Flat V Weir

The Flat V weir is long-based weir with a triangular longitudinal profile and a transverse symmetrical V-shaped crest having small side-slopes.

Data

Theory and Guidance

Introduction

The Flat V weir is long-based weir with a triangular longitudinal profile and a transverse symmetrical V-shaped crest having small side-slopes. It is a common structure in waterways in the UK and elsewhere. It conforms to BSI and ISO standards.

The hydraulics of the Flat V weir are well understood, following research by White at the Hydraulics Research Station (White W.R. (1966), White W.R. (1968), White W.R. (1971)) and documented in BS/ISO 4377 (2002), Herschey R.W. (1985) and Bos (1989).

The Flat-V weir resembles a Crump weir in section (in direction of flow, i.e. 1:2 upstream face; 1:5 downstream face), and has a mildly sloping V-shape (typically 1:10, 1:20 or 1:40) in the plane normal to the flow direction. Research by White W.R. (1968) has provided some alternatives for a downstream sloping face of 1:2, and Herschey R. W. (1985) provides adjustment factors for a non-vertical walled approach channel, although neither of these form part of either of the aforementioned Standard.

Connectivity

The unit is connected as per other weir units, although remote nodes are available from which to calculate areas, and hence the velocity in the approach and downstream channels. If the weir unit is not connected, directly or remotely, to a channel section (from which areas can be obtained), the area is calculated using the dimensions of the weir unit (breadth × depth above bed [given by p values]).

It is noted that the head-discharge relationship may be very sensitive to changes in velocity, particular at high velocity flows, so the choice of an appropriate section is important.

General equations of flow

The general form of the discharge relationship can be given as

where:
m is the slope of the V (1 [vertical]:m [horizontal]);

Z is a shape function dependent on the channel side-slope; its value is unity when flow is wholly within the V; when the water surface lies above the top of the V,

C_dr is a drowned flow reduction factor (set to one if flow is free (modular));

H_e is the total effective upstream head (=h₁+av²/2g-k_h), m;

n is the channel side slope (1 [vertical]:n [horizontal]);

h^' is the depth of the V (=b/(2m)), m;

h₁ is the upstream gauged head, m above crest;

b is the breadth of the weir crest (at the top of the V), m;
v is the upstream velocity, ms^-1;

a is the coriolis energy coefficient, and

k_h is the head adjustment factor due to water properties, m.

Two alternative methods are described in the BS/ISO standard to calculate discharge; one based on total head, the other based on gauged head using a coefficient of approach velocity factor. The former method is used in Flood Modeller.

Tables for values of C_d and k_h for different slopes are given in the table below.

Further studies by White W.R. (1968) have considered a flat-V weir with a downstream face slope of 1:2, although this does not form part of the BS/ISO standard. Furthermore, no data exists for the 1:40 V-slope, the regions of applicability are more restricted and the drowned flow conditions are much more complex and harder to quantify

Drowned Flow

The BS/ISO standard provides drowning reduction factors for non-modular flow, based on downstream tappings, in the absence of crest tappings. Although less accurate than the crest tapping method, downstream tappings are those available within Flood Modeller. For the 1:5 downstream face slope, the drowning function is given as

for

for,

where H_1e and H_2e are the effective upstream and downstream heads, respectively.

This implies a modular limit of approximately 0.73, although this quantity is in reality a function of H_1e/h'. Bos lists the values of the modular limit ranging from 0.67 (H_1e/h'»0.5) to 0.78 (H_1e/h'»1.5); Flood Modeller will interpolate for the modular limit between these two values of H_1e/h', and use the given values as lower and upper bounds of modular limit outside of them.

For the 1:2 downstream face slope, the modular limit values are generally lower, and more complex. The values for modular limit and drowning function are derived from Figures 21 and 22, respectively, in White W.R. (1968).

Reverse Flow

For reverse flow, the same equations apply, with the nodes reversed. It is noted, however, that in the case of the 1:5 downstream face slope, this is not ideal, since the geometry of the weir with respect to the direction of flow will have changed.

Notes

• Cross slopes of less than 1:10 are undesirable, and validated results are not provided for these.
• The 1:2 downstream face slope is not part of the BS/ISO Standard, and the flow equations for this structure are less well understood; no data is provided in the literature for such with a 1:40 V-slope.
• For V-slopes other than 1:10, 1:20 or 1:40, parameters are either interpolated or taken from those relating to the nearest published value.
• Unit state: The drowned flow reduction factor is output in the unit state.

Limits of applicability

1:5 downstream face slope

• The ratio h'/p₁ should not exceed 2.5
• The ratio H_1e/p₂ should not exceed 2.5 for the when flow is within the V.
• The ratio H_1e/p₂ should not exceed 8.2 (1:20 and 1:40 V-slopes) or 4.2 (1:10 V-slope) when flow is out of the V.
• For further limits of applicability, the user is referred to the BS/ISO Standard.

1:2 downstream face slope

• The ratio h'/p₁ should not exceed 1.6 (1:20 V-slope) or 1.5 (1:10 V-slope).
• The ratio H_1e/p₂ should not exceed 1.6 (1:20 V-slope) or 1.5 (1:10 V-slope), when flow is within the V.
• The ratio H_1e/p₂ should not exceed 3.2 (1:20 V-slope) or 2.7 (1:10 V-slope), when flow is out of the V.
• The out of V ratio H_1e/h' should not exceed 2 (1:20 V-slope) or 1.8 (1:10 V-slope).

Modes of flow

Reverse flow modes:

These are the same conditions and equations as the forward flow modes, with the nodes reversed; mode number = forward flow mode + 4, i.e.:

Mode 5 - Free reverse flow - within 'V'

Mode 6 - Drowned reverse flow - within 'V'

Mode 7 - Free reverse flow - out of 'V'

Mode 8 - Drowned reverse flow - out of 'V'

Note

To prevent flow from extending into an indefinitely increasing (with respect to depth) width of channel, it is recommended to apply the Elevation of Bank Top field if the channel side slopes are non-vertical. Above this level, flow will be contained with vertical walls, and not and indefinite continuation of the side slope.

Datafile Format

Line 1 - Keyword 'FLAT-V WEIR' [comment]. NB Only first four characters read.

Line 2 - Label1, Label2, Label3, Label4

Line 3 - Cc, b, zc, r, m, n, ds_fslope, alpha, zbank

Line 4 - p1, p2

Example

FLAT-V WEIR
UNIT076     UNIT076     UNIT077     UNIT078
       1.0       0.0       0.0                 0.0       0.0         5       1.2
     0.000     0.000