Labyrinth Weir

A labyrinth weir is a structure designed to convey large flows at low heads by increasing the effective length of the weir crest with respect to the channel breadth. Although generally less efficient than other weir types of the same effective length, the available increase in weir length more than compensates for this.

Data

Theory and Guidance

Introduction

A labyrinth weir is a structure designed to convey large flows at low heads by increasing the effective length of the weir crest with respect to the channel breadth. Although generally less efficient than other weir types of the same effective length, the available increase in weir length more than compensates for this.

Much research has been performed into the discharge-head relationships of labyrinth weirs, with those developed by Tullis et al. [i] being widely accepted. These use a standard weir equation based on total head, with a variable coefficient of discharge based on a function of labyrinth angle and H/P, where H is the total upstream head, and P is the depth of weir crest relative to the upstream bed level. The labyrinth angle (α) is that subtended by the weir with the direction of flow, with a linear weir being represented by the case α=90˚.

As the upstream head increases, the labyrinth weir becomes less efficient, hence the coefficient of discharge decreases. Although the equations are not verified for H/P > 0.9, the coefficient of discharge (c ) values must logically approach those of a linear weir for large H; the values of c are therefore extrapolated to this end for large H.

For drowned weir flow, the weir begins to act as a linear weir, with appropriate drowning reduction factor applied to the free flow weir equation. Two formulations of the drowning factor are available - the Villemonte formulation, which applies a reduction factor from when the downstream water level exceeds crest level, and the standard Flood Modeller drowning factor, which applies once the downstream to upstream head ratio exceeds the predefined modular limit. Research by Tullis et al .(2007)[ii] suggests the Villemonte relationship tends to overestimate the upstream head for drowned flow; using the same findings, the Flood Modeller relationship, if anything, may underpredict the upstream head.

Weir Equations

Mode 0 - dry crest

Conditions:

h₁≤0

h₂≤0

Equation:

Q=0

Mode 3 - free weir flow

Conditions:

h₁>0

h₂<mh₁

Equation:

Mode 4 - drowned weir flow

Conditions:

h₁>0

h₂≥mh₁

Equation:

(Villemonte; used if m≤0)

(alternative formulation; used if m>0)

Where

Q - discharge over weir (m³s^-1)

c_d - coefficient of discharge; calculated as a function of H₁/P and α. See Tullis et al. (1995) for full formulation.

L - effective length of weir (m)

g - acceleration due to gravity (=9.81ms^-2)

H₁ - total upstream head, relative to weir crest (m)

P - depth of weir crest above upstream bed (m)

f_dr - drowned flow reduction factor

h₁ - upstream water level above weir crest (m)

h₂ - downstream water level above weir crest (m)

Weir geometry

Note: The calculated values of c_d are heavily dependent on the ratio H/P as well as the labyrinth weir angle α, so it is imperative that these values are entered correctly.

The effective length of the weir, L, is defined as the total length along the weir crest.

If the effective length is not entered by the user, it may be calculated from the constituent components, namely

L1 - length along crest of one arm of the labyrinth (m), or B - length of labyrinth in the direction of flow (m).

N - number of labyrinth cycles - need not be an integer, but normally should be a multiple of one half.

D - width of labyrinth apex exterior (m), or A – width of labyrinth apex interior (m)

t - thickness of labyrinth weir (m)

W - breadth of channel at weir section (e.g. from left to right bank) (m).
NB Thickness and apex widths may both be zero; positive values must be entered for the other components.

If the upstream and/or downstream nodes of a labyrinth weir are not connected directly to a river (or other channel) section, the use of remote nodes is highly recommended. This allows Flood Modeller to calculate an area, and hence velocity, in order to calculate the velocity and total heads. If remote nodes are not specified in such instances, the upstream flow area will be estimated using the channel breadth, W , and depth of weir above bed, P.

Alternatively, the velocity head may be ignored by selecting the "Use static head" option. This may be particularly relevant when the labyrinth weir is being used to model discharge over a spillway from a reservoir, for instance.

Additional Options

1. Drowned flow formulation (see above) - the option to select between the Villemonte and alternative formulations of drowned flow reduction factor is governed by the choice of modular limit (set to zero for Villemonte; between 0 and 1 for the alternative formulation).
2. The "Use static head" option may be used to force the weir equation to use static head in preference to total head (see above).
3. It is occasionally possible that no solution is possible given the above formulation, being based on total head, which has a minimum value > 0. It may in such cases be necessary to reduce the calculated value of c_d by setting the lower limit of c_d (cdlim) to be lower than Lsinα.

Other applications

Oblique weirs - an oblique weir can be thought of a special case of labyrinth weir with n=½.

Horseshoe weirs - although not specifically validated by the research of Tullis et al. (1995), a similar principle applies, in that the effective weir length is longer than the channel breadth, but behaves more like a linear weir as the water level increases. It is suggested to enter the effective length of the weir, L , the breadth of channel, W , and labyrinth angle may be approximated by L sinα= W .

The unit state output for a labyrinth weir is the drowned flow reduction factor.

_[i] Tullis, J.P., Amanian, N. and Waldron, D. (1995), Design of Labyrinth Spillways, J Hyd Eng 121 3, pp247-255

_[ii] Tullis, B.P., Young, J.C. and Chandler, M.A. (2007), Head-Discharge Relationship for Submerged Labyrinth Weirs, J Hyd Eng 133 3, pp248-254

Datafile Format

Line 1 - Keyword 'LABYRINTH WEIR' [comment]. NB Only first nine characters read.

Line 2 - Label1, Label2, [Label3], [Label4]

Line 3 - P1, alpha, W, N, B, A, t, zc, L

Line 4 - ccf, m, cdlim, L1, D, shflag

Example

LABYRINTH WEIR    Avon Spillway
LabyU       LabyD
     3.010      27.5   135.360    10.000       0.0       0.0       0.0       0.0  
     1.000     0.700     0.000    13.230              STATIC

_[1] Remote labels are strongly recommended if the adjoining unit is not a channel section

_[2] Only used if effective labyrinth weir length is not specified

_[3] L1 may be used as an alternative to B (and vice versa )

_[4] D may be used as an alternative to A (and vice versa )

_[5] If total labyrinth length is not specified, then the length may be derived from the constituent components