Mathematical model & methodology for optimal hydraulic design of labyrinth spillway

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RESEARCH ARTICLE | AU GU ST 0 6 20 24
Mathematical model & methodology for optimal hydraulic
design of labyrinth spillway 
Bhalchandra V. Khode; Nalini Vaidya ; Sujesh D. Ghodmare; Prafull Wadhai
AIP Conf. Proc. 3139, 110001 (2024)
https://doi.org/10.1063/5.0224522
09 August 2024 09:23:30

Mathematical Model & Methodology for Optimal Hydraulic
Design of Labyrinth Spillway
Bhalchandra.V.Khode a) , Nalini Vaidyab), Sujesh D Ghodmarec) and Prafull
Wadhaid)
G H Raisoni College of Engineering, Nagpur, Maharashtra, India
a) bhalchandra.khode@raisoni.net
b) Corresponding author: nalini.vaidya@raisoni.net
c)sujesh.ghodmare@raisoni.net
d)prafulla.wadhai@raisoni.net
Abstract: A spillway generally includes a type of regulator structure placed normally placed at right angles to the
direction of flow. It is possible to increase the capacity of an existing spillway if the length the spillway crest is increased
or discharge coefficient is increased or the operating head is increased. The combination of these approaches can also be
adopted. This type of weir is more economical. It can be considered as the realistic solution for enhancing the discharge
without any increase in width of the structure. The concept of labyrinth weir is specifically suitable at sites where crest
width and up-stream water surface of weir are limited. It is also suitable if larger discharge capacities are actually
required. Constructing a labyrinth weir in an existing spillway is useful to increases the crest lengths which can also
increase the discharge ability maintaining the same operating head. This type spillway may also result in providing
additional reservoir storage capacity as compared to more gated spillway. Labyrinth weirs may be the attraction in order
to fulfill required design flood criteria; shape and patterns of flow have discouraged its usage in the past. The current
development of design may results in the extensive approval of structure in mega projects like interlinking river project in
India. In this paper, an approach of mathematical modeling has been suggested for best hydraulic design of labyrinth
weir.
Keywords: Labyrinth spillway, Floods, Hydraulics structure, Head to height ratio.
INTRODUCTION
Existing reservoirs and dams are required to reanalyze in order to ensure that the structures meet the present
criteria of dam safety. Recent advances in the fields of hydrology indicate that many old reservoirs which are safe
with exclusive design may not be adequately pass the predicted maximum flood. This may result in overtopping or
dams failure leading to alteration of dam. In many cases enlargement of dams can be suggested in order to meet
increasing downstream water demands. It is also required to make provision of surplus head for hydropower plant
and to increase flood storage-capacity of the existing reservoirs.
For any reservoirs with the overflow control as major function along and level of water surface is limited, it can
be suggested to built labyrinth spillway (Fig.1). It has a elevated crest that provides surplus storage. It has elevated
crest along with increase length. This provides larger discharge capacity length resulting in greater discharge
capability. This type of weirs is also used to water quality control through the aeration of the flow and also serve as
fall structure on canal system. It has to be noted that a provision of labyrinth weirs can help as energy dissipaters. At
the same time it maintain uniform flow-depth in canal in comparison to the conventional fall structure
Second Online International Conference on Research Frontiers in Sciences
AIP Conf. Proc. 3139, 110001-1–110001-7; https://doi.org/10.1063/5.0224522
Published under an exclusive license by AIP Publishing. 978-0-7354-5018-9/$30.00
110001-1
09 August 2024 09:23:30

FIGURE 1. Concept of Labyrinth Weir
GEOMETRIC AND HYDRAULIC VARIABLES OF LABYRINTH WEIR LAYOUT
FIGURE 2. Shows the parameters commonly used define a trapezoidal labyrinth weir.
LITERATURE REVIEW
The investigations related to systematic hydraulic presentation of labyrinth weir was conducted by Hay and
Taylor [1]. The concept of flow magnification for labyrinth weir was proposed. User-friendly method of data for
design of labyrinth weir proposed by neglecting the hypothetical linear weir was proposed [1][2]. A design curve
was presented coefficient of rate of flow as the function of L/W and ratio of H/P. USBR verified model of labyrinth
spillway was recommended for Hyrum Dam as well as Ute Dam [6]. The discrepancy between the result of Darvas
and Hay and Taylor due to variance in head definition was noted at that time [1]. Darvas [2] expect their curves are
for ogee spillway, and the coefficient of rate of flow were developed by Megalhaes and Lorena [7]. The Hydraulics
data from flume studies and specific models was determined by Lux [3] [4]. The dimensional analysis suggested an
expression for the flow rate of the labyrinth weir was used. The wide range design of labyrinth spillway and
performance was presented by Amanian.N [5]. The wide range of experimentation on performance of the labyrinth
weir carried out by Tullis et.al. The crest coefficient curves in basic way as related to previous investigators [9].
Also a methodology to design a labyrinth weir, using the basic equation was proposed. Improving Performance of
Low Head Labyrinth Weir was also proposed [11]. Experimental investigations on flow over labyrinth weir using
side wall angles 6, 8, 10, 16 21, 26, 30 degree of trapezoidal labyrinth weir and linear weir keeping the height
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constant at P=0.10 and P=0.15m was carried out by Khode et al. [12] [14] [15] [16] [17]. Model study performed by
Ghare et.al [13] . For Flow Energy Dissipation over Labyrinth.. The optimal hydraulic design of labyrinth weir for
range of 6 ,8 10, 20 and 35 degree and Design curves for optimal value of crest coefficient are obtained accordingly
[12][13]. The relation between (Ht/P) & Crest coefficient dimensionless parameters (Ht/P) are also presented
[12][13].
FACTOR AFFECTING THE PERFORMANCE
It has been observed from models under consideration that as the ratio (H/P) increases, the flow magnification
drops for a labyrinth weir. In other words it can be stated that as head rises discharge coefficient continuous to
deceases. It also results increased capacity of the spillway ultimately equal to that of linear weir having crest length
equivalent to apron width. The flow over labyrinth weir is complicated due to interference of the jet at the up-stream
peak of the labyrinth weir. At large flows, the nappe from neighboring crests strikes to-gather. This results in
generation of a non-aerated nappe and also in reduction the coefficient of discharge of the weir.
Labyrinth angle (α) is another crucial parameter which affects the volume as well as the layout of labyrinth weir.
The smaller angle design is preferred if reservoir height to be inadequate at flows smaller than the projected flow.
Whereas if the out-flow is to be limited at low reservoir elevation the greater angle is preferred. A greater angle to
be preferred for a spillway replacement if width of existing apron is sufficient. It is advisable to use the full width to
curtail the changes to the up-stream and down-stream criteria. Generally the maximum labyrinth angle adopted
which would pass the required discharge. It would also give the layout for the given apron width.
It has also been noted that an inside apex width (A) affects spillway capacity to certain extent. Its provision
differentiates between a trapezoidal and a triangular labyrinth. The Effective length of the labyrinth weir is reduced.
This drops the spillway capability. Therefore it is recommended to have the value of ‘A’ as small as possible. The
apex width is provided to avoid the effect of nappe interference in the labyrinth upstream and downstream condition.
It also influences the hydraulic functioning of labyrinth. It has to be noted that the flow from downstream of the
labyrinth should be supercritical in order to avoid submergence effects.
Aspect ratio (w/p) also influences the hydraulic efficiency of the labyrinth. It is suggested that the accepted value
of the vertical ratio should not be less than 2 (For the case of trapezoidal plan) and not less than 2.5 (For in the case
triangular plan from weirs).
DESIGN CONSIDERATION
Tullis et al (1995) performed wide range of experimental data on labyrinth weir and obtained curves for crest
coefficient in easy way as related to prior study. The linear weir equation with altered Cd values is proposed to be
used. Hence the capacity of labyrinth weir is given by
2/3
2
3
2HgLCQ d
=
(1)
Where Cd:- Coefficient of discharge
L:- actual weir.length
H:- Total crown on crest
Figure.3 shows the design curves proposed by Tullis et.al (1995)
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FIGURE 3. Design Curves (Tullis et. Al data)
Using experimental findings, the crest coefficient Cd for different α in the range of 60 to 350 have been
represented by the expression
4
5
3
4
2
321 EAEAEAEAACd ++++=
(2)
In which
P
H
E=
and coefficient A1, A2, A3 and A4 are listed in Table-1. Eq.2 was compared with field
measurement for 0.1<H/P<0.9. The standard deviation between the measured and estimated values for a labyrinth
angle from 6 to 18 degree was less than +/-3.0%. The data from a labyrinth angle from 250 to 350 are the values
obtained by interpolation with accuracy of about +/-10%.
Table-1. Coefficient developed for Labyrinth weir used for spillway
Sidewall
Angle
A1
A2
A3
A4
A5
Limitation
60
0.49
-0.24
-1.20
2.17
-1.03
80
0.49
1.08
-5.27
6.79
-2.83
120
0.49
1.06
-4.43
5.18
-1.97
150
0.49
1.00
-3.57
3.82
-1.38
180
0.49
1.32
-4.13
4.24
-1.50
250
0.49
1.51
-3.83
3.40
-1.05
350
0.49
1.69
-4.05
3.62
-1.10
900
0.49
1.46
-2.56
1.44
0
E<0.70
Observing the curve (Fig-3) for discharge coefficient for labyrinth weirs, it has been observed that each curve
has a peak initial stage with the long recession limb afterwards. It is suggested that while design the layout of the
labyrinth weir, the value of Cd should be nearer to the peak value. Another regression analysis for optimum values
of Cd for the curve for H/P, side wall angle (α) as well as combined function of H/P with side wall angle (α) is
developed and are shown in Fig .4, and Fig.5
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FIGURE 4. Optimum of Cd =f(H/P) FIGURE 5. Optimum value of Cd=f(α)
The optimum Cd relevant to H/P is expressed as
Cd = -0.7688(H/P)2 + 1.3055(H/P) + 0.3579 (3)
R2 = 0.9947
The optimum Cd in terms of side wall angle is given by
Cd = 0.229α3 - 0.8061α2 + 0.9244α + 0.4064 (4)
R2 = 0.9732
The optimum Cd relevant to side wall (Labyrinth ) angle (α) and H/P given by
Cd= 0.3965+0.946*(H/P) -0.009*α (5)
R2 = 0.9884
Where,
H/P = Ratio between Head to weir Height
α = Side wall (Labyrinth) angle in radian
It is hereby suggested to use Eq.1 in combination with Eq.3,Eq.4, Eq.5 so as to decide the length of labyrinth
weir,L, appropriate to site specific condition
DESIGN METHODOLOGY.
Based on the analysis the steps that should be considered for design shall be either triangular or trapezoidal weir
are as follows.
1. Estimate the total width (w) of approach channel and height (P) of crest of weir based on condition of site
2. Maximum rate of flow to pass over the spillway is calculated using hydrological data.
3. For the reservoir the maximum allowable head is to be specified.
4. Then the desired crest shapes is to be selected. It has been observed that the quarter round shape of crest has
a higher flow rate coefficient in comparison with the other shapes.
5. The either of triangular or trapezoidal plan forms can be chosen. The length of the upstream as well as
downstream peak should be kept small as far as possible for trapezoidal plan forms. These peak are being
used for stability of construction.
6. Referring the earlier steps, total head, H/p is to be calculated.
7. Assume the reasonable number of spillway repeat ions and w/p (Aspect ratio,) less than 2.5 and 2 for the
triangular and trapezoidal labyrinth weirs respectively.
8. For the particular ratio of weir height i.e. H/p, optimal value of ‘Cd’ to be obtained using Eq.(3,4,5) &
necessary effective length (L) of control structure to be obtained using the Eq.(1).
Fig.4 Optimum Value of Cd=f(H/P)
Cd = -0.7688(H/P)2 + 1.3055(H/P) + 0.3579
R2 = 0.9947
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5
H/p
Discharge Coefficient (Cd)
cd
Fig.5 Optimum Value of Cd=f(α)
Cd = 0.229α3 - 0.8061α2 + 0.9244α + 0.4064
R2 = 0.9732
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Side w all angle (Alpha) in radian
Discharge Coefficient (Cd)
cd
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9. Various layout with the Side wall angle with respect to Maximum Cd value, by considering length of
labyrinth apron (B), the appropriate layout for the approach channel and carriage channel condition to be
finalized.
CONCLUSIONS
Based on the mathematical modeling for an optimal design of Labyrinth weir (Trapezoidal and Triangular weir)
following conclusions can be drawn.
1. Labyrinth weir shall find more solicitation with to its integral benefits in link with stability of structure and flow
magnification in case of upcoming river linking project in India.
2. Labyrinth weir can provide an economically viable flood discharging structure.
3. The design will be particularly suitable at sites where the limited head or width the spillway is restricted by the
topography.
4. A free-flow labyrinth can be more reliable when compared with traditional gated spillways,. The design is also
an effective means of increasing the flood discharge and storage capacity of existing dams.
5. Analysis of the data from design curve has been conducted to work out a new mathematical model for optimal
coefficient of discharge for the labyrinth weir.
6. The design methodology for the labyrinth weir suggested in this paper to determine the most economical layout
of the labyrinth weir which will also exhibit better hydraulic functioning.
REFERENCES
1. N. Hay, and G. Taylor, (1970) “Performance and design of labyrinth weir.” Journal of Hydraulics
Engineering. ASCE. 96 (11), pp.2337-2357.
2. L. A. Darvas (1971) “Performance and design of labyrinth weir.” Journal of Hydraulics Engineering. ASCE.
97 (8), pp. 1246-1251.
3. F. Lux III (1984). “Discharge characteristics of labyrinth weirs.” Proc. ASCE Hydr. Div Specialty Conf.
ASCE. New York. N. Y.
4. Lux and Hinchliff (1985). “Discharge construction of labyrinth spillway.” Transactions of 15th congress of
international Committee on large Dam. Lausanne. Switzerland pp. 249-274.
5. N. Amanian (1987) “ Performance and design of Labyrinth spillway”, MSc thesis Utah University, Logan,
Utah.
6. GR-82-13“Hydraulic model study of Hyrum dam auxiliary labyrinth spillway”May-1983. U.S.B.R.
7. Pinto Mangalhaes A. (1985,). “Labyrinth weir spillway” Transactions of 15th congress of international
Committee on large Dam. Lausanne. Switzerland pp.385-407.
8. F. G. Tacail, B. Evans. and A. Babb.(1990),“ Case study of Labyrinth spillway.” Candian Journal of Civil
Engineering,17,1-7.
9. S. P. Tullis, N. Amanian and D. Waldron (1995) “Design of labyrinth spillway”. J. Hydr. Engrg. ASCE 121
(3) pp. 247-255.
10. D. R. Waldron (1994). “Design of labyrinth weir.” M.Sc thesis Uthah University, Logan, Uthah.
11. B.P.Tullis,C.M.Willmore and J.S.Wolfhope (2005),“Improving Performance of Low Head Labyrinth Weir” J.
Hydr. Engg. ASCE
12. B.V. Khode, A. R. Tembhurkar, P. D. Porey, R. N. Ingle (2011),“ Experimental Studies On Flow Over
Labyrinth Weir”, American Society of Civil Engineering, Journal of Irrigation and Drainage Engineering ISSN
0733-9437, IR1943-4774.
13. A.D.Ghare A.S.Pote, B.V.Khode.(2014)‘Experimental model for Computation of Flow Energy Dissipation
over Labyrinth Weirs Journal of Disaster Advances Vol. 7 (1) . pp 10-16.
14. B.V. Khode, A. R. Tembhurkar, P. D. Porey, R. N. Ingle ,(2010)., “Improving Discharge Capacity Over
Spillway By Labyrinth Weir” Journal of World Applied Science 10(6),709-714
15. B.V. Khode, A. R. Tembhurkar, P. D. Porey, R. N. Ingle, ,(2011). “Determination of Crest Coefficient For
Flow Over Trapezoidal Labyrinth Weir,” Journal of World Applied Science 12 (3),324-329
16. B.V. Khode, A. R. Tembhurkar, P. D. Porey, R. N. Ingle, (2011) “Experimental Studies On Determination of
Discharge Coefficient For Flow Over Trapezoidal Labyrinth Weir,” International Journal of Civil Engineering
Research & Technology.vol 4(1),47-54)
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17. B.V. Khode, A. R. Tembhurkar.,(2010).“Evaluation and Analysis of Crest Coefficient for Labyrinth weir,”
Journal of World Applied Science 11 (7),835-839
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