ISO 4377:2002

INTERNATIONAL ISO
STANDARD 4377
Third edition
2002-10-01
Hydrometric determinations — Flow
measurement in open channels using
structures — Flat-V weirs
Déterminations hydrométriques — Mesure de débit dans les canaux
découverts au moyen de structures — Déversoirs en V ouvert

Reference number
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ii ISO 2002 – All rights reserved

Contents Page
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols . 3
5 Characteristics of flat-V weirs . 4
6 Installation . 4
7 Maintenance . 7
8 Measurement of head(s) . 8
9 Discharge relationships . 12
10 Computation of discharge . 25
11 Uncertainties in flow measurement . 27
12 Examples . 30
Annex
A Velocity distribution. 35
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ISO 2002 – All rights reserved iii

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO
member bodies). The work of preparing International Standards is normally carried out through ISO technical
committees. Each member body interested in a subject for which a technical committee has been established has
the right to be represented on that committee. International organizations, governmental and non-governmental, in
liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical
Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3.
Draft International Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this International Standard may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
International Standard ISO 4377 was prepared by Technical Committee ISO/TC 113, Hydrometric determinations,
Subcommittee SC 2, Notches, weirs and flumes.
This third edition cancels and replaces the second edition (ISO 4377:1990), which has been technically revised to
give a rigorous version of the basic discharge equation for a weir operating under drowned flow conditions. The
successive approximation method for calculating discharges is reintroduced.
Annex A forms a normative part of this International Standard.
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iv ISO 2002 – All rights reserved

INTERNATIONAL STANDARD ISO 4377:2002(E)
Hydrometric determinations — Flow measurement in open
channels using structures — Flat-V weirs
1 Scope
This International Standard describes the methods of measurement of flow in rivers and artificial channels under
steady or slowly varying conditions using flat-V weirs (see Figure 1).
Annex A gives guidance on acceptable velocity distribution.
2 Normative references
The following normative documents contain provisions which, through reference in this text, constitute provisions of
this International Standard. For dated references, subsequent amendments to, or revisions of, any of these
publications do not apply. However, parties to agreements based on this International Standard are encouraged to
investigate the possibility of applying the most recent editions of the normative documents indicated below. For
undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC
maintain registers of currently valid International Standards.
ISO 772, Hydrometric determinations — Vocabulary and symbols
ISO/TR 5168, Measurement of fluid flow — Evaluation of uncertainties
Guide to the expression of uncertainty in measurement (GUM), BIPM, IEC, IFCC, ISO, IUPAC, INPAP and OIML
3 Terms and definitions
For the purposes of this International Standard, the terms and definitions given in ISO 772 apply.
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ISO 2002 – All rights reserved 1

Key
1 Head gauging section
2 Upstream tapping
3 Stilling wells
4 Crest tapping
5Flow
6 Downstream tapping
100 mm
7 above stilling basin level
8 Permissible truncation
a �
10H but equal or greater than 3H
max
b �
25H but equal or greater than 3H
max
Figure 1 — Triangular profile flat-V weir
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4Symbols
The following is a list of symbols used, with the corresponding units of measurement.
a
Symbol Meaning Units
A Area of cross-section of flow m
B Width of approach channel m
b Crest width m
C Coefficient of discharge Non-dimensional
D
C Drowned flow reduction factor Non-dimensional
dr
C Coefficient of approach velocity Non-dimensional
v
e Uncertainty in head measurement m
h
e Uncertainty in gauge zero m
h,0
e Uncertainty in head correction factor m
k,h
g Gravitational acceleration (standard value) m/s
H Total head above lowest crest elevation m
H Maximum total head above crest elevation m
max
h Gauged head above lowest crest elevation m
h Separation pocket head m
p
h Effective separation pocket head relative to lowest crest elevation m
pe
� �
h ,H Difference between lowest and highest crest elevations m
K ,K Constants Non-dimensional
1 2
k Head correction factor m
h
mmCrest cross-slope (1 vertical: horizontal) Non-dimensional
n Number of measurements in a set Non-dimensional
p Difference between mean bed level and lowest crest elevation m
Q Discharge m /s
s Standard deviation of the mean of several head readings m
h
v Mean velocity at cross-section m/s
v Mean velocity in approach channel m/s
a
X Percentage uncertainty in discharge coefficient Non-dimensional
C,D
X Percentage uncertainty in coefficient of velocity Non-dimensional
C,v
X Percentage uncertainty in drowned flow reduction factor Non-dimensional
C,dr
X Percentage uncertainty in head measurement Non-dimensional
h
X Percentage uncertainty in discharge measurement Non-dimensional
Q
Z Z Shape factors Non-dimensional
h, H
α Coriolis energy coefficient Non-dimensional
Subscript
1 denotes upstream value
2 denotes downstream value
e denotes “effective” and implies that corrections for fluid effects have been made to the quantity
a
In cases where the subscript of a symbol also contains a subscript, it is house style to write the second subscript on the same
line after a comma.
Thus e is written e , and X is written X
k k,h C C,dr
h dr
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ISO 2002 – All rights reserved 3

5 Characteristics of flat-V weirs
The standard flat-V weir is a control structure the crest of which takes the form of a shallow V when viewed in the
direction of flow.
The standard weir is of a triangular profile with an upstream slope of 1 (vertical): 2 (horizontal) and a downstream
slope of 1:5. The cross-slope of the crest line shall not be steeper than 1:10. The cross-slope shall lie in the range of
0 to 1:10 and, at the limit when the cross-slope is zero, the weir becomes a two dimensional triangular profile weir.
The weir can be used in both the modular and drowned ranges of flow. In the modular flow range, discharges depend
solely on upstream water levels and a single measurement of upstream head is sufficient. In the drowned flow range,
discharges depend on both upstream and downstream water levels, and two independent head measurements are
required. For the standard flat-V weir, these are:
— the upstream head;
— the head developed within the separation pocket which forms just downstream of the crest or, as a less accurate
alternative, the head measured just downstream of the structure.
The flat-V weir will measure a wide range of flows and has the advantage of high sensitivity at low flows.
Operation in the drowned flow range minimizes afflux at very high flows. Flat-V weirs shall not be used in steep rivers,
particularly where there is a high sediment load.
There is no specified upper limit for the size of this structure. Table 1 gives the ranges of discharges for three typical
weirs.
The flat-V weir can be designed to minimize the obstruction to the passage of fish in the appropriate flow range. Flow
is concentrated towards the centre of the downstream face and this provides the depth of water required by migratory
fish.
Table 1 — Ranges of discharge
Elevation of crest above
Width Range of discharge
Crest/cross-slope
bed
ratio
mm m /s
0,2 1:10 4 0,015 to 5
0,5 1:20 20 0,030 to 180 (within maximum head of 3m)
1,0 1:40 80 0,055 to 630 (within maximum head of 3m)
6 Installation
6.1 Selection of site
6.1.1 The weir shall be located in a straight section of the channel, avoiding local obstructions, roughness or
unevenness of the bed.
6.1.2 A preliminary study of the physical and hydraulic features of the proposed site shall be made, to check that it
conforms (or can be constructed or modified to conform) to the requirements necessary for measurement of
discharge by the weir. Particular attention shall be paid to the following:
a) the adequacy of the length of channel of regular cross-section available (see 6.2.2.2);
b) the uniformity of the existing velocity distribution (see annex A);
c) the avoidance of a steep channel (see 6.2.2.6);
d) the effects of increased upstream water levels due to the measuring structure;
e) the conditions downstream (including such influences as tides, confluences with other streams, sluice gates, mill
dams and other controlling features, such as seasonal weed growth, which might cause drowning);
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f) the impermeability of the ground on which the structure is to be founded and the necessity for piling, grouting or
other means of controlling seepage;
g) the necessity for flood banks, to confine the maximum discharge to the channel;
h) the stability of the banks, and the necessity for trimming and/or revetment;
i) the uniformity of the approach channel section;
j) the effect of wind on the flow over the weir, especially when it is wide and the head is small and when the
prevailing wind is in a transverse direction.
6.1.3 If the site does not possess the characteristics necessary for satisfactory measurements, or if an inspection of
the stream shows that the velocity distribution in the approach channel deviates appreciably from the examples
shown in Figure 2, the site shall not be used unless suitable improvements are practicable.
� � � �
α α
left left
a) − 1 × 100 = 6,9 % b) − 1 × 100 = 9,0 %
α α
right right
� � � �
α α
left left
c) − 1 × 100 = 12,3 % d) − 1 × 100 = 1,2 %
α α
right right
� � � �
α α
left left
e) − 1 × 100 = 0,6 % f) − 1 × 100 = 0,9 %
α α
right right
Figure 2 — Examples of velocity profiles in the approach channel
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6.2 Installation conditions
6.2.1 General requirements
The complete measuring installation consists of an approach channel, a weir structure and a downstream channel.
NOTE 1 The condition of each of these three components affects the overall accuracy of the measurements. Installation
requirements include such features as the surface finish of the weir, the cross-sectional shape of the channel, channel roughness
and the influence of control devices upstream or downstream of the gauging structure.
NOTE 2 The distribution and direction of velocity can have an important influence on the performance of a weir (see 6.2.2 and
annex A).
NOTE 3 Once a weir has been installed, any physical changes in the installation will change the discharge characteristics;
recalibration will then be necessary.
6.2.2 Approach channel
6.2.2.1 If the flow in the approach channel is disturbed by irregularities in the boundary, e.g. large boulders or rock
outcrops, or by a bend, sluice gate or other feature which causes asymmetry of discharge across the channel, the
accuracy of gauging may be significantly affected. The flow in the approach channel shall have a symmetrical
velocity distribution (see annex A.) This can be achieved by providing a long, straight approach channel of uniform
cross-section.
6.2.2.2 A minimum required length of straight approach channel shall be five times the width of the water surface at
maximum flow, provided flow does not enter the approach channel with high velocity via a sharp bend or angled
sluice gate.
NOTE 1 This figure refers to the distance upstream of the head-measuring position.
NOTE 2 A greater length of uniform approach channel is desirable if it can be readily provided.
6.2.2.3 In a natural channel where it is uneconomic to line the bed and banks with concrete for this distance, and if
where the width between the vertical walls of the lined approach to the weir is less than the approach width of the
natural channel, the banks shall be profiled to give a smooth transition from the approach channel width to the width
between the vertical side walls. The unlined channel upstream of the contraction shall nevertheless conform to
6.2.2.1 and 6.2.2.2.
6.2.2.4 Vertical side walls constructed to effect a narrowing of the natural channel shall be symmetrically aligned
with the centre line of the channel and curved to a radius not less than 2H as shown in Figure 1. The tangent
max
point of this radius nearest to the weir crest shall be at least H upstream of the head measurement section. The
max
height of the side walls shall be chosen to contain the design maximum discharge.
6.2.2.5 In a channel where the flow is free from floating and suspended debris, good approach conditions can also
be provided by suitably placed baffles formed of vertical laths. No baffle shall be nearer to the point at which the head
is measured than 10 times the maximum upstream head.
6.2.2.6 Under certain conditions, a hydraulic jump may occur upstream of the measuring structure, for example if
the approach channel is steep. Provided the wave created by the hydraulic jump is at a distance upstream of no less
than 20 times the maximum upstream depth, flow measurement is feasible, subject to confirmation that an even
velocity distribution exists at the gauging station.
6.2.2.7 Conditions in the approach channel can be verified by inspection or measurement for which several
methods are available such as current meters, floats, velocity rods, or concentrations of dye, the last being useful in
checking conditions at the bottom of the channel. A complete and quantitative assessment of velocity distribution can
be made by means of a current meter. The velocity distribution shall comply with the requirements of annex A,
clause A.5.
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6.3 Weir structure
6.3.1 The structure shall be rigid and watertight and capable of withstanding flood flow conditions without damage
from outflanking or from downstream erosion. The weir crest shall be straight when viewed from above and at right
angles to the direction of flow in the upstream channel. The geometry shall conform to the dimensions given in
clause 5 and Figure 1.
The weir shall be contained within vertical side walls, and the crest width shall not exceed the width of the approach
channel (see Figure 1). Weir blocks may be truncated but their horizontal dimensions shall not be reduced in the
direction of flow to less than H and 2H , upstream and downstream of the crest line respectively, where
1max 1max
H is the maximum upstream total head, expressed in metres, relative to the lowest crest elevation.
1max
6.3.2 The weir and the approach channel as far as the upstream tapping point shall be constructed with a smooth
non-corrodible material. A good surface finish is important near the crest but can be relaxed a distance along the
profile of 0,5H upstream and downstream of the crest line.
max
The crest shall be formed by an embedded stainless steel insert with bevelled edges to conform with the surface of
the weir block. The insert shall be in the form of a removable bar-section, typically 15 mm by 100 mm lying along the
◦
downstream face of the weir with its 15 mm edge bevelled at 52,1 to align it with the upstream face at a gradient of
1:2.
6.3.3 In order to minimize uncertainty in the discharge, the following tolerances are acceptable:
a) crest width ± 0,2 % (with a maximum of 0,01 m);
b) upstream slope ± 1,0 %;
c) downstream slope ± 1,0 %;
d) crest cross-slope ± 1,0 %;
e) point deviations from the mean crest line of the crest width.
± 0,2 %
NOTE Laboratory installations will normally require higher accuracy.
6.3.4 The structure shall be measured upon completion and mean dimensional values and their standard deviations
at 95 % confidence limits computed. The former are used for computation of discharge and the latter are used to
obtain the overall uncertainty of a single determination of discharge (see 11.6).
6.4 Downstream conditions
Conditions downstream of the structure are an important factor controlling the tailwater level. This level is one of the
factors which determines whether modular or drowned flow conditions will occur at the weir. It is essential, therefore,
to calculate or observe tailwater levels over the full discharge range and make decisions regarding the type of weir
and its required geometry in light of this evidence.
7 Maintenance
Maintenance of the measuring structure and the approach channel is important to enable accurate measurements to
be made. The approach channel shall be kept clean and free from silt and vegetation for at least the distance
specified in 6.2.2.2. The float wells, tappings and connecting pipework shall also be kept clean and free from
deposits.
The weir structure shall be kept clean and free from clinging debris and care taken in the process of cleaning to avoid
damage to the weir crest.
The weir crest shall be inspected for erosion damage regularly. If the mean effective radius of the crest exceeds
5mm then refurbishment shall be considered.
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ISO 2002 – All rights reserved 7

Erosion lowers the zero datum and affects the coefficient of discharge at low flows (see 8.3 and clause 9). In such
cases, the metal crest shall be removed, dressed and refitted.
If conditions are modular when maintenance is carried out, a useful check on the satisfactory operation of a crest
tapping is to ensure that the readings accord with the specification given in 9.5.
8 Measurement of head(s)
8.1 General
Where spot measurements are required, the heads can be measured by vertical gauges, hooks, points, wires or tape
gauges. Where continuous records are required, recording gauges shall be used.
NOTE As the size of the weir and head decreases, small discrepancies in construction and in the zero setting and reading of the
head measuring device become of greater relative importance.
8.2 Gauge wells
8.2.1 It is common practice to measure the upstream head in a gauge well to reduce the effects of water surface
irregularities.
Periodic checks on the measurement of the head in the approach channel shall be made.
Where the weir is designed to operate in the drowned flow range, a separate gauge well shall be used to record the
piezometric head. This develops within the separation pocket which forms immediately downstream of the crest or in
the channel downstream of the structure.
8.2.2 Gauge wells shall be vertical and of sufficient height and depth to cover the full range of water levels. In field
installations they shall have a minimum height of 0,3 m above the maximum water levels expected. Gauge wells shall
be connected to the appropriate head measurement positions by means of pipes.
8.2.3 Both the well and the connecting pipe shall be watertight. Where the well is provided for the accommodation
of the float of a level recorder, it shall be of adequate size and depth.
8.2.4 The invert of the pipe shall be positioned at a distance of no less than 0,06 m below the lowest level to be
gauged.
8.2.5 Pipe connections to the upstream and downstream head measurement positions shall terminate either flush
with, or at right angles to the boundary of the approach and downstream channels. The channel boundary shall be
plain and smooth (equivalent to carefully finished concrete) within a distance 10 times the diameter of the pipes from
the centreline of the connection. The pipes may be oblique to the wall only if it is fitted with a removable cap or plate,
set flush with the wall, through which a number of holes are drilled. The edges of these holes shall not be rounded or
burred. Perforated cover plates are not recommended where weed or silt are likely to be present.
8.2.6 The static head at the separation pocket behind the crest of the weir shall be transmitted to its gauge well by
one of the following:
a) an array of tapping holes set into a plate covering a cavity in the crest of the weir block;
b) the underside of the plate supporting a manifold into which the static head is communicated via an array of feed
tubes;
c) a horizontal conduit leading from the cavity through the weir block beneath the crest and terminating at the gauge
well;
d) a flexible transmission tube to communicate static head within the manifold to the gauge well;
e) a watertight seal around the transmission tube to prevent static head within the cavity from influencing the static
head transmitted from within the manifold.
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The static head within the manifold may be at a different pressure because of leakage around the periphery of the
cover plate.
These arrangements minimize the occurrence of silting within the communication path between the separation
pocket and the gauge well and provide for the effective purging of the pipework by the occasional backflushing of the
system. For this purpose a volume of water shall periodically be introduced into the gauge well.
Figure 3 shows the general arrangement for the crest-tapping installation. The size and disposition of the crest-
tapping holes is given in Table 2.
8.2.7 Adequate additional depth shall be provided in wells to avoid the danger of floats, if used, grounding either on
the bottom or on any accumulation of silt or debris.
The gauge well arrangement may include an intermediate chamber of similar size and proportions as the approach
channel, to enable silt and other debris to settle out where it may be readily seen and removed.
8.2.8 The diameter of the connecting pipe or width of slot to the upstream well shall be sufficient to permit the water
level in the well to follow the rise and fall of head without appreciable delay. Care should be taken however not to
oversize the pipe, in order to ensure ease of maintenance and to damp out oscillations due to short period waves.
NOTE No firm rule can be laid down for determining the size of the connecting pipe to the upstream well, because this is
dependent on a particular installation, e.g. whether the site is exposed and thus subject to waves, and whether a larger diameter
well is required to house the floats of recorders.
Table 2 — Arrangements for crest tappings
Crest width
b
Crest tapping holes
m
0,30 to 0,99 1,00 to 1,99 2,00 to 3,99 > 4,00
Hole diameter (mm) 5 5 10 10
Hole pitch (mm) 25 25 40 50
Number of tapping holes 3 5 7 9
Offset of centre hole from centre line of weir 0,1b 0,1b 0,1b 0,1b
Distance of the array of holes downstream of the crest (mm) 10 15 20 20
Bore diameter of manifold feeder tubes (mm) 5 5 10 10
Bore diameter of transmission tube (mm) 15 20 25 30
8.3 Zero setting
8.3.1 Accurate initial setting of the zeros of the head measuring devices with reference to the level of the crest and
subsequent regular checks of these settings is essential.
8.3.2 An accurate means of checking the zero at frequent intervals shall be provided. Bench marks, in the form of
horizontal metal plates, shall be set up on the top of the vertical side walls and in the gauge wells. These shall be
accurately levelled to ensure their elevation relative to crest level is known.
Instrument zeros can be checked with respect to these bench marks without the necessity of re-surveying the crest
each time. Any settlement of the structure may, however, affect the relationships between crest and bench mark
levels and it is advisable to make occasional checks on these relationships.
8.3.3 A zero check based on the water level (either when the flow ceases or just begins) is susceptible to serious
errors due to surface tension effects and shall not be used.
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ISO 2002 – All rights reserved 9

a) Cross-section through one crest tapping and showing part of the weir block
b) Downstream view with section through the manifold (item 3)
Figure 3 — Arrangements for crest tappings
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c) View of the underside of the crest plate
Key
1 Crest tappings
2 Feed tubes communicating crest head to the manifold (some shown as single lines only)
3 Manifold [section in view b)]
4 Cavity in the crest of the weir block
5 Conduit leading to a gauge well
6 Transmission tube (other end sealed within the conduit but communicating head in the manifold to the gauge well)
7 Holes for screw-mounting the crest plate onto the weir block
Figure 3 — Arrangements for crest tappings (continued)
8.3.4 Values for the crest cross-slope, m, and the gauge zero can be obtained by measuring the crest elevation at
regular intervals along the crest line. A best fit straight line is positioned through the measured points for each side of
the weir, and the intersection of these lines is the gauge zero level. The mean of the crest cross-slopes (m) for the
two sides is used in the discharge formulae. For field installations the use of standard levelling techniques is
recommended but precise micrometer or vernier gauges shall be used for laboratory installation.
8.4 Location of head measurement sections
8.4.1 The approach flow to a flat-V weir is three dimensional. Drawdown in the approach to the lowest crest
elevation is more pronounced than in other positions across the width of the approach channel. This results in a
depression in the water surface immediately upstream of the lowest crest position. Further upstream this depression
�
is less pronounced and at a distance of 10 times the V-height, 10H , the water surface elevation across the width of
�
the channel is constant. To achieve an accurate assessment of the upstream head, the tapping shall be set 10H
�
upstream of the crest line. H =b/2m is the difference between lowest and highest crest elevation, in metres.
However, if this distance is less than 3H the tapping shall be set 3H upstream of the crest to avoid drawdown
max max
effects.
8.4.2 If other considerations necessitate siting the tapping closer to the weir, then corrections to the discharge
coefficients are necessary if . In all cases, a reduction in the coefficient is applicable and the percentage
H /p > 1
1 1
reductions depend on the tapping point location. The value of H /p is given in Table 3.
1 1
8.4.3 Flat-V weirs can be used for gauging purposes in the drowned flow range if a tapping is incorporated at the
crest. The centre position of the 10 crest tapping holes (see 8.2.6) shall be offset laterally from the position of the
lowest crest elevation a distance of 0,1 times the total crest width (see Figure 1 and Table 2).
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ISO 2002 – All rights reserved 11

8.4.4 Alternatively flat-V weirs can be used for gauging purposes in the drowned flow range if a downstream tapping
is incorporated.
NOTE This method is not as accurate as the method described in 8.4.3.
�
The downstream tapping shall be 25H or 3H , whichever is the greater, downstream of the crest line and set at a
max
level 100 mm above the downstream bed level.
Table 3 — Corrections to discharge coefficients
H /p
1 1
01 23
L
Correction
%
�
10H 0,0 0,0 0,0 0,0
�
8H 0,0 0,0 0,3 0,6
�
6H 0,0 0,0 0,6 0,9
�
4H 0,0 0,0 0,8 1,2
H is the upstream total head relative to lowest crest elevation, expressed in metres;
p is the height of lowest crest elevation relative to upstream bed level, expressed in metres;
L is the distance of upstream head measurement position from crest line, expressed in metres.
9 Discharge relationships
9.1 Equations of discharge
9.1.1 In terms of total head, the basic discharge equation for a flat-V weir operating under modular flow conditions
is:
√
5/2
Q = 0,8C gmZ H (1)
De H 1e
where
Q is the total discharge expressed in cubic metres per second (m /s);
C is the effective coefficient of discharge in the modular range;
De
g is the gravitational acceleration (standard value) expressed in metres per second squared (m/s );
mmis the mean crest cross-slope (1 vertical: horizontal);
Z is the shape factor;
H
H is the effective upstream total head relative to lowest crest elevation expressed in metres (m).
1e
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Alternatively, the discharge equation may be expressed in terms of gauged head by introducing a coefficient of
velocity dependant upon the weir and flow geometries:
√
5/2
Q = 0,8C C gmZ h (2)
De v h 1e
where
C is the coefficient of velocity;
v
Z is the shape factor;
h
h is the effective upstream gauged head relative to lowest crest elevation expressed in metres (m).
1e
9.1.2 In terms of total head, the basic discharge equation for a flat-V weir operating under drowned flow conditions
is:
√
5/2
Q 0,8C C gmZ H (3)
=
De dr H 1e
where C is the drowned flow reduction factor.
dr
The corresponding gauged head equation is:
√
5/2
Q = 0,8C C C gmZ h
(4)
De v dr h 1e
Values for the modular coefficient of discharge, C , are given in Table 4.
De
Table 4 — Summary of recommended coefficients, limitations and tolerances
Crest cross-slope
Flat-V weirs
1:40 or less 1:20 1:10
a)H /H � 1,0
1 ∆
a a a
Modular coefficient C 0,625 0,620 0,615
De
Head correction factor, k 0,000 4 m 0,000 5 m 0,000 8 m
h
Uncertainty in coefficient, X ± 3,0 % ± 3,2 % ± 2,9 %
C,De
b
Modular limit 65 % to 75 % 65 % to 75 % 65 % to 75 %
� � �
Other limitations H /p � 2,5 H /p � 2,5 H /p � 2,5
1 1 1
H /p � 2,5 H /p � 2,5 H /p � 2,5
1 2 1 2 1 2
� � �
Upstream tapping 10H 10H 10H
�
b)H /H > 1,0
a a a
Modular coefficient, C 0,630 0,625 0,620
De
Head correction factor, k 0,000 4 m 0,000 5 m 0,000 8 m
h
Uncertainty in coefficient, X ± 2,5 % ± 2,8 % ± 2,3 %
C,De
b
Modular limit 65 % to 75 % 65 % to 75 % 65 % to 75 %
� � �
Other limitations H /p � 2,5 H /p � 2,5 H /p � 2,5
1 1 1
H /p � 8,2 H /p � 8,2 H /p � 4,2
1 2 1 2 1 2
� � �
Upstream tapping 10H 10H 10H
a
Computations under non-modular conditions are based on C = 0,631, C = 0,629 and C = 0,620 respectively.
De De De
b
See 9.5.
©
ISO 2002 – All rights reserved 13

9.2 Effective heads
Effective heads are obtained by reducing observed values by a small constant amount which corrects for fluid
property effects. Thus:
h =h −k (5)
1e 1 h
and
αv
a
H =H −k =h + (6)
−k
1e 1 h 1 h
2g
Values for the head correction factor, k , are given in Table 4. The value of the Coriolis energy coefficient, α, shall be
h
checked on site by measuring the velocity distribution at the section where the head is measured. At the design stage
the value of α shall be taken as 1,2.
9.3 Shape factors
Shape factors are introduced into discharge equations for flat-V weirs because the geometry of flow changes when
the discharge exceeds the V-full condition. Thus:
�
when ,h Z =Z = 1,0
(7)
h H
�
when ,h >h
� 5/2
Z =[1,0−(1,0−h /h ) ] (8)
h 1e
and
� 5/2
Z =[1,0−(1,0−H /H ) ]
(9)
H 1e
where
� �
h (=H =b/2m) is the difference between the lowest and highest crest elevations, expressed in metres;
b
is the crest width, expressed in metres.
� �
Values of Z , Z in terms of h /h and H /H are given in Table 5.
h H 1e 1e
9.4 Coefficient of velocity
�
9.4.1 The coefficient of velocity, C , is related to the modular coefficient of discharge, , the ratio h /p and the
C
v De 1
�
ratio .h /h
1e
9.4.2 The coefficient of velocity, C , occurs in equations (2) and (4), together with the shape factor, Z . As
v h
�
indicated in 9.3 this shape factor is a function of h /h , one of the factors affecting C . It is convenient to present
1e v
� �
data for the product C Z in terms of h /p and h /h since C and Z are not required separately. Numerical
v h 1 1e h
v
values of this product are given in Table 6.
©
14 ISO 2002 – All rights reserved

� �
Table 5 — Evaluation of Z and Z in terms of h /h and H /H
h H 1e 1e
�
h /h or
1e
0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09
�
H /H
1e
0,0 to 0,9 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000
1,0 1,000 1,000 1,000 1,000 1,000 1,000 0,999 0,999 0,999 0,998
1,1 0,998 0,997 0,996 0,996 0,995 0,994 0,993 0,992 0,991 0,990
1,2 0,989 0,987 0,986 0,985 0,984 0,982 0,981 0,979 0,978 0,976
1,3 0,974 0,973 0,971 0,969 0,968 0,966 0,964 0,962 0,960 0,958
1,4 0,956 0,954 0,952 0,950 0,948 0,946 0,944 0,942 0,940 0,938
1,5 0,936 0,934 0,932 0,929 0,927 0,925 0,923 0,921 0,918 0,916
1,6 0,914 0,912 0,909 0,907 0,905 0,903 0,900 0,896 0,898 0,895
1,7 0,891 0,889 0,887 0,884 0,882 0,880 0,877 0,875 0,873 0,871
1,8 0,868 0,866 0,864 0,861 0,859 0,857 0,855 0,852 0,850 0,848
1,9 0,846 0,843 0,841 0,839 0,837 0,834 0,832 0,830 0,828 0,825
2,0 0,823 0,821 0,819 0,817 0,814 0,812 0,810 0,808 0,806 0,804
2,1 0,801 0,799 0,797 0,795 0,793 0,791 0,789 0,787 0,784 0,782
2,2 0,780 0,778 0,776 0,774 0,772 0,770 0,768 0,766 0,764 0,762
2,3 0,760 0,758 0,756 0,754 0,752 0,750 0,748 0,746 0,744 0,742
2,4 0,740 0,738 0,736 0,734 0,732 0,731 0,729 0,727 0,725 0,723
2,5 0,721 0,719 0,717 0,716 0,714 0,712 0,710 0,708 0,707 0,705
2,6 0,703 0,701 0,699 0,698 0,696 0,694 0,692 0,691 0,689 0,687
2,7 0,685 0,684 0,682 0,680 0,679 0,677 0,675 0,674 0,672 0,670
2,8 0,669 0,667 0,665 0,664 0,662 0,661 0,659 0,657 0,656 0,654
2,9 0,653 0,651 0,649 0,648 0,646 0,645 0,643 0,642 0,640 0,639
3,0 0,637 —— ———— ———
� �
NOTE To evaluate Z or Z from this table, the appropriate value of h /h or H /H is inserted as a combination of the
h H 1e 1e
values in the first column and in the first row (above the horizontal rule).
�
EXAMPLE The value of Z corresponding to h /h = 2,23 is given at the intersection formed by the horizontal line from 2,2
h 1e
with the vertical line from 0,03, and Z is therefore = 0,774.
h
9.5 Conditions for modular/drowned flow
The modular limit for flat-V weirs is not single valued as in the case of a two-dimensional weir, i.e. a weir with a
horizontal crest line. In the case of the flat-V weir the modular limit in terms of H /H is (70± 5) % depending on
2e 1e
�
the ratio H /H . H is calculated in the same way as H .
1e 2e 1e
Under modular flow conditions the value of H /H is less than or equal to (70± 5) % and will depend on the
2e 1e
nature of the downstream channel. The value of h /H is, however, constant in the modular flow range and is
pe 1e
independent of conditions in the downstream channel. The modular value of h /H always lies within the range
pe 1e
(40± 5) % and a check on this value during the modular flow conditions provides a sound method for determining
whether the crest tapping is performing satisfactorily. If the value of this ratio is not within this range the installation
should be checked for leakage around the tapping plate and/or general blockage of the system.
©
ISO 2002 – All rights reserved 15

� �
Table 6 —C Z in terms of h /p and h /h
v h 1 1e
�
h /p
�
h /h
1e
0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6
0,05 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000
0,10 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000
0,15 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000
0,20 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000
0,25 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,000 1,001 1,001 1,001 1,001
0,30 1,000 1,000 1,000 1,000 1,000 1,000 1,001 1,001 1,001 1,001 1,001 1,001 1,001
0,35 1,000 1,000 1,000 1,000 1,001 1,001 1,001 1,001 1,001 1,002 1,002 1,002 1,002
0,40 1,000 1,000 1,000 1,001 1,001 1,001 1,002 1,002 1,002 1,002 1,003 1,003 1,003
0,45 1,000 1,000 1,001 1,001 1,002 1,002 1,003 1,003 1,003 1,004 1,004 1,004 1,005
0,50 1,000 1,001 1,001 1,002 1,002 1,003 1,004 1,004 1,004 1,005 1,005 1,006 1,006
0,55 1,000 1,001 1,001 1,002 1,003 1,004 1,005 1,005 1,006 1,007 1,007 1,008 1,008
0,60 1,000 1,001 1,002 1,003 1,004 1,005 1,006 1,007 1,008 1,009 1,009 1,010 1,011
0,65 1,000 1,001 1,003 1,004 1,005 1,006 1,008 1,009 1,010 1,011 1,012 1,013 1,013
0,70 1,001 1,002 1,003 1,005 1,007 1,008 1,010 1,011 1,012 1,013 1,015 1,016 1,017
0,75 1,001 1,002 1,004 1,006 1,008 1,010 1,012 1,013 1,015 1,016 1,018 1,019 1,020
0,80 1,001 1,003 1,005 1,008 1,010 1,012 1,014 1,016 1,018 1,020 1,021 1,023 1,024
0,85 1,001 1,004 1,007 1,009 1,012 1,015 1,017 1,020 1,022 1,024 1,025 1,027 1,029
0,90 1,001 1,004 1,008 1,011 1,015 1,018 1,021 1,023 1,026 1,028 1,030 1,032 1,034
0,95 1,002 1,005 1,009 1,014 1,017 1,021 1,024 1,027 1,030 1,033 1,035 1,037 1,039
1,00 1,002 1,006 1,011 1,016 1,020 1,025 1,028 1,032 1,035 1,038 1,040 1,043 1,045
1,05 1,002 1,007 1,013 1,018 1,023 1,028 1,032 1,036 1,039 1,042 1,045 1,048 1,050
1,10 1,001 1,006 1,012 1,019 1,024 1,029 1,034 1,038 1,042 1,046 1,049 1,052 1,054
1,15 0,997 1,004 1,011 1,017 1,024 1,029 1,034 1,039 1,043 1,047 1,050 1,053 1,056
1,20 0,993 1,000 1,007 1,015 1,021 1,028 1,033 1,038 1,042 1,047 1,050 1,054 1,057
1,25 0,986 0,994 1,003 1,011 1,018 1,024 1,030 1,036 1,040 1,045 1,049 1,052 1,056
1,30 0,979 0,988 0,997 1,005 1,013 1,020 1,026 1,032 1,037 1,042 1,046 1,050 1,053
1,35 0,971 0,980 0,990 0,999 1,008 1,015 1,022 1,027 1,033 1,038 1,042 1,046 1,050
1,40 0,962 0,972 0,983 0,992 1,001 1,009 1,016 1,022 1,028 1,033 1,037 1,041 1,045
1,45 0,953 0,963 0,974 0,985 0,994 1,002 1,009 1,016 1,022 1,027 1,031 1,036 1,040
1,50 0,943 0,954 0,966 0,976 0,986 0,995 1,002 1,009 1,015 1,020 1,025 1,030 1,034
1,55 0,932 0,944 0,957 0,968 0,978 0,987 0,995 1,001 1,008 1,013 1,018 1,023 1,027
1,60 0,922 0,934 0,947 0,959 0,969 0,978 0,987 0,994 1,000 1,006 1,011 1,016 1,020
1,65 0,911 0,924 0,938 0,950 0,961 0,970 0,978 0,986 0,992 0,998 1,004 1,008 1,013
1,70 0,900 0,914 0,928 0,940 0,952 0,961 0,970 0,977 0,984 0,990 0,996 1,001 1,005
1,75 0,889 0,904 0,918 0,931 0,942 0,952 0,961 0,969 0,976 0,982 0,988 0,993 0,997
1,80 0,878 0,893 0,908 0,922 0,933 0,943 0,953 0,960 0,968 0,974 0,980 0,985 0,989
1,85 0,867 0,883 0,898 0,912 0,924 0,935 0,944 0,952 0,959 0,966 0,971 0,977 0,981
1,90 0,856 0,873 0,889 0,903 0,915 0,926 0,935 0,943 0,951 0,957 0,963 0,968 0,973
1,95 0,845 0,863 0,879 0,893 0,906 0,917 0,926 0,935 0,942 0,949 0,955 0,960 0,965
2,00 0,835 0,852 0,869 0,884 0,896 0,908 0,917 0,926 0,933 0,940 0,946 0,952 0,957
©
16 ISO 2002 – All rights reserved

� �
Table 6 —C Z in terms of h /p and h /h (continued)
v h 1 1e
�
h /p
�
h /h
1e
0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6
2,05 0,824 0,842 0,859 0,874 0,887 0,899 0,909 0,917 0,925 0,932 0,938 0,944 0,949
2,10 0,814 0,833 0,850 0,865 0,878 0,890 0,900 0,909 0,916 0,923 0,930 0,935 0,940
2,15 0,804 0,823 0,841 0,856 0,869 0,881 0,891 0,900 0,908 0,915 0,921 0,927 0,932
2,20 0,794 0,813 0,831 0,847 0,861 0,872 0,883 0,892 0,900 0,907 0,913 0,919 0,924
2,25 0,784 0,804 0,822 0,838 0,852 0,864 0,874 0,883 0,891 0,899 0,905 0,911 0,916
2,30 0,774 0,795 0,813 0,830 0,843 0,855 0,866 0,875 0,883 0,891 0,897 0,903 0,908
2,35 0,764 0,785 0,804 0,821 0,835 0,847 0,856 0,867 0,875 0,883 0,889 0,895 0,900
2,40 0,755 0,776 0,796 0,812 0,827 0,839 0,850 0,859 0,867 0,875 0,881 0,887 0,893
2,45 0,746 0,768 0,787 0,804 0,819 0,831 0,842 0,851 0,860 0,867 0,874 0,880 0,885
2,50 0,737 0,759 0,779 0,796 0,811 0,823 0,834 0,843 0,852 0,859 0,866 0,872 0,878
2,55 0,728 0,751 0,771 0,788 0,803 0,815 0,826 0,836 0,844 0,852 0,859 0,863 0,870
2,60 0,720 0,742 0,763 0,780 0,795 0,808 0,819 0,828 0,837 0,844 0,851 0,857 0,863
2,65 0,711 0,734 0,755 0,772 0,787 0,800 0,811 0,821 0,829 0,837 0,844 0,850 0,856
2,70 0,703 0,726 0,747 0,765 0,780 0,793 0,804 0,814 0,822 0,830 0,837 0,843 0,849
2,75 0,695 0,719 0,740 0,757 0,772 0,785 0,797 0,806 0,815 0,823 0,830 0,836 0,842
2,80 0,687 0,711 0,732 0,750 0,765 0,778 0,790 0,799 0,805 0,816 0,823 0,829 0,835
2,85 0,679 0,703 0,725 0,743 0,758 0,771 0,783 0,792 0,801 0,809 0,816 0,822 0,828
2,90 0,671 0,696 0,718 0,736 0,751 0,764 0,776 0,786 0,795 0,802 0,809 0,816 0,822
2,95 0,664 0,689 0,711 0,729 0,744 0,758 0,769 0,779 0,788 0,796 0,803 0,809 0,815
3,00 0,657 0,682 0,704 0,722 0,738 0,751 0,762 0,773 0,781 0,789 0,796 0,803 0,809
3,05 0,649 0,675 0,697 0,716 0,731 0,744 0,756 0,766 0,775 0,783 0,790 0,797 0,802
3,10 0,642 0,668 0,690 0,709 0,725 0,738 0,750 0,760 0,769 0,777 0,784 0,790 0,796
3,15 0,636 0,662 0,6
...