

still

surface

is

computed

from

the

readings

in

step

5

and

6

and the

computation

in

step

7;

(9)

finally,

the

zero

reading

on

the

permanent

gage

is

computed

as

its

reading

on

the

still

water

surface

plus

the

computed

distance

from

the

vertex

to

the

still

surface.

An

obvious

advantage

of

this

method

is

that

it

refers

the

gage

zero

to the

geometrical

vertex

which

is

defined

by

the

sides

of

the

notch.

Studies

of

Flow

of

Water

Over

Weirs

and

Dams

GEOLOGICAL

SURVEY

WATER-SUPPLY

PAPER

1617

This

volume

was

published

as

separate

chapters

A

-B

UNITED

STATES

DEPARTMENT

OF

THE

INTERIOR

JAMES

G.

WATT,

Secretaiy

GEOLOGICAL

SURVEY

Doyle

G.

Frederick,

Acting

Director

CONTENTS

[Letters

designate

the

chapters]

(A)

Studies

of

Flow

of

Water

Over

Weirs and

Dams-Discharge

Characteristics

of

Embankment-Shaped

Weirs,

by

Carl

E.

Kindsvater.

(B)

Studies

of

Flow

of

Water

Over

Weirs

and

Dams-Discharge

Characteristics

of

Triangular-Notch

Thin-Plate

Weirs,

by

John

Shen.