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 triangular-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 international standard. The equation and coefficients have been quoted extensively 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 kh __________________________________________________ 16

An alternative for kh ______________________________________________ 19

Evaluation of Ce _________________________________________________ 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 kh _________________________________________ 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 kh illustrated with Numachi, Kurokawa, and Hutizawa

data (v = 90°) _____________________________ 17

3. Evaluation of kh illustrated with Lenz data _____________ 18 

4-11. Values of:

4. kh as a function of v __________________ 20

5. k as a function of v _____________________- 21

6. Ce, notch angles approximately 120° __________ 23

7. Ce, notch angles approximately 90° _________ 24

8. Ce, notch angles approximately 60° ________ 25

9. Ce, notch angles approximately 45° ______ 26

10. Ce, notch angles approximately 27° _______ 27

11. Ce, notch angles of 10° and 13° _______________ 28

12. Coefficient of discharge Ce as a function of v ___________ 29

13. Coefficient of discharge Ce 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 he 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)

ft2 (square feet)

Length

0.30480 

25.4

Area 

0.09290

To obtain SI units

m (meters)

mm (millimeters)

m2 (square meters)
CONTENTS VII

Multiply inch-pound units By To obtain SI units

Volume 

ft3 (cubic feet) 0.02832 m3 (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

Ce coefficient of discharge (based on effective head)

E relative error

g acceleration due to gravity

h piezometric head referred to vertex of notch

he effective head

j exponent in equation 9

h adjustment quantity

hh 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 recommended and requirements for precise measurements are described. Limits of applicability 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 frequently 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 accuracy 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 experimental 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 parBl
B2 FLOW OF WATER OVER WEIRS AND DAMS

ticular set of experimental data. None provides a comprehensive solution, even for a single liquid. Deficiencies in these formulas are concealed 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 solution. 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), rectangular, 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 channel. 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 generally 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, rectangular, 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 integrated 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 dimensionally 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 second; C is the nondimensional coefficient of discharge; g is the acceleration 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 nondimensional, would remain unchanged in numerical value).

Whereas C is assumed to be a coefficient of contraction in the traditional derivation, in actuality it is an experimentally determined coefficient 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>9t p 9 pt 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 widthcontraction 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 surface 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 related to the absolute magnitude of h 

alone. Thus, both R and W in equation 3 can be replaced with the quantity 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

he = h+kh, (5)

in which he 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, he, 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 successfully 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 Ce 

and kh, is proposed as a new, comprehensive discharge formula for triangular-notch, thin-plate weirs.2 Equation 7 defines Ce 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 experiments were made at a pond in an open field near Belfast. His purpose in making the tests was to acquire experimental evidence to support his proposal that triangular-notch weirs be used instead of rectangular 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 appreciable 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 instrumentation 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 conclusion that the coefficient of discharge was a constant. His results indicated 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 experiments, 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. Experiments 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 vertical 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 produce 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, industrial 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 University (Greve, 1932). A total of 16 different weirs, with notch angles varying 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 comprehensive 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' 

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 

69 

65 

71 

72 

71 

77 

71 

74 

74 

69 

72 

74 

80

To

69 

71 

73 

79 

67 

73 

73 

72 

84 

75 

77 

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 expression 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 experiments 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 

18 

24 

19 

33

Range of head (ft)

From

0.23 

.16 

.20 

.26 

.18 

.24 

.24

To

0.50 

.60 

.55 

.70 

.70 

.70 

.88

Range of temperatures

From

61 

54 

56 

51 

54 

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=(pgh2)/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

« m- 0.170 ,1(.v (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 removed, and C can be computed with the equation

C=0.560+ > (16) ha

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 discussion 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 supplementary 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.^) £- (MWithin 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 triangularnotch weirs, some are of interest because they were especially concerned with certain details of the discharge characteristics. For example, 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 ranging from 39° F to 165° F, Switzer concluded that the effect on the 

coefficient of discharge was less than 2 percent and "inappreciable compared 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 formula which incorporated the kinematic viscosity in a correction coefficient.

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 ft0- 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 vertices 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 mention 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 equipment and technique. More significantly, however, conclusions drawn 

from the experiments are believed to differ because of certain inadequacies in traditional methods of analyzing the data.

It is reasonable to assume that experimental equipment was adequate 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 related to the measurement 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 instability which are related to the 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 investigation was quite limited. For this reason, his proposed general formulas 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" implications of the phenomenon were ignored. It is believed that the effectivehead concept (eq 5) provides a reasonable and practical method of accounting for the effects related to 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 kh

The coefficient Ce in equations 6 and 7 is described as a function of 

geometric ratios (hIP, PIE, and 6) when he , the "effective head," is determined from equation 5,

he =h+kh. (5)

In this equation, h is the measured head on the notch and k^ is an experimentally 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 experimental 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 related to the effective 

head concept has been demonstrated in a previous study 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 experiments 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 interest 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 experimental 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»

A

L 

^

A 

A f

i

r*- ~~^

C as a

 AB 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 kh 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 related

3 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 clearly defined correlation with P. Figure 25 shows Ce plotted as a function 

of hIP. The value of k^ which was used in the computations of Ce for this 

figure was 0.003 foot. Evidence of the lack of correlation between Ce 

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 kh 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 related to h 

and 6.

The lower part of figure 3 shows Ce 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 Ce for notch angles between 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 Ce. 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 kh

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 triangularnotch weirs, a quantity added to the head is simultaneously an adjustment 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°.
0.024

50 60 70 

VALUE OF 0, IN D

FIGURE 4. -Value of kh as a function of 
DISCHARGE OVER TRIANGULAR-NOTCH THIN-PLATE WEIRS B21

» To <o E .- c > N

CO 1= fc P C

31- -s

D O

8

w .

in ^

HI C4H

DC °

O C!

ill O

O -«

UJ *0

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< 13

l

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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°, he can be computed as

(23)

sn

Which of these alternatives is the more convenient is conjectural. 

Hereafter in this report, he is computed with values of k^ from figure 4.

EVALUATION OF Ce

Figures 6 to 11, inclusive, show values of Ce computed from results of 

the tests by Barr, Cone, Yarnall, Greve, Lenz, and Numachi, 

Kurokawa, and Hutizawa. In the figures, Ce is plotted as a function of 

h/P. The data cover a long range of notch angles from 120° to 10°. Included 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 computing Ce for each set of tests are the values plotted in figure 4. The 

average values of Ce 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 practical applications, it is apparent that Ce is virtually independent of h/P 

and PIE. Thus, within this range Ce 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 

Ce determined from figures 6 through 11. The dashed portions of the 

summary curves for Ce (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 

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0.59

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VALUE OF hIP 

FIGURE 6.-Value of Ce, notch angle approxim
LL 

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0.62

0.61

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Sym

V 

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+ 

A 

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;^,

0.05 0.10 0.15 

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VALUE OF hIP 

FIGURE 7.-Value of Ce, notch angle approxim
DISCHARGE OVER TRIANGULAR-NOTCH THIN-PLATE WEIRS B25

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Symbol 

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0.05 0.10 0.15

VALUE OF h/P 

FIGURE 9.-Value of Ce , notch angle approximat
DISCHARGE OVER TRIANGULAR-NOTCH THIN-PLATE WEIRS B27

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Symbol 

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05 0.10 0.15 0.20 0.2

VALUE OF hIP 

FIGURE 11.-Value of Ce , notch angles of 10
DISCHARGE OVER TRIANGULAR-NOTCH THIN-PLATE WEIRS B29

T3 s

O

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 obtained from the experiments on 90° notches made by Numachi, 

Kurokawa, and Hutizawa.

Values of Ce were computed from the results of the NumachiKurokawa-Hutizawa tests, using kh= 0.003 foot. Smooth curves interpolated from the plotted data were cross-plotted to obtain the family of 

curves showing Ce 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 unsubstantiated. 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 Ce values from those tests, plotted in figure 7, are all within 1 percent 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 combined 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 Ce 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

I

o I

8 S 5

6 c> o

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 Thomson 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 formulas 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 Ce 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 Ce 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, convenient means of measuring the flow of liquids. Classic discharge formulas 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 experimental data are sufficient to substantiate a practical solution which 

is based on dimensional analysis and experimental coefficients. Limitations 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 he, 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

0.56

1.0 1.2 

VALUE OF h, 

FIGURE 15. -Comparison of formulas for notch angle
0.64

0.63

0.62

0,57

1.0 1.2 

VALUE OF h, IN F

FIGURE 16.-Comparison of formulas for notch angle
0.64

0.63

0.57

1.0 1.2 

VALUE OF h, IN F

FIGURE 17. -Comparison of formulas for notch angle
0.65

0.64

0.2 0.4 0.6 0.8 1.0 1.2 

VALUE OF h

0.58

0.57

FIGURE 18. -Comparison of formulas for notch ang
0.68

0.67

0.61

0.60

0.2 0.4 1.0 1.2 

VALUE OF h

FIGURE 19.-Comparison of formulas for notch angle
DISCHARGE OVER TRIANGULAR-NOTCH THIN-PLATE WEIRS B39

In general, the coefficient of discharge, Ce , 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, Ce is a function of 6 alone (fig. 12). In this range, values of Ce 

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 Ce 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: Engineering (London), v. 89, (April 8 and 15), p. 435 ff.

Burgess, J. S., and White, W. R., 1966, The triangular-profile (Crump) weir-two-dimensional 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 verticalnotch 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 notchesrectangular, 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 Tribune 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 triangular 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 measurements is restricted to steady, free, and fully ventilated flows. The 

recommended coefficients are applicable for water only, in the approximate range of temperatures from 40° F to 85° F.

General specifications for a satisfactory weir installation are described 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 observation that head-measurement difficulties and errors result from surges 

and waves which occur in the approach channel when the velocity of approach 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 accuracy for head measurements. These limits are separate from restrictions 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 related to 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 kk

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 he 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 he in equation 6.

EFFECT OF APPROACH-CHANNEL CONDITIONS

Specifications for a satisfactory weir installation include the requirement 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, approach velocities are negligible, and the shape of the channel is insignificant. 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 accurately. 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 manometer. Hook and point gages preferably are used in a stilling well. 

However, if approach velocities and surface disturbances in the approach channel are negligible, the gages can be mounted over the headwater surfaces. Stilling wells and manometers are connected to 

piezometers located in the floor or walls of the channel. To prevent errors 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 deflection 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 acceptable 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 temporary 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 approach 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 bottom 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.