SPILLWAY REHABILITATION: A LABYRINTH WEIR DESIGN TOOL

Brian M. Crookston Ph.D., Postdoctoral Researcher, Utah Water Research Laboratory, Logan, Utah

Blake P. Tullis, Ph.D., Associate Professor, Utah State University, Logan, Utah

ABSTRACT

A design example is presented to illustrate the labyrinth weir hydraulic design and analysis method developed by the authors. This design method is based upon experimental results from physical models tested at the Utah Water Research Laboratory. It is valid for trapezoidal labyrinth weirs with sidewall angles from 6 to 35, quarter-round and half-round crest shapes, and it includes various design parameters and hydraulic conditions that affect flow performance, including weir geometry, orientation (e.g., reservoir vs. channel), cycle configuration, cycle efficiency, tailwater submergence, local submergence, nappe aeration conditions, nappe vibration, nappe instability or flow surging, and artificial aeration (vents, nappe breakers). The validity of this method is established by comparing predicted results to data from previously published labyrinth weir studies.

INTRODUCTION

General Overview

Labyrinth weirs (see Fig. 1) provide an increase in crest length for a given channel or spillway width, relative to linear weirs. The additional crest length increases weir discharge for a given upstream head. As a result of their versatility and hydraulic performance (large discharges at relatively low heads) labyrinth weirs have become widely used globally as headwater control structures, energy dissipaters, flow aerators, and spillways (including spillway rehabilitation) in a variety of applications. Due to the unique geometric shape, the flow passing over a labyrinth weir is three-dimensional and complex, and exact solutions for head-discharge relationships are not readily determined analytically. However, Eq. (1) and empirically determined labyrinth weir dimensionless discharge coefficients, Cd(), can be used to determine the head-discharge relationship.

23232

Tc)(d HgLCQ = (1)

In Eq. (1), Q is the labyrinth weir flow rate, Lc is the centerline length of the weir crest, g is the acceleration constant of gravity, and HT is the total upstream head defined as HT = V2/2g + h (V is the average cross-sectional velocity and h is the piezometric head upstream of the weir relative to the weir crest elevation, see Fig. 1).

Fig. 1. Labyrinth weir schematic, including flow parameters, geometric parameters, and crest shapes

Previous Labyrinth Weir Studies

Published research on the hydraulic performance of labyrinth weirs began as early as 1940 and is still of interest to researchers and designers today. A number of noteworthy research studies have been conducted over the past 50 years that have contributed to the evolution of labyrinth weir design; a selection is presented in Table 1. The Tullis et al. (1995) design method is currently widely accepted in the USA and presents an intuitive spreadsheet-based labyrinth weir design program [the Cd() data is also used in a spreadsheet presented by Falvey (2003)]. This method is for trapezoidal, quarter-round crest applications and utilizes an effective weir length, Le, as the characteristic weir length in Eq. (1) to partially account for apex influence on discharge capacity. However, apex influence is more accurately examined via nappe interference and is therefore an unnecessary complexity. Willmore (2004) found the Tullis et al. (1995) = 8 data to be incorrect and discovered a minor mathematical error in the geometric calculations. Also, the = 6 Cd() data display a different trend and are significantly lower than the adjacent curves.

Table 1. Notable labyrinth weir design methods

Study Crest Type

1 Taylor (1968), Hay and Taylor (1970) Sh HR

Triangular Trapezoidal Rectangular

2 Darvas (1971) LQR Trapezoidal

3 Hinchliff and Houston (1984) Sh QR Triangular

Trapezoidal

4 Lux and Hinchliff (1985) QR Triangular Trapezoidal 5 Magalhes and Lorena (1989) WES Trapezoidal

6 Tullis et al. (1995) QR Trapezoidal

7 Melo et al. (2002) LQR Trapezoidal

8 Tullis et al. (2007) HR Trapezoidal

9 Lopes et al. (2006, 2008) LQR Trapezoidal

10 Crookston (2010) QR HR Trapezoidal

QR Quarter-round (Rcrest=tw/2), LQR Large Quarter-round (Rcrest = tw), HR Half-round, Sh Sharp, WES Truncated Ogee The experimental data for the Tullis et al. (1995) method were limited to 6 18, with the = 25 and 35 curves linearly interpolated (with the aid of = 90 weir data). Objective

The purpose of this study was to improve current labyrinth weir hydraulic design and analysis tools by providing new insights, information, and experimental results. This is to be accomplished by utilizing the experimental results from physical modeling to provide a design optimization and analysis program and supportive hydraulic information (e.g., artificial aeration, nappe stability). The design program (see Crookston 2010) developed during this study is similar to the design procedure presented by Tullis et al. (1995) with the addition of the following: new Cd() data sets for quarter-round and half-round crests, a user-specified footprint size (channel width, W, and apron length, B), cycle efficiency (), nappe aeration conditions and behaviors, aeration device placement, and tailwater submergence effects. This method utilizes Lc instead of the effective crest length presented by Tullis et al. (1995).

EXPERIMENTAL METHOD

32 labyrinth weir physical models were rigorously tested at the Utah Water Research Laboratory (UWRL), located in Logan, Utah, USA (Crookston, 2010). Labyrinth weirs were fabricated from High Density Polyethylene (HDPE) sheeting. A laboratory flume (4 ft x 48 ft x 3 ft) and an elevated headbox (24 x 22 ft x 5 ft deep) were used for experimental investigations. All models were placed on an elevated horizontal platform (level to 1/64 in). The flume facility included a ramped upstream floor transition, which was reported by Willmore (2004) to have no influence on discharge capacity. In the head box test facility, the discharge channel downstream of the weir was relatively short (~4 in) and terminated with a free overfall. Where a rounded abutment wall inlet was

Table 2. Physical model test program

Model P Lc-cycle Lc-cycle/w w/P N Crest Type . Orientation ( ) () () (mm) (mm) ( ) ( ) ( ) ( ) ( ) ( ) 1 6 0 304.8 4,654.6 7.607 2.008 2 HR Trap Inverse

2-3 6 0 304.8 4,654.6 7.607 2.008 2 QR HR Trap Normal 4 6 0 203.2 3,075.5 7.607 2.008 5 HR Trap Projecting

5-7 6 10, 20, 30 203.2 3,075.5 7.607 2.008 5 HR Trap Arced & Projecting 8 6 0 203.2 3,075.5 7.607 2.008 5 HR Trap Flush 9 6 0 203.2 3,075.5 7.607 2.008 5 HR Trap Rounded Inlet

10-11 8 0 304.8 3,544.9 5.793 2.008 2 QR HR Trap Normal 12-13 10 0 304.8 2,879.1 4.705 2.008 2 QR HR Trap Normal 14-15 12 0 304.8 2,435.1 3.980 2.008 2 QR HR Trap Normal

16 12 0 203.2 634.6 4.705 2.008 5 HR Trap Projecting 17-19 12 10, 20, 30 203.2 634.6 4.705 2.008 5 HR Trap Arced & Projecting

20 12 0 203.2 634.6 4.705 2.008 5 HR Trap Flush 21 12 0 203.2 634.6 4.705 2.008 5 HR Trap Rounded Inlet

22-23 15 0 304.8 1,991.4 3.254 2.008 2 QR HR Trap Normal 24 15 0 152.4 1,991.4 3.254 4.015 2 QR Trap Normal 25 15 0 152.4 995.7 3.254 2.008 4 QR Trap Normal 26 15 0 304.8 995.7 3.254 1.019 4 QR Trap Normal

27-28 20 0 304.8 1,548.1 2.530 2.008 2 QR HR Trap Normal 29-30 35 0 304.8 983.5 1.607 2.008 2 QR HR Trap Normal 31-32 90 - 304.8 1,223.8 1.000 4.015 - QR HR - - Linear cycle configuration was used for all model orientations unless Arced is specified Based upon the outlet labyrinth cycle geometry

Fig. 2. Tested labyrinth weir orientations

modeled, the radius was set to the cycle width (Rabutment = w). Details of the physical model test program are summarized in Table 2 and Figure 2. Model test flow rates were determined from calibrated orifice meters in the flume supply piping, differential pressure transducers, and a data logger. The test program evaluated the performance of aeration vents (1 per sidewall) and wedge-shaped nappe breakers in a variety of locations (upstream apex, weir sidewall, downstream apex).

Experimental data were collected under steady-state conditions. A large number of head-discharge data points were collected for all tested weir geometries, and a system of checks was established wherein at least 10% of the data were repeated to ensure accuracy and determine measurement repeatability. Q measurements were recorded for 5 to 7 minutes with the data logger to determine an average flow rate, and h (0.006 in) was determined with a stilling well equipped with a point gauge. Digital photography and high-definition (HD) digital video recording were used extensively to document the hydraulic behaviors of the tested labyrinth weirs. Testing included velocity measurements and a dye wand to examine flow patterns. Observations also noted nappe aeration conditions and behavior, nappe stability, nappe separation point, nappe interference, areas of local submergence, and any harmonic or recurring hydraulic behaviors for all tested.

EXPERIMENTAL RESULTS

Overview

A new hydraulic design method, based upon the experimental results of this study, was developed for trapezoidal labyrinth weirs with a quarter- or half-round crest shape (see Crookston 2010). It is based upon Eq. (1) and utilizes experimentally determined Cd() values to calculate head-discharge relationships for labyrinth weirs located in a channel or reservoir application. The design table format introduced by Tullis et al. (1995) has been adapted to incorporate additional design information and parameters (e.g., nappe breakers, nappe behavior, cycle efficiency, and submergence). The following example illustrates how to use this new method to hydraulically design a labyrinth weir. The design example also includes corresponding design information from Crookston (2010). Design Example

A labyrinth weir is being considered as the control structure of a spillway. The allowable spillway width is ~100 ft; the spillway is required to pass ~16,500 cfs (e.g., 75% of PMP); and due to urban development, it is desirable to limit the upstream pool elevation. This information is entered into Table 3. This section also includes desired apron and crest elevations and downstream total head (relative to crest) for labyrinth weir submergence calculations (e.g., located on a river with high tailwater effects). Basic labyrinth weir geometry is input into the second section of the design method, shown in Table 4. Due to the geometric complexities of labyrinth weirs, a number of iterations will likely be necessary to determine a satisfactory labyrinth weir design. For this example, the selected initial values are a quarter-round crest shape, a sidewall angle of 8, a weir height of 12 ft, the number of cycles N = 4, and a channel width less than 100 ft. The wall thickness (at the crest) and the apex width have been selected to be geometrically similar to the physical models tested in this study. This section also provides the option to include an aeration device (e.g., nappe breakers, vents). Depending upon the nappe aeration behavior and the potential for debris, it may

Table 3. User-defined hydraulic conditions input section of design method

Symbol Value Units Notes

Design Flow Qdesign 16,557 ft3/s Input g = 32.174 ft/s2 Design Flow Water Surface Elevation H 582.00 ft Input Approach Channel Elevation Hapron 563.00 ft Input Crest Elevation Hcrest 575.00 ft Input Unsubmerged Total Upstream Head HT 7.00 ft Input (Pies. Head + Vel. Head Losses) Downstream Total Head Hd 2.75 ft Input (Pies. Head + Vel. Head) above weir crest

Table 4. Geometric user-defined input section of design method

Symbol Value Units Notes

Angle of Side Legs 8 ~ 6 - 35 0.140 (rad) Width of labyrinth (Normal to Flow) W 95.96 ft Input or W = Nw Number of Cycles N 4 - Input or N = W/w Wall Height P 12.00 ft P ~ 1.0HT Thickness of Wall at Crest tw 1.50 ft tw ~ P/8 Inside Apex Width A 1.50 ft A ~ tw Crest Shape Crest Shape Quarter - Quarter or Half

Nappe Breakers / Vents - None - Breakers, Vents, None

or may not be desirable to include a nappe breaker (see Fig. 3). This optional structural feature will be discussed further in the nappe behavior section of this design method. The third section of this design method (see Table 5) calculates various labyrinth weir ratios and geometric dimensions, including the headwater ratio and discharge coefficient. This study developed Cd() vs. HT/P for 6 35 for trapezoidal labyrinth weirs with quarter- and half-round crest shapes (see Crookston 2010). For convenience, curve-fit equations were provided, and linear interpolation is recommended for values not tested. Fig. 4 presents the experimental Cd() vs. HT/P results, and Eq. (2) corresponds to the quarter-round = 8 curve fit equation (see Crookston 2010 for additional Cd() equations).

Cd (8) = 0.03612HTP

2.576 HTP

0.4104

+ 0.1936 QR, = 8 (2)

At the design flow rate, this labyrinth weir has HT/P = 0.58 and Cd(8) = 0.304; a full rating curve for this proposed labyrinth weir is presented in Fig. 5. The apron length (B) is ~ 67 ft, a cycle is 24 ft wide (w), and the labyrinth provides nearly 551 ft of total crest length. As previously mentioned, this method provides hydraulic design information for quarter- and half-round crest shapes. The increase in discharge capacity associated with using a half-round crest rather than the quarter-round crest for the 8 labyrinth weir

Fig. 3. Nappe breaker, located on the downstream apex

Table 5. Free-flow calculations section of design method

Symbol Value Units Notes

Headwater Ratio HT/P 0.583 - Labyrinth Weir Discharge Crest Coefficient Cd() 0.304 - Cd() = f(HT/P, , Crest Shape, Aeration) Total Centerline Length of Weir Lc 550.77 ft Lc = 3/2Qdesign/[(Cd()HT3/2)(2g)1/2] Centerline Length of Sidewall lc 66.04 ft lc = (B-tw)/cos() Length of Apron (Parallel to Flow) B 66.90 ft Input or B = [Lc/(2N)-(A+D)/2]cos()+tw Cycle Width w 23.99 ft w = 2lcsin()+A+D Outside Apex Width D 4.108 ft D = A+2twtan(45-/2) Cycle Width Ratio w/P 2.00 - ~2 w/P ~4 Relative Thickness Ratio P/tw 8.00 - Apex Ratio A/w 0.063 - < 0.08 Cycle Efficiency 1.742 - = Cd()Lc/(wN) # of Nappe Breakers or Vents - none - Breakers on DS Apex, Vents on Sidewall Linear Weir Discharge Coefficient Cd(90) 0.829 - Cd(90) = f(HT/P, , Crest Shape) Length of Linear Weir for equivalent Q Lc-linear 201.71 ft Lc-linear = 3/2Qdesign/[(Cd(90)HT3/2)(2g)1/2]

ranges from 0 to 15%, as shown in Fig. 6, over the range of HT/P values evaluated, Table 5 also presents the length of linear weir (Lc-linear) that would be required to pass the equivalent discharge as the labyrinth weir (same crest shape) at the same value of HT. In this example, the labyrinth weir reduces the required channel width (W) by ~106 ft, relative to the linear weir, and provides nearly 3 times the weir length. The cycle width ratio (w/P), the relative thickness ratio (P/tw), and the apex ratio (A/w) correspond to the physical models tested in this study. The predictions of this

Fig. 4. Cd vs HT/P for quarter round trapezoidal labyrinth weirs ( = 8 and 15)

Fig. 5. Predicted head-discharge rating curve for proposed labyrinth weir geometry

Fig. 6. Hydraulic efficiency crest shape comparison for trapezoidal labyrinth weir, = 8

design method may deviate from actual weir performance if these limits, listed in Table 5, are exceeded. The cycle width ratio (w/P), the relative thickness ratio (P/tw), and the apex ratio (A/w) correspond to the physical models tested in this study. The predictions of this design method may deviate from actual weir performance if these limits, listed in Table 5, are exceeded. Cycle Efficiency

When trying to optimize a labyrinth weir geometry for a given channel width, it is useful to note that Cd(), which is proportional to the discharge per unit weir length, decreases with decreasing . Conversely, Lc increases with decreasing . To characterize the combined influence of these two counter influences on discharge capacity, a new parameter is introduced: cycle efficiency (), where = Cd()Lc/(wN). compares the discharge efficiencies of different cycle geometries for a given channel width; discharge efficiency increases with increasing values of . The proposed 8 labyrinth weir has = 1.742. As a comparison in the design example, assume that an = 15 labyrinth weir alternative is also being considered (w = 24 ft, W = 96 ft). At the same HT/P condition listed in Table 5 (HT/P = 0.583) for the 8 labyrinth weir, = 1.443 for the 15 labyrinth, which corresponds to a 17% reduction in discharge (Q = 13,723 cfs) relative to the 8 weir. To pass the design flow rate of 16,577 cfs, the 15 labyrinth weir requires an HT/P value of 0.704 [determined using Eq. (1) and the Cd(15) data presented in Fig. 4], which represents a 20.8% increase in HT relative to the 8. Although a higher upstream head is required to pass the design discharge, it is worth noting that the 15 weir length is ~44% shorter (Lc = 310 ft) than the 8 weir and the apron length is 34% shorter (B = 44 ft). In addition to the hydraulic performance of the weir, a proper feasibility study would also include economic and other considerations in selecting the most appropriate labyrinth weir design.

Approach Flow Influence

The discharge rating curve of a labyrinth weir can be influenced by the approach flow conditions (e.g., reservoir approach flow, spillway entrance geometry). The Cd() data used in the labyrinth weir design method are based on channelized approach flow conditions. If a labyrinth weir is installed in a reservoir application, the head-discharge relationship will be influenced by how and where the labyrinth weir is laid out (e.g., projecting into the reservoir, at the upstream end of the discharge channel with abutment wall transitions, etc.) Experimental results from Crookston (2010) showed that the outer 1-2 cycles on either end of the labyrinth weir can be affected by the abutments in a reservoir application, resulting in a reduced discharge capacity relative to the channelized approach condition.

Assume that the proposed 4-cycle, 8 labyrinth weir is located in a reservoir, and features rounded abutment walls. According to the data in Fig. 7, at a dimensionless upstream head (HT/P) of 0.583 (HT = 7ft), the discharge capacity of the reservoir labyrinth weir (15,729 cfs) will be approximately 5% less than Qdesign (16,557 cfs). Note that Figure 7 was developed by comparing 5-cycle labyrinth weir data for various reservoir applications to the same weir geometry in a channel application. The percent reduction in discharge efficiency will decrease as the cycle number increases. As needed, the design table can be used to predict a new HT/P ratio and discharge coefficient to address this adjustment and satisfy the design requirements (Q = 16,557 cfs / 0.95, HT = 7.4 ft, Cd() = 0.294). Relative discharge data (Q-Res/Q-Channel) for additional inlet geometries and arced labyrinth weirs can be found in Crookston (2010) to make similar rating curve adjustments.

Fig. 7. Hydraulic efficiency comparison for labyrinth weir with rounded inlet (reservoir) and located in a channel

Nappe Behavior

In addition to the head-discharge data, nappe behavior should also be considered in an effort to reduce the potential for undesirable pressure fluctuations, noise, vibrations, and flow surging. The range of nappe aeration conditions labyrinth weirs can experience include: clinging, aerated, partially aerated, or drowned. A clinging nappe is attached to the downstream face of the weir wall and is hydraulically more efficient than an aerated

nappe; the aerated nappe has an air cavity underneath. A partially aerated nappe is not fully aerated along the entire weir crest and/or the air cavity may be transient (spatially and temporally) in nature. A drowned nappe occurs at relatively higher HT/P and is characterized by a thick nappe without an air cavity. The range of HT/P associated with the various nappe aeration conditions for the 8 and 15 quarter-round crest are illustrated in Fig. 4 and summarized in Table 6. Nappe aeration behaviors for trapezoidal labyrinth weirs, 6 35, with quarter- and half-round crest shapes are documented in Crookston (2010).

Table 6. Labyrinth weir nappe behaviors

Quarter-Round Crest Shape (HT/P) = 8 = 15

Clinging none none Aerated 0.057-0.288 0.052-0.256 Partially Aerated 0.288-0.364 0.256-0.508 Drowned >0.364 >0.508 Nappe Vibration

as Eqs. (3) (5). As previously defined, HT represents the free-flow (unsubmerged) upstream total head. H* represents the total upstream head under submerged conditions (H* HT). Hd is the total downstream head; all heads are relative to the weir crest elevation. Calculated results are shown in Table 7, including tailwater submergence levels (S). Tailwater submergence resulted in H* (7.22 ft) increasing 3.2% relative to HT.

12008.00332.0*

24

+

+

=

T

d

T

d

T HH

HH

HH

53.10

T

d

HH

(3)

2174093790 .

HH.

H*H

T

d

T

+

=

5.353.1

T

d

HH

(4)

dHH =*

Td

HH5.3

(5)

Table 7. Tailwater submergence section of design method

Symbol Value Units Notes

DS/US Unsubmerged Head Ratio Hd/HT 0.39 - Submerged Head Discharge Ratio H*/HT 1.032 - Eqs. (3) (5), from Tullis et al. (2007) Submerged Upstream Total Head H* 7.222 ft Submergence S 0.381 - S = Hd/H* Submerged Weir Discharge Coefficient Cd-sub 0.290 - Cc()(HT/H*)3/2

The design method and support data are limited to the geometries (Table 2) and hydraulic conditions tested in this study (e.g., 0.05 HT/P 0.9). However, this method can be conservatively applied to geometrically comparable labyrinth weir geometries and similar flow conditions. The design method may be used as a first-order approximation for HT/P > 1.0, based on the general trends observed in the available supporting data from the current study [Model 13 tested to HT/P = 2.0 (QR), see Table 2]. Nevertheless, a hydraulic model study is recommended to confirm design method predictions, especially for hydraulic conditions and labyrinth weir geometries that deviate from the tested physical models.

CONCLUSION

The design method developed by Crookston (2010) and illustrated here is a useful tool for determining labyrinth weir hydraulic performance; it can also be used to estimate the hydraulic performance for labyrinth weir geometric configurations and/or operating conditions not specifically included in available labyrinth weir design data. The design example presented in this paper illustrates how to use the equations, figures, tables, and information presented by Crookston (2010). This design method is valid for trapezoidal labyrinth weirs, 6 35, with quarter-round and half-round crest shapes. It includes channel and reservoir applications,

linear and arced cycle configurations, tailwater submergence from Tullis et al. (2007), nappe aeration conditions, nappe vibration, nappe instability (flow surging), and artificial aeration. Crookston (2010) established the validity of this method by comparing predicted results to data from previously published labyrinth weir studies. This method may be applied to geometrically comparable labyrinth weir geometries and flow conditions (not evaluated during physical modeling) as a first order approximation; however, a hydraulic model study is recommended to confirm design method predictions.

ACKNOWLEDGEMENTS

This study was funded by the State of Utah and the Utah Water Research Laboratory.

AUTHORS

Brian M. Crookston is a postdoctoral researcher at the Utah Water Research Laboratory, located in Logan, Utah. Labyrinth weirs were the research subject for his Ph.D. dissertation, and he has over 6 years of experience performing hydraulic model studies. Blake P. Tullis is an associate professor at Utah State University. He has over 15 years of experience performing hydraulic model studies at the Utah Water Research Laboratory.

NOTATION

A Inside apex width A/w Apex ratio Sidewall angle B Length of labyrinth weir (Apron) in flow direction Cd() Labyrinth weir free-flow discharge coefficient, data from current study Cd(90) Linear weir discharge coefficient Cd-sub Labyrinth weir submerged discharge coefficient D Outside apex width Efficacy Cycle efficiency g acceleration constant of gravity H Design flow water surface elevation h depth of flow over the weir crest H* Submerged upstream total head H*/HT Submerged head discharge ratio Hapron Approach channel elevation Hcrest Elevation of labyrinth weir crest Hd Total tailwater head downstream of labyrinth weir (above the weir

crest) Hd/H* Downstream/upstream unsubmerged head ratio

HT Unsubmerged total upstream head on weir HT/P Headwater ratio Lc Total centerline length of labyrinth weir lc Centerline length of weir side wall Lc-cycle Centerline length for a single labyrinth weir cycle Lc-cycle/w Magnification ratio, M Lc-linear Centerline length for a linear weir Le Total effective length of labyrinth weir N Number of labyrinth weir cycles P Weir height P/tw Relative thickness ratio Q Discharge over weir Qchannel Discharge specific to a labyrinth weir located in a channel Qdesign Design flow rate specified in design method QHR Discharge specific to a labyrinth weir with a half-round crest QQR Discharge specific to a labyrinth weir with a quarter-round crest Qres Discharge specific to a labyrinth weir located in a reservoir Rcrest Radius of crest shape Rabutment Radius of rounded abutment S Submergence level tw Thickness of weir wall Cycle arc angle V Average cross-sectional flow velocity upstream of weir W Width of channel w Width of a single labyrinth weir cycle w/P Cycle width ratio

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