1C. 397

GEOLOGICAL SURVEY CIRCULAR 397

DISCHARGE CHARACTERISTICS 

OF BROAD-CRESTED WEIRS
UNITED STATES DEPARTMENT OF THE INTERIOR 

Fred A. Seaton, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

GEOLOGICAL SURVEY CIRCULAR 397

DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

By H. J. Tracy

Washington, D. C., 1957

Free on application to the Geological Survey, Washington 25, D. C.
SYMBOLS

6 = weir width, in feet, normal to the direction of flow; also, width 

of contracted opening

B = width, in feet, of rectangular channel 

C = coefficient of discharge 

F = Froude number, V / J/yh 

g = acceleration of gravity

h = height of water surface, in feet, above the crest of the weir 

measured at a section just upstream from the beginning of acceleration of the fluid as it approaches the weir

H = energy head, in feet, A + V0 2 l%9, where V0 is the mean 

velocity of the fluid at the section of approach

k = mean height of boundary-roughness projections, in feet, above 

the nominal level

L - weir length, in feet, parallel to the direction of flow

P = height of weir, dn feet

q discharge per foot of width

Q = discharge, in cubic feet per second

r radius of founding, in feet, at upstream crest

R = Reynolds number, V h p/p.

t = height of the water surface, in feet, above the crest of the 

weir, at a section just downstream from the point where the 

flow has reestablished its normal regime after passing over 

the weir

V = velocity of flow, in feet per second 

W = Weber number, V I J/cr / ph 

y = depth of water on weir crest

7 = specific weight 

p = mass density 

/i = viscosity 

cr = surface tension
DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

By H. J. Tracy

CONTENTS

Page

Symbols ................................. Faces

Abstract. ...................................

Introduction.................................

Control section.............................

Dimensional analysis. .......................

External flow pattern. .......................

Discharge coefficient .......................

Definition................................

Effect of parameters influencing the coefficient ...................................

Weir length ............................

Page

Weir height ............................. 7

Nappe form ............................. 7

Reynolds number ........................ 8

Rounded weir entrance ................... 9

Boundary roughness...................... 10

Shape ratio.............................. 11

Sloping upstream and downstream faces .... 11

Submergence............................ 11

Bibliography ................................ 15

Published reports. ......................... 15

Unpublished reports. ....................... 15

ILLUSTRATIONS

Figure 1. Definition sketches of broad-crested weirs with vertical faces and horizontal crest. ........

2. Dimensionless water-surface profiles for broad-crested weirs with vertical faces and hori- 

	zontal crest......................................................................

3. Discharge coefficients for broad-crested weirs with vertical faces and horizontal crest,

	square entrance...................................................................

4. Effect of the wetted-underneath nappe form on the discharge coefficient...................

5. Effect of the depressed nappe form on the discharge coefficient..........................

6. Effect of the detached nappe form on the discharge coefficient ...........................

7. Variation of the discharge coefficient with scale of weir ..................................

8. Discharge coefficients for broad-crested weirs with vertical upstream face^ ..............

9. Discharge coefficients for broad-crested weirs with upstream-face slope of 1/2:1. .........

10. Discharge coefficients for broad-crested weirs with upstream-face slope of 1:1 ...........

11. Discharge coefficients for broad-crested weirs with upstream-face slope of 2:1 ...........

Page

6

7

8

9

10

11

12

13

14

ABSTRACT

Discharge characteristics of broad-crested weirs 

defined by laboratory tests are described. Broadcrested weirs are classified as short, normal, or 

long according to the form of the water-surface profile over the weir. The discharge equation is obtained 

by dimensional analysis, and the coefficient of discharge is related to dimensionless ratios that describe 

the geometry of the channel and the relative influence 

of the forces that determine the flow pattern. The effect of these various ratios on the discharge coefficient is shown by the use of existing laboratory data. 

No new experimental work is involved.
DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

INTRODUCTION

Frequently, so-called indirect methods of discharge 

measurement are the only practicable means of obtaining the magnitude of a peak flood flow past a given 

site. These determinations are based on the watersurface profile, usually defined from high-water 

marks, and upon the geometry and hydraulic .characteristics of the channel. If a transition structure, 

such as a dam or abrupt channel constriction, is used 

as the measuring device, the geometry of the structure also affects the water-surface profile for a given 

discharge. For this reason, a knowledge of the headdischarge relation for a given weir or spillway may 

prove invaluable for the determination of an important flood peak which could not be otherwise measured.

A flow determination involving a weir or spillway is 

classified as indirect if the structure is uncalibrated, 

and the head-discharge relation must be obtained from 

a comparison with other similar calibrated weirs. 

Needless to say, the number of possible weir forms 

is almost unlimited.

This report deals only with the broad-crested weir 

form. Definition sketches of this weir are shown 

in figure 1. This study is based on existing data on 

flow over broad-crested weirs, and no new experimental work is involved. It has not been possible to 

answer many questions which arise. For example, 

sufficient data have not been published to define the 

effect of angularity of crest to direction of flow, of 

gates and piers, of end abutments, and so on. The 

question of submergence has also been considered in 

only a very general manner.

The term "broad-crested weir" is generally poorly 

defined; usually, this weir has been classified only 

with respect to the geometry of the structure itself. 

Almost universally, a weir is called broad-crested if 

it has a more or less horizontal crest of finite length 

in the direction of flow. This definition is not entirely 

adequate, because at sufficiently high head-to-length 

ratios the nappe tends to spring clear of the weir 

crest, and the structure no longer performs as a 

broad-crested weir. At the opposite extreme, for 

very small head-to-length ratios, the weir crest becomes a reach of open channel in which frictional resistance predominates, and for which the discharge 

is more properly evaluated by one of the open-channel 

flow formulas than by a weir formula. It is thus clear 

that any definition of a broad-crested weir must include the head acting on the weir. This is considered 

in some detail on page 4.

CONTROL SECTION

For broad-crested weirs it is usually assumed that 

the flow will be critical on the weir crest. Traditionally, the weir discharge is determinable from a single 

depth measurement on the weir crest and the equation

g = (1)

This elementary method, however, is not satisfactory when an accurate discharge determination is required. The flow depth does not correspond to that 

given by equation 1 everywhere on the weir crest. 

The location of the control, or critical depth, section, 

is not constant, but varies with discharge, weir geometry, and crest roughness.

The inlet geometry is of primary importance in the 

location of the critical-flow section. If the weir entrance is not rounded, separation will occur just downstream from the entrance, and the critical section 

will coincide with the maximum elevation of the separation surface. This is a zone of curvilinear flow, 

for which, for a given specific energy, the discharge is not necessarily the same as for parallel 

flow. Equation 1 applies only to rectilinear motion, and may not be used otherwise.

If, on the other hand, the weir entrance is sufficiently well rounded to prevent separation, the control section is shifted downstream. If this weir is of 

great enough length, a central region of essentially 

parallel flow will form which is free from the curvilinear effects at the two ends. If the fluid were frictionless, the depth of flow would be critical everywhere in this region. Actually, an appreciable 

boundary layer is formed in which are concentrated 

the viscous shear losses which are almost entirely 

responsible for the slope of the total-head line. By 

defining the discharge-displacement thickness of this 

boundary layer as the distance to which the boundary 

would have to be displaced to satisfy the equation of 

continuity if the velocity in the boundary layer were 

taken equal to the uniform, potential velocity outside 

the layer, and using this displacement thickness as 

an effective correction to be applied to the computed 

critical depth referenced to the weir crest, it seems 

possible that the traditional approach to the treatment 

of flow over a broad-crested weir with rounded entrance could be reconciled with that which has actually 

been found to exist. However, largely due to instrumentation difficulties, few boundary-layer measurements have been made in water, so that little progress 

has been made in this direction.

These considerations, in the light of present knowledge, restrict the use of the broad-crested weir as a 

critical depth meter whereby the discharge may be 

determined as a function of a single depth measurement. It may be pointed out, however, that all of 

these factors influencing the location of the criticaldepth section and the shape of the water-surface profile over the weir are functions of weir geometry, 

weir roughness, and head on the weir. These parameters, likewise, control the head-discharge relation of the weir. Lacking a generalized method to 

permit the location and evaluation of the critical-flow 

depth on the weir, a functional form of the discharge 

coefficient must be sought that is based upon a combination of these variables into significant dimensionless ratios which adequately describe the pattern of 

flow. This is the method to be used here.

DIMENSIONAL ANALYSIS

The following variables are sufficient to describe 

the flow characteristics of a level, broad-crested, 

nonsubmerged weir having vertical upstream and 

downstream faces and located in a long, smooth, 

horizontal, rectangular channel. In functional notation,

f(h or p> V, y, p, fj., P, B, L, It, r) = 0.

The dimensions listed here are shown in figure 1. 

All symbols used are defined at the beginning of the 

report (facing p. 1).

There are 3 independent fundamental dimensions 

and 10 physical quantities involved in equation 2. By 

choosing A, V, and 7 as repeating variables, and combining these in succession with the remaining quantities, seven dimensionless ratios may be formed. 

Equation 2 thus becomes
DIMENSIONAL ANALYSIS

-Total-head line

(a) Submerged flow, square upstream crest

Nominal boundary layer

(b) Free flow, rounded upstream crest

Figure 1. --Definition sketches of broad-crested weirs with vertical faces and horizontal crest.
DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

C

V VHP A. A *. * JL\ 

Ygh /j, ' P ' B ' L ' k ' h)

= 0. (3)

It will be observed that the first of these ratios in 

the left member of equation 3 has the form of a Froude 

number. Under the assumed flow conditions, however, the weir is a control structure , and the Froude 

number is not an independent variable; that is to say, 

V and h cannot be varied independently of each other. 

For this reason, this characteristic number is best 

described as a coefficient of discharge and rewritten 

in the form C = Q/

The second ratio is recognized as the Reynolds 

number. Rearranging equation 3,

If, in the original array of variables, the surface 

tension of the fluid had been included as one of the 

variables influencing the flow pattern, the Weber 

number would have been produced as an additional 

dimensionless ratio. Except when the head is very 

small, the effect of surface tension is negligible and 

has hence been omitted from consideration altogether.

Thus, in equation 4, the coefficient of discharge is 

shown as a function of the Reynolds number and a 

number of independent length ratios involving the head 

and the physical dimensions of the weir. An explicit 

relation between the discharge coefficient and these 

variables is, of course, not known.

In the following sections of this report, the effect 

of each of these combinations of dimensionless variables upon the discharge coefficient will be investigated to the extent that it is possible with the data 

available.

EXTERNAL FLOW PATTERN

It will be helpful, before proceeding to an analysis 

of the variation of the coefficient of discharge as outlined in the preceding section, to consider some aspects of the external flow pattern of the broad-crested 

weir as evidenced by measured water-surface profiles. 

For this purpose, data by Prentice (1935) and Woodburn (1932) that represent two basic weir forms have 

been utilized. Sufficient generality is provided by 

these data to illustrate several significant points.

First, in figure 2A t the water -surface profile for 

flow over a level, broad-crested weir with vertical 

upstream and downstream faces is shown. The approach channel is rectangular and horizontal. The 

weir breadth is the same as the channel width. The 

weir surfaces are of planed wood or smooth concrete. 

The downstream nappe is fully aerated, and the weir 

entrance is square. The profile for the second weir 

form is shown in figure 26. This weir is similar in 

all respects to that of figure 2A, except that the entrance is rounded.

These profiles have been made dimensionless by 

dividing both the depth of flow over the weir crest and 

the distance from the beginning of the weir by the piezometric head, h , on the crest. The righthand ends 

of these curves represent the dimensionless flow 

depth at the downstream weir brink.

With the exception of two curves (those for an h/L 

ratio of 0. 40 and 0. 073)^ the profiles for the weir "with 

the square entrance (fig. 2,4} exhibit strikingly similar configuration characteristics. In the acceleratedflow zone in the proximity of the entrance, the ratio

of depth to head decreases rapidly, its magnitude varying somewhat with the value of A/P. Separation occurs 

at the sharp entrance to the crest, and the control section lies in the body of fluid above this separation zone. 

The flow becomes supercritical beyond this point. 

After passing through this crest zone, the flow begins 

to decelerate slightly, due to boundary resistance 

and to some expansion and consequent diffusion downstream from the separation zone. This deceleration 

is manifest in a slowly rising water-surface profile 

over the approximately parallel-flow region of the 

weir crest. Some distance downstream from the weir 

entrance, the profiles tend to superimpose, and the 

A/P effect is no longer distinguishable.

If the weir crest is sufficiently long, the depth must 

eventually again approach the critical on the weir 

crest as the flow continues to decelerate with increased distance from the entrance. At this critical 

value, the specific head is the minimum at which 

this rate of flow is physically possible. A further 

energy loss, of necessity, causes an adjustment of 

depth and leads to the formation of a standing-wave 

pattern typical of flow at or near critical depth. This 

standing-wave configuration first appears when the 

weir length is roughly 12. 5 times as great as the 

head on the weir, or at an h/L ratio of about 0. 080. 

All profiles for smaller ratios (not shown) of head to 

length show similar standing-wave characteristics.

If, on the contrary, the weir crest is relatively 

short and the head is large, the flow over the weir 

will be entirely curvilinear. In the upstream crest 

zone, this profile is somewhat lower vertically than 

that for the weir of great enough length to permit a 

zone of approximately parallel flow to form. This 

curvilinear profile appears when the h/L ratio becomes greater than about 0. 40. This bounding figure 

is, of course, not precise, but varies to some extent 

with the ratio, of A to P. Curves for h/L values 

greater than 0. 40 are not shown on figure 2A. It is 

evident, however, that as h/L becomes larger, the 

water-surface profile must approach that of the upper 

water surface for a thin-plate sharp-crested weir.

Thus, broad-crested weirs may be divided into 

three groups with respect to the flow profile: short 

weir--entirely curvilinear flow; normal weir curvilinear flow zones at the beginning and at the end of 

the weir crest, with a central zone of essentially 

parallel flow; and long weir--flow with a standingwave pattern on the crest of the weir.

Approximate limits for these weirs, in terms of the 

h/L ratio, have already been suggested. The weir is 

short if h/L is larger than about 0.40, and is long if 

h/L is less than about 0.08. Between these limits, the 

weir is normal. For the purpose of this report only 

the general limits are of importance.

It was suggested earlier that the flow is critical at 

a point near the downstream end of the weir at the 

transition between a normal and a long weir. It may 

be pointed out, as evidence of this, that the depth of 

flow at the downstream brink with respect to the weir 

head is remarkably constant. This depth for all the 

profiles of figure ZA at the brink is about 0. 43*. The 

ratio of the brink depth to criticat-flow depth for the 

free overfall has been fairly well established by 

Rouse (1936) as 0.715. Then* in terms of A, the critical depth on the erest of the weir is 0.43& / 0^715 = 

0. 60%, which is in good agreement with the maximum 

depth approached by the profiles of figure 2A near the 

downstream end of the weir for h/L ratios near 0. 08.
EXTERNAL FLOW PATTERN

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DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

The effect on the water-surface profile of rounding 

the weir entrance may be obtained from a comparison 

of figures 2A and 2B. The profiles for the two weir 

forms have the same general shape. The profiles for 

the weir of figure 2B, however, are displaced upward 

vertically. Further, the range of h/L for the normal 

weir of figure 2B is much smaller than its counterpart of figure ZA.

The energy losses in the separation zone just downstream from the beginning of the weir with the square 

upstream entrance may be estimated from these two 

figures. Assuming that the flow is critical at a point 

just upstream from the drawdown zone over the downstream crest exit for both weirs at the limit of the 

normal weir, y c for the weir with the square entrance 

is about 0. 58A and for the weir with the rounded entrance is 0.67A. The difference between these two 

depths represents the energy losses in the separation zone if it may be assumed that the resistance 

losses along the weir crests are approximately equal 

for both weirs, and if the reference head, A , may be 

considered to be roughly equal to the total head, H . 

This difference is 0. 09A.

DISCHARGE COEFFICIENT 

Definition

In a preceding section, the coefficient of discharge 

was written in the form C = Q /h>Yff A 3/2J. It will be

more convenient to define this coefficient in terms of 

the total head, H , on the weir and to combine the term 

yj with the coefficient of discharge. Thus C = Q/(bH3/2). 

The precepts of dimensional analysis are not violated 

by using H instead of A, Because of the dependency of 

V and A, the coefficient thus defined is functionally 

related to the same variables as the coefficient based 

upon the piezometric head, A.

Effect of Parameters Influencing the Coefficient 

Weir Length

For the weir with a square entrance the dimensionless variable most important in fixing the value 

of the coefficient of discharge is the ratio h/L. 

The coefficient of discharge is shown as a function 

of this ratio in figure 3, which is defined from the 

data of several investigators. Here, each curve is 

identified by the name of the person responsible 

for the experimental work, and the source is given 

in the bibliography. The individual points through 

which these mean curves are drawn are not shown, 

but an idea of the magnitude of the scatter of these 

points about each curve may be obtained from a 

typical plot such as figure 7, for which the experimental data are shown.

400

180

3.60

MO

120

3.00

Z80

2.60

2.40

KMtncr

U&GS. tats at Cornell

Bazin

1.0 L5 2.0

Figure 3. --Discharge coefficients for broad-crested weirs with vertical faces and horizontal crest,

square entrance.
DISCHARGE COEFFICIENT

Weir Height

In order to define the effect of a change in P upon 

the discharge coefficient, the deviations of the test 

points from the curves of figure 3 were plotted against 

the ratio A/P . No correlation between these deviations and the h/P ratio was found. It was therefore 

concluded that the effect of the A/P ratio is adequately compensated for by the use of the total head 

in computing the discharge coefficient. This is, in 

large part, the reason for substituting H for A in computing the discharge coefficient, and has been found to 

be advantageous even though the selection of a discharge coefficient from a chart where the coefficient 

is a function of the total head involves a solution requiring successive approximations.

The normal weir profiles (fig. 2A ) show almost 

complete geometric similarity, except for some 

slight variation due to approach velocity. This geometrical similitude suggests that the discharge coefficient should then be very nearly constant in the 

range ofh/L for the normal weir. This appears even 

more reasonable when one* considers that the control 

section is just downstream from the entrance and 

over the separation zone, and that, because of this, 

the weir length does not affect the discharge. The lack 

of effect of h/P on the weir coefficient may perhaps 

be explained by the fact that the separation profile, 

as well as the surface profile, is affected by this 

variable, and that a compensation in the flow area 

at the critical section is obtained. These remarks, 

for less reason, have also been extended to the long

weir. It is conceivable, of course, that the weir 

could be long enough, or that the crest could be 

roughened to such an extent, so that the upstream 

channel control could be submerged by the downstream channel control. This possibility has not 

been considered.

In any event, the discharge coefficients for the long 

and the normal weir are shown as having a constant 

value in figure 3.

Nappe Form

The literature contains many references to the importance of the nappe form in influencing the discharge 

coefficient. Given a nonaerated, nonsubmerged broadcrested weir, the nappe may assume any of the following forms, depending upon the head. These four forms 

are listed in their order of occurrence as the discharge 

increases.

1. Adherent to the downstream face of the weir.

2. "Depressed"; that is, some of the air under the 

nappe is entrained by the moving fluid and replaced 

by water, leaving the remaining air at less than atmospheric pressure.

3. "Wetted underneath"; all the air underneath the 

nappe is entrained and replaced by water, and the 

space formerly occupied by air is filled with eddying 

fluid.

4.20

4.00

3.80

3.60

3.40

3.20

3.00

Curve defined by BozW* datq for aerated 

nappe; L- 0.656*

Plotted^ polnte repreeent 

wetted-underneath napp* form; L-0.656'

2£0

1.0

JL

(.5 2.0

Figure 4. --Effect of the wetted-underneath nappe form on the discharge coefficient.
DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

4.20

4.0O

3.80

3.60

3.40

3.20

3.00 /.

Curve defined by Bozin'i data far 

nappei L» 0.666*

Plotted point* repreeent BcBhft teete with 

depreeeed nappe form; L-0.656'

2.80

2.60

0.5 1.0 1.5 ^0

Figure 5. --Effect of the depressed nappe form on the discharge coefficient.

4. Free altogether of the flat crest of the weir and 

in contact with the weir only at the upstream face.

These nappes were named by Bazin (1896), who was 

among the first to investigate their effect on the discharge over the weir. The adhering nappe is usually 

of negligible importance and occurs only when the 

head is extremely low. The point at which one nappe 

form disappears and the next appears is somewhat 

indeterminate. These limits also change according 

to whether the discharge is increasing or decreasing, 

and are dependent upon the shape of the weir itself.

Complete data for the establishment of these transition ranges does not exist. The data of Bazin does, 

however, enable a comparison to be made between the 

coefficient of discharge for an aerated nappe and the 

coefficients for the wetted-underneath and depressed 

nappes, as shown in figures 4 and 5, respectively. In 

these figures, the plotted data represent the coefficient of discharge for these nappe forms, and the 

curves represent the coefficient of discharge for 

weirs with aerated nappes. These tests were made 

on level broad-crested weirs with vertical upstream 

and downstream faces. Apparently, from these 

figures, the overall change in the discharge coefficient due to a change from the aerated to the depressed 

or wetted-underneath nappe forms is on the order of 1 

or 2 percent. This is almost inconsiderable for any except very precise discharge determinations.

The point at which the nappe becomes detached at 

the upstream crest entrance and springs clear of the

weir crest varies somewhat, depending upon the completeness of aeration of the space beneath the downstream nappe, the h/P ratio, and upon the sharpness 

of the entrance; it usually occurs suddenly. The tests 

made by Bazin indicate that it may occur at an h/L 

as low as 1. 5, or as high as 2.0, for aerated weirs. 

Once the nappe springs clear of a broad-crested weir, 

it is usual to assume that the weir acts as a thin-plate, 

sharp-crested weir. In figure 6, the tests made by 

Bazin on level broad-crested weirs with vertical 

faces, square entrances, and detached nappes are 

compared with the curve defined by Bazin's data for 

thin-plate, sharp-crested weirs., ft will be noted that 

the agreement of the plotted points with the curve is 

somewhat less than ideal; this may possibly be attributed to the lack of true sharpness of the entrance 

to the br^ad-crested weir.

Reynolds Number

A definite variation in the coefficient of discharge 

with scale ratio is to be expected. This so-called 

scale effect is the result of the action of the forces 

due to the viscosity of the fluid. The Reynolds number is used to describe the relative importance of 

the viscous forces; the larger the Reynolds number, 

the less important the influence of viscosity upon the 

flow pattern. The Reynolds number of equation 3, for 

a given fluid, is proportional to A3' 2 , or to the discharge per foot of width of the weir. From this, it is 

obvious that the Reynolds numbers of model and prototype can be equal only at the same scale. It is thus 

usually impracticable to satisfy the Reynolds law
DISCHARGE COEFFICIENT

4.00

3.80

3.60

3.40

3.20

3.00

2.80

2.60

2.40

Curve del ined by Bozin data for 

sharp-crested weirs

thln-plo1e

Plotted points represent Bazin's- tests on level, brood-crested weirs with vertical faces end square 

upstream crest after nappe hod become detached from the horizontal crest] series 66 94

ai 0.2 0.3 0.4 0.5 0.6 0.7 O8

h+P

Figure 6. --Effect of the detached nappe form on the discharge coefficient.

completely. Generally, however, it is possible to 

obtain Reynolds numbers in the laboratory sufficiently 

high to include the range in which the effects of fluid 

viscosity may be ignored.

Two noteworthy studies have been made to determine the effect of scale on the discharge coefficient. 

These experiments were made on nappe-shaped weirs. 

The first, reported by Eisner (1933), describes tests 

made on models at scales of 1:70, 1:35, 1:17.5, 1:8.75, 

and 1:4.375. An increase in discharge coefficients was 

noted with an increase in model scale; this effect was 

attributed by Eisner to the relative roughness of the 

boundary surf ace--absolute roughness being the same 

in each of the several models.

From measurements made on geometrically similar 

models, of the nappe-shaped Pickwick Landing spillway at scales of 1:200, 1:100, and 1:500, Kirkpatrick 

(1955) concluded that the scale of the model does not 

affect the relation between head and discharge coefficient. He found no consistent deviation of the discharge coefficient at the three scales and therefore 

attributed the small variations to experimental error.

For the broad-crested weir, Bazin's experiments 

furnish sufficient data to determine the Reynolds number effect qualitatively. In figure 7 are plotted the results of Bazin's measurements on weirs of rectangular 

shape and square upstream entrances, and at various 

scale ratios. For the lowermost curve the scale is 8 

times as large as that for the uppermost curve, for 

the next curve 4 times as large, and for the third curve

twice as large. The scale of P is not considered in 

these tests as it has been shown to have no effect on 

the discharge coefficient. For Bazin's tests the coefficient of discharge decreases with increasing scale, 

or with increasing Reynolds number.

With a very few exceptions, these plotted points 

represent tests on short weirs. The heads for the 

uppermost curve are very small, so that the deviation 

of these points may perhaps be ascribed primarily to 

effects of surface tension. The slight variation between the remaining three curves is due to some differences in energy required to maintain the eddy motion in the separation zone as the weir scale increases. 

For the rounded-entrance weir, this trend is probably 

reversed; that is, it would be expected that the coefficient will increase with increased scale, and for the 

same reasons that the Eisner coefficients were found 

to increase.

Because the U. S. Geological Survey tests at Cornell (see fig. 3), made at a much greater scale than 

Bazin's testsj agree fairly well with the lowermost 

of Bazin's curves on figure 7, it is assumed that for 

weirs at least as large as Bazin's largest, the Reynolds number is large enough so that the effects of 

fluid viscosity may be ignored. Most structures found 

in the field are of much greater size. For these, the 

the lowermost curve of figure 7 should be used to 

select a discharge coefficient.

Rounded Weir Entrance

The effect of rounding the upstream entrance to 

the weir is to increase the discharge coefficient by
10 DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

removing the cause of the flow separation at the crest 

entrance, which is the primary source of energy loss. 

This variation in the discharge coefficient has been 

shown to be a function of r/h in equation 2, but sufficient data to completely define the function do not 

exist. Woodburn ran a series of tests in which he systematically varied the upstream crest-entrance radius 

of rounding from 0 to 8 inches in 2-inch increments. 

The discharge coefficients for the 2- , 4- , 6- , and 

8-inch roundings (r/h > 0.15) superpose and are about 

9 percent greater than the corresponding coefficients 

for the square entrance (r/h = 0). i

This data may be interpreted by analogy with similar 

data for the contracted opening. Kindsvater and Carter 

(1955) demonstrated that the discharge coefficient for 

contracted openings increased almost linearly with the 

values for r/b from r/b = 0 to r/b = O.U; the coefficient remained constant after r/b exceeded 0. 14. A 

similar variation in the coefficient of discharge for 

orifices has been demonstrated in the literature.

For the lack of a more exact procedure, it is suggested that a discharge coefficient be chosen for the 

rounded-entrance weir equal to 1. 09 times the coefficient for the square entrance if the value of r/h is 

greater than 0. 14. For intermediate values of r/h the 

adjustment may be interpolated between the extremes 

of unity and 1.09.

Boundary Roughness

One of the most troublesome aspects of model testing is the difficulty involved in duplicating in the model

the surface resistance of the prototype. For a given 

test structure having a given boundary roughness, the 

energy losses due to boundary resistance are a function of a characteristic Reynolds number and the relative roughness. As the scale is decreased, the relative 

roughness increases, and the boundary shear force 

thus becomes disproportionately large if the boundary 

surfaces are of the same material for the various 

scales.

The relative roughness is usually described by relating the mean height of the boundary projections 

above the nominal surface level to some significant 

length. Here, the ratio h/k is used, where k is the 

mean roughness height. Thus, only if the relative 

roughness and the Reynolds number are equal in 

model and prototype is there assurance that the relative effect of boundary resistance is the same at both 

scales. It is generally impractical, if not impossible, 

to arrange equal values of either K or A/ft. Requirements of similitude are compromised by either ignoring the difference in relative roughness or by 

extension to the scale of the prototype by the use of 

some method based on experience.

Here, again, definitive data for a determination of 

the effect of boundary roughness are not at hand. Unless the model is extremely small, it is usual to 

apply the results from it to much larger scale structures. In general, if the influence of boundary roughness is considered separate from other effects, the 

discharge coefficient should increase slightly as the 

scale of the model increases.

bH

%ww 

3.80 

3.60 

3.40

320 

f

3.00

£80 

£60 

2.40

 -

h

-+-

__.

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T"~

H

P

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4-

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L

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ft

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.;

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X +

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^

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^

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p

X

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'^'

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^

^^

^

Bin M

 

t

&

^

4M X 

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" 113

^"

~*i

^

.93,9- 

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 ^B7^

 -*'«» 

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-"

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I L-0. 

L»l

.1

164* 1

666' 

31' 1

»« 1.15 

»»£46

^46

1

i

> OS 10 . 15 2JO

Figure 7. --Variation of the discharge coefficient with scale of weir. (Data from Bazin, 1896.)
DISCHARGE COEFFICIENT 11

Shape Ratio

The remaining variable of equation 3, A/B , may be 

interpreted as a shape parameter. It was not found to 

be significant in the range for which test data are 

available.

Sloping Upstream and Downstream Faces

In figures 8-11, the discharge coefficient for level, 

broad-crested weirs having various combinations of upstream- and downstream-face slope is shown. The test 

data are from the experiments of Bazin.

In each of these figures, the upstream-face slope is 

constant. Thus, figure 8 represents the discharge coefficients for weirs having a vertical upstream-face 

slope, figure 9 the coefficients for weirs with an upstream-face slope of 1/2:1, figure 10 the discharge coefficient for a weir with an upstream-face slope of 1:1, 

and figure 11 the coefficient for a weir with an upstream-face slope of 2:1. The families of carves shown 

on each of these figures represent the discharge coefficients for various downstream-face slopes. Interpolated curves are shown as dashed lines.

Submergence

Submerged weirs (fig. I A) are beyond the scope of 

this report. Generally, it appears that the nonsubmerged-weir coefficients may be applied to weirs for 

which the ratio t/h is equal to or less than 0.85.

4.00

 Strict 144 Downstream slop* 2:1 

157 

149

2.40

Figure 8. --Discharge coefficientsfor broad-crested weirs with vertical upstream face. (Data from Bazin, 1896.)
12 DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

4.00

3.80

Downstream slope 

as indicated in

 Series 147 Downstream slope hi 

148

Figure 9. --Discharge coefficients for broad-crested weirs with upstream-face 

slope of 1/2:1. (Data from Bazin, 1896. )
DISCHARGE COEFFICIENT 13

4.0 Or

£80

£60

£40*-

 Series 174 Domnsfreom slope I: I 

167

« 176 2:1 

177

Y 171 3:J

+ (72 5:1 

(80

0.5 1.0 1.5 £0

Figure 10. --Discharge coefficients for broad-crested weirs with upstream-face 

slope of 1:1. (Data from Bazin, 1896.)
14 DISCHARGE CHARACTERISTICS OF BROAD-CRESTED WEIRS

4.00

Ommttr«am slope

'-* as indicated in explanation

3.40

3.20

Q

bH 1

3.00

1:1 

2:1 

3:1 

5H

2.80

2.60

 Series 168

* 175

i 170

178

179

i 173 

181

slope 1:1 

2:1

5:1

£40

0.5 1.0 1.5 2.0

Figure 11. --Discharge coefficients for broad-crested weirs with upstream-face 

slope of 2:1. (Data from Bazin, 1896.)
BIBLIOGRAPHY

Published Reports

Bazin, M. H., 1896, Experiences nouvelles sur 

1'ecoulement en deversoir: Annales des Fonts et 

Chausse'es, v. 7, ser. 7.

Boss, Paul, 1929. Berechnung der Abflussmengen 

und der Wasserspiegellage bei Absturzen und 

Schwellen unter besonderer Berucksichtigung der 

dabei auftretenden Zusatzspannungen: Wasserkraft 

und Wasserwirtschaft, nos. 2-3, p. 13-14, 28-33.

Doeringsfeld, H. A., and Barker, C. L., 1941, Pressure-momentum theory applied to the broad-crested 

weir: Am. Soc. Civil Engineers Trans., v. 106, 

p. 934.

Eisner, Franz, 1933, Uberfallversuche in verschiedener Modellgrosse: Mitt. Preussichen Vers.uchsanst, 

Wasserbau und Schiffbau [Berlin], no. 11.

Forchheimer, Philipp, 1930, Hydraulik: B. G. Teubner, 

Leipzig and Berlin, p. 373-410.

Mailer, Rudolf, 1929, Fehlerquellen bei der tiberfallmessung: Mitt. Hydraul. Inst. Techn. Hochschule 

Munschen, no. 3.

Horton, R. E., 1907, Weir experiments, coefficients, 

and formulas: U.S. Geol. Survey Water-Supply 

Paper 200.

Ippen, A. T., 1950, in Rouse, Hunter (ed.), Engineering hydraulics: New York, John Wiley and Sons, Inc., 

p. 526-27.

Keutner, Chr., 1934, Strbmungsvorgange an breitkronigen WehrkSrpern und an Einlaufbauwerkern: 

Der Bauingenieur, v. 15, p. 366.

Kindsvater, C. E., and Carter, R. W., 1955, Tranquil flow through open-channel constrictions: Am. 

Soc. Civil Engineers Trans., v. 120, p. 955.

King, H. W., 1954, Handbook of hydraulics: New 

York, McGraw-Hill Book Co., 4th ed., p. 5-1 to 

5-15.

Kirkpatrick, K. W., 1955, Discharge coefficient for 

spillways at TVA dams: Am. Soc. Civil Engineers 

Separate 626.

Moore, W. L., 1943, Energy loss at the base of a 

free overfall: Am. Soe. Civil Engineers Trans., 

v. 109, p. 1343.

Musterle, Th., 1930 r Abfiussberechnungen bei Wehren 

mit breiter Krone mit Hilfe des Impulssatzes: Die 

Wasserwirtschaft, no. 21.

Rouse, Hunter, 1933, Verteilung der Hydraulischen 

Energie bei einem lotrechten Absturz: Munich and 

Berlin, R. Oldenburg.

______ 1936, Discharge characteristics of the free

overfall: Civil Eng. v. 6, no. 4, p. 257.

United States Bureau of Reclamation, 1948, Studies 

of crests for overfall dams: Boulder Canyon Proj., 

Part 6, Bull. 3.

Woodburn, J. G., 1932, Tests of broad-crested 

weirs: Am. Soc, Civil Engineers Trans., v. 96, 

p. 387.

Yarnell, D. L., 1930, Flow of water over railway 

and highway enbankments: Public Roads, v. 2, 

April, p. 30.

Unpublished Reports

Delleur, J. W., 1955, Boundary layer development 

in open channels: Columbia Univ., Civil Eng. Dept., 

Ph. D. thesis.

Guernesey, Robert, Jr., 1940, The boundary layer 

in a broad-crested weir: Columbia Univ., Civil 

Eng. Dept., M. S. thesis.

Krimgold, D. D., and Perko, F. L., 1936, Investigation of the laws of similitude as applied to flow 

of water over broad-crested weirs: Univ. of Calif., 

Civil Eng. Dept., B. S. thesis.

Joe, A. K., and Thomson, G. C., 1937, A report on 

the similarity of models of broad-crested weirs: 

Univ. of Calif., Civil Eng. Dept., B. S. thesis.

Prentice, T. H., 1935, Hydraulics of the broadcrested weir: Columbia Univ., Civil Eng. Dept., 

M. S. thesis.

Schmidt, M. O. , 1941, Tests of a flat-crested weir: 

Univ. of Wisconsin, M. S. thesis.

Sigurdsson, Gunnar, 1955, Discharge characteristics 

of an embankment-shaped weir: Georgia Inst. of 

Technology, Civil Eng. Dept., M. S. thesis.

Sullivan, A. B. , and Davis, J»W», 1936, A report on 

discharge coefficients for broad-crested weirs: 

Univ. of Calif., Civil Eng. Dept., B. S. thesis.