Title Numerical Analysis on the Hydraulic Characteristics of Piano Key Weir( 本文(Fulltext) )
Author(s) LE ANH TUAN
Report No.(Doctoral
Degree) 博士(農学) 甲第770号
Issue Date 2021-09-17
Type 博士論文
Version ETD
URL http://hdl.handle.net/20.500.12099/82781
※この資料の著作権は、各資料の著者・学協会・出版社等に帰属します。
Numerical Analysis on the Hydraulic Characteristics of Piano Key Weir
(
ピアノキー堰の水理学的特徴に関する数値解析)
2021
The United Graduate School of Agricultural Science, Gifu University
Science of Biological Environment (Gifu University)
LE ANH TUAN
Numerical Analysis on the Hydraulic Characteristics of Piano Key Weir
(
ピアノキー堰の水理学的特徴に関する数値解析)
LE ANH TUAN
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LIST OF TABLES
Table 1.1 Impacts of Enhanced Dam safety requirements on spillway capacity in various countries .... 5
Table 2.1 The World register of PKWs ... 13
Table 3.1 Experimental detail dimensions ... 32
Table 3.2 Grid Convergence Index (GCI) results ... 34
Table 4.1 PKW models with varying overhangs lengths dimensions ... 48
Table 4.2 Grid convergence index (GCI) results... 50
v
LIST OF FIGURES
Figure 1.1 Some of the World’s oldest dams ... 2
Figure 1.2 Cause of Dam Failure ... 3
Figure 2.1 PKW unit and main geometric notation ... 16
Figure 2.2 Different types of PKW ... 16
Figure 2.3 3-D streamlines for low and high heads of (a) H/P = 0.08 and (b) H/P = 0.25 ... 17
Figure 2.4 Distribution summary of geometrical and hydraulic parameters from 34 PKWs ... 18
Figure 2.5 Comparison of rating curves for PKW and straight sharp crested weir ... 23
Figure 3.1 PKW (left) with geometrical parameters; RL weir (right) ... 29
Figure 3.2 Geometry of the RL weir (left) and PKW (right) and boundary conditions ... 32
Figure 3.3 Comparison of discharge coefficients estimated by equations of different authors with physical and CFD results ... 34
Figure 3.4 Percentage of total discharge in the inlet, outlet and sidewall of PKW ... 36
Figure 3.5 Percentage of total discharge in the inlet, outlet and sidewall of RL weir ... 37
Figure 3.6 Curvilinear abscissa S ... 38
Figure 3.7 Comparison between PKW and RL weir flow rate distribution along the crest ... 41
Figure 3.8 Weir side section views (H/P = 0.45): PKW (left); RL weir (right) ... 42
Figure 3.9 Weir plan views (H/P = 0.45): PKW (left); RL weir (right) ... 42
Figure 4.1 PKW unit and main geometric notation ... 46
Figure 4.2 Layouts of PKW models with varying overhang lengths ... 49
Figure 4.3 Comparison between physical model and numerical model (from Machiels, 2012) ... 51
Figure 4.4 Discharge amplification ratios versus H/P for each model ... 52
Figure 4.5 Discharge amplification ratios versus Bo/Bi for each water head ... 53
Figure 4.6 Comparison of the discharge provided by PKWs with various Bo/Bi ratios ... 53
Figure 4.7 Free surface profile in the middle of the inlet key for H/P = 0.149 ... 55
vi
Figure 4.8 Free surface profile in the middle of the inlet key for H/P = 0.307 ... 55
vii
NOTATIONS
a = correction factor
Ai = fractional areas open to flow along axes ANFIS = Adaptive Neuron Fuzzy Inference System ANN = artificial neural network
ASCE = American Society of Civil Engineers b = correction factor
B = lateral (side) crest length [m]
Bi = downstream overhang crest length [m]
Bo = upstream overhang crest length [m]
C1ε = constant in the ε equation (=1.44) C2ε = constant in the ε equation (=1.92) Cd = discharge coefficient of crest CdW = discharge capacity of weir CFD = Computational Fluid Dynamics
Cs = discharge coefficient of sharp crested weir Cμ = empirical constant (=0.99)
D = grid size [mm]
DQ = the increase in discharge [m3.s-1] e = the length of the spillway [m]
FAVOR = Fractional Area/Volume Obstacle Representation fi = viscous acceleration [m2.s-1]
Fs = factor of safety
g = gravitational acceleration (=9.81[m2.s-1]) G = turbulence energy production rate
viii GCI = Grid Convergence Index
H = the height of the water above the sill of the spillway [m]
ICOLD = Internaltional Commission of Large Dams IDF = Inflow Design Flood [m3.s-1]
KWi = coefficient that describes the influence of the inlet key
KWo = coefficient that decribes the decrease of side crest length induced by the outlet key flow and the side nappe interference
L = total crest length [m]
Lu = the developed crest length [m]
mo = discharge coefficient of linear sharp crested weir Nu = number of Piano Key Weir unit
p = correction factor P = weir height [m]
Pd = dam height below the PKW [m]
Pe = mean side wall height [m]
Pi = inlet height [m]
PK = Piano key PKW = piano key weir PKWs = Piano Key Weirs
PMF = Probable Maximum Flood [m3.s-1] Po = outlet height [m]
q = specific discharge capacity [m3.s-1.m-1]
qd = specific discharge on the downstream crest [m3.s-1.m-1] QPKW = discharge capcity of PKW
Qs = discharge capacity of sharp crested weir [m3.s-1] qs = specific discharge on the lateral crest [m3.s-1.m-1]
ix qu = specific discharge on the upstream crest [m3.s-1.m-1]
r = ratio between discharge capacity of Piano key weir and sharp crested weir R = regression coefficient
RANS = Reynolds-Average Navier-Stokes RCC = roller compacted concrete
RL = rectangular labyrinth RNG = Re-Normalization Group Ro = parapet wall height
SDF = Spillway Design Flood [m3.s-1] SEF = Spillway Evaluation Flood [m3.s-1] Si = inlet key slope
So = outlet key slope
SST = Shear-Stress Transport t = time [s]
Ts = wall thickness [m]
ui = velocity component along the axes
USCOLD = United States Commission of Large Dams VF = fractional volume open to the flow
VOF = Volume of Fluid w = correction factor W = width of the weir [m]
Wi = width of inlet key [m]
Wo = width of outlet key [m]
Wu = width of a unit [m]
α = parameter that characterizes the influence of the inlet key slope on the side crest discharge efficiency
x
β = parameter that characterizes the influence of the in let key slope on the side crest discharge efficiency
δ1 = threshold related to the outlet key slope in Equation 2.20 δ2 = threshold related to the outlet key slope in Equation 2.20 μi = turbulent viscosity coefficient
ρ = density [kg.m-3]
σk = turbulent Prandtl number for k (=1.0) σε = turbulent Prandtl number for ε (=1.3)
xi
ACKNOWLEDGEMENTS
Firstly, I would like to express my sincere gratitude to my main supervisor Professor Ken HIRAMATSU of Faculty of Applied Biological Sciences, Gifu University, for the continuous support of my Ph.D study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better supervisor for my Ph.D study.
Besides my main supervisor, I would like to thank the rest of my co-supervisors: Professor Tatsuro NISHIYAMA from Faculty of Applied Biological Sciences, Gifu University and Professor Fumitoshi IMAIZUMI from Faculty of Agriculture, Shizuoka University for their insightful comments and encouragement, but also for the hard question which incented me to widen my research from various perspectives. Next, I also extend my appreciation to Professor NGUYEN Thu Hien from Thuyloi University, Vietnam, for being my cooperative supervisor.
My sincere thanks also goes to Professor Takeo ONISHI of Faculty of Applied Biological Sciences, Gifu University, for providing me an opportunity to join his weekly lab’s seminar, Professor Fusheng LI, Professor Yongfeng WEI and Professor Yasushi ISHIGURO for accepting and supporting me towards the success of BWEL program at Gifu University.
I thank my fellow lab-mates, my friends in Gifu University for all the fun we have had in the last three years.
Last but not the least, I would like to thank my family: my parents and to my wife and daughter for supporting me spiritually throughout writing this thesis and my life in general.
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2 1.1 Dams and Spillways
Dams and their reservoirs play an important role in social and economic development as they help supply seasonal water needs or generate renewable energy. Whilst dams and impoundments have been built for thousands of years, most large dams (defined as having a wall higher than 15 meters) have been built in the last 60 years and according to current estimates there are now around 58,000 of these large dam worldwide. However, dams are structures that differ from most other infrastructures as a result of the fact that the expected useful life of a properly engineered and maintained dam, can easily exceed 100 years (American Society of Civil Engineers & U.S. Committee on Large Dams, 1975;
ICOLD Bulletin No. 99, 1995). In fact, properly maintained, the actual life span can be much longer.
For example, the Bahman and Mizan Dams were built in Iran in the first and fourth century A.D.
respectively, and are still in operation.
Figure 1.1 Some of the World’s oldest dams
However, dam failures can, and do, occur. For example, in the spring of 1889, the largest dam failure incident in North American history took place. Following a period of heavy rains, the 22-m high South Fork dam, located just upstream of Johnstown, Pennsylvania, broke, releasing over 20 million cubic yards of water and debris into a narrow valley, killing more than 2,200 people. Over a century later, also following a period of unprecedented rainfall, Canada’s most significant dam safety event took place during the devastating Saguenay floods of 1996. In this case, eight dams were overtopped.
3
In 1975 a study performed by ASCE/USCOLD showed that there were four general causes of dam failure as in Figure 1.2 (ICOLD Question No. 91, 2009).
Figure 1.2 Cause of Dam Failure
The results of an assessment of the causes of dam failure as reported in (ICOLD Bulletin No 83, 1983) also shows that overtopping is a key dam failure risk for embankment and masonry dams.
(ICOLD Bulletin No 83, 1983) “Deterioration of Dams” reports that failures due to defective spillway performance occurred in 27 to 37 percent of recorded accidents with about half of these failures occurring as a result of operating errors rather than from underestimating the required spillway capacity.
In the responses to (ICOLD Question No. 90, 2009), it was noted that hydrological and hydraulic issues were responsible for a third of the reported dam failures. It is noteworthy that 36 of 58 reports responding to (ICOLD Question No. 90, 2009), “Upgrading of Dams” focused on the important issue of hydrological and hydraulic risks, pointing to the overarching interest in this topic by the dam industry.
In the twentieth century dams grew increasingly large and closer to populated centers. For these reasons, it becomes more important to better account for the risks a dam poses to the public and establish mechanisms to minimize the risks to “as low as reasonable practicable”. Returning to the matter of public perception of dam safety risks, management of this perception is complicated by the fact that, in general, a single incident that results in a large number of fatalities is perceived as being much more problematic than cumulative losses arising from a series of smaller incidents, even if the cumulative
4
impact of the smaller events far exceeds the major event. Clearly, any dam failure, whatever the cause, has the potential to be this catastrophic single incident in which many lives are lost.
With this perspective, the importance of reducing the risks from overtopping, the single most common cause of dam failure, is clear.
The original design of a dam may be reevaluated due to the availability of new information, the refinement of a certain design criteria or guidelines, or as part of a regular dam safety program. During this process, the design flood may be revised, resulting in a flood that is larger than was used for the original design. In many cases, analysis may show that the revised flood will result in the dam being overtopped due to insufficient reservoir storage and/or release capabilities.
Hydrological information was comparatively limited when dams were designed and built during the first half of the twentieth century. As discusses by (Wosnik et al., n.d.) in the United States, spillways were sized prior to 1970 using Spillway Design Flood (SDF), which is calculated by transposing an actual storm that occurred nearby and centering it over the reservoir under consideration.
Since the Seventies, the SDF criteria in the US was replaced by the Probable Maximum flood (PMF) concept, which is defined as the flood that may be expected from the most severe combination of critical meteorological and hydrological conditions that are reasonably possible in the region.
As more long-term hydrological data is gathered and processed, the inflow design flood (IDF) used for many existing dams and spillways is being reviewed, usually resulting in an increase in the required discharge capacity, creating new dam safety challenges both in the United States and around the world.
In addition to regulatory requirements, the effects of climate change can have a significant effect on flood handling requirements at some sites. For example, in Norway, climate change could result in a 30 percent increase in regulatory requirements, impacting 69 percent of the 411 dams in the Norwegian dam safety classes 3 and 4.
The 70 m high Ovre Kalvvatn rockfill dam was constructed in 1979 in Norway. In 2008, changing dam safety requirements resulted in the need to increase the discharge capacity by over 350 percent to comply with new dam safety requirements and forecast climate change effects. A case example,
5
outlining the path from the development of hydrological studies and dam safety assessments to the implementation of necessary measures to improve flood handling by the Povodi Vltavy state enterprise, provides details of some of the activities needed to satisfy modern dam safety requirements. In this example, increases in the required discharge capacity for three example dams ranged from 170 percent to 370 percent. These increases were achieved through a variety of measures, including gated and un- gated options. Table 1.1 provides a summary of some of the examples in various countries. Clearly, evolving dam safety standards worldwide have a significant impact on the requirements for discharge capacity at existing dams, highlighting the importance of finding cost effective measures to achieve these requirements.
Table 1.1 Impacts of Enhanced Dam safety requirements on spillway capacity in various countries
Dam name Year
constructed Country % increase in Required capacity
Up-rated discharge capacity
Ovre Kalvvatn 1979 Norway 352 556
Kilen 1989 Norway 100 800
Orlik 1950’s Czech 168 5300
Rimov 1970’s Czech 209 900
Zaskalska Unknown Czech 369 118
Paradela 1958 Portugal 122 1143
Salamonde 1953 Portugal 166 2828
Canicada 1955 Portugal 182 3762
Edensforsen 1958 Sweden 146 2060
Conastoga 1958 Canada 140 2100
6 1.2 Spillway Enhancement
Adding spillway capacity typically involves either a loss of storage or construction of gates with direct or indirect costs in the range of thousands of U.S. dollars.m-3.m-1.
Raising the reservoir level above the original design flood level can be a low-cost way to increase the discharge capacity of an existing spillway. Embankment improvements associated with this approach may include:
- Raising the crest by steepening the upper slopes of an earthfill dam (usually practical only for new construction)
- Adding a crest parapet or
- Improving the crest’s imperviousness and resistant to wave erosion.
Raising the reservoir level might also require that the stability of the concrete structures be enhanced by remedial measures such as anchoring.
Lempérière and Vigny, n.d. provides an excellent discussion on some of these methods that include:
- Improvement of the embankment crest
- Addition of a device, such as a labyrinth weir, to increase free-surface discharge - Installation of a fuse device
- Design of an embankment to safely overtop
According to Lempérière and Vigny, n.d., an approximate formula to compute the additional discharge capacity gained by allowing reservoir level to raise by 1 m may be derived from the following rule of thumb:
12
3
DQ eH (1.1)
where:
DQ = the increase in discharge, in m3.s-1; e = the length of the spillway, in meters;
H = the height of the water above the sill of the spillway, in meters.
7
When the length of the dam is on the order of 5 to 10 times the width of the spillway and the allowable rise in reservoir above the spillway sill is between 2 and 4 m, the costs of providing this additional capacity are usually a few hundred U.S. dollars per additional m3.s-1.
Labyrinth weirs are relatively thin concrete structure with a total crest length about four times the structure length, a nappe depth about half the wall height, and a discharge capacity roughly double that of a traditional weir. They can be constructed for about the same cost per m3.s-1 of flow as crest raising (Lempérière and Vigny, n.d.). The main drawback of a traditional labyrinth weir is that requires a large amount of space. For this reason, the traditional labyrinth weir cannot be built on top of a gravity dam or on most spillway structures.
Fuse gates are engineered structures designed to overturn at a specified reservoir level and are typically activated by an uplift chamber connected by a well to the reservoir. Fuse gates have been used for almost 20 years in countries such as Australia, France, India, South Africa, Switzerland, and the United States. They can be an attractive alternative for large spillways and discharges, offering discharge capacities of up to 100 m3.s-1.m-1 (Lempérière and Vigny, n.d.). They also can be designed in a labyrinth shape, combining the benefits of a labyrinth and fuse device.
Earth fill fuse plugs have been used in many countries, offering discharge capacities as high as several thousand m3.s-1.
In general, embankment fuse plugs tend to be relatively expensive compared with the other non- traditional methods but are significantly less expensive than a gated structure.
Fuse plugs can require a considerable amount of space and there are questions about their long term reliability due to the possible cementation of the fill or densification over time owing to natural consolidation or the effects of vehicular traffic. In addition, deployment would result in significant sediment discharges that may be environmentally unacceptable.
Flashboards are another device used by owners of small dams to allow reservoirs to be maintained at or near full supply level. Typically, most flashboards are wooden boards supported by vertical steel pipes set in the sill of the dam. They are either removed by hand before the flood season or designed to
8
fail at pre-determined reservoir levels through bending of the supports. Although imprecise, they can be very practical for increasing the storage of small dams by a meter of less. However, in the event of an unexpected extreme flood events, fixed flashboards can and have led to overtopping as was the case in the 1996 Saguenay floods in Canada where flashboards could not be removed due to unsafe conditions in at least one case.
Embankment dams can be converted to emergency spillways by placing engineered riprap, commercially available erosion protection products of roller compacted concrete (RCC) on the downstream slope. Lempérière and Vigny, n.d. reports that RCC has been used at about 100 low dams in the United States. For an overtopping depth of 3 m, each 1-m3.s-1 increase in flow requires 2 to 3 m3 of RCC at a cost of several hundred U.S. dollars.
1.3 Piano Key Weirs
In this context, Piano Key weir (PKW) was developed (Leite Ribeiro, Bieri, et al., 2012). This type of spillway is a further development of labyrinth weir, which was first initiated by Hydrocoop in collaboration with Biskra University (Algeria), the Hydraulic Laboratory of Elictricité de France (France) and Roorkee University (India) (Lempérière and Ouamane, 2003) that uses cantilevered apexes to restrict its basis length (Blanc and Lempérière, 2001) and permits its installation on top of existing structures such as concrete gravity dams and arch dams (Barcouda et al., 2006) and internal ramps in the cycles, thus reducing the forces exerting on the lateral walls and hence the structural cost (G. Paxson et al., 2013). The recent studies have shown that this weir can provide more than four times as much discharge capacity as a traditional ogee-crested weir can at a defined hydraulic head and crest length on the dam (Ouamane and Lempérière, 2006).
As it allows an important increase of the release capacity compared with traditional weirs capacity and as its geometry allows an easy use on dam crest, the PKW is a relevant tool to increase the safety level of an existing dam.
The PKW are thus of main interest both for the rehabilitation of existing structures, and for the
9
development of new equipment. It stays, however, of main importance to understand their hydraulic and structural working, and to optimize their design.
1.4 Numerical Model
It is difficult to determine how long physical flow modeling has been used in engineering applications.
Until now, all existing prototype PKWs in the world were being built using scaled up physical modeling.
Sometimes, conducting hydraulic research of these physical models is impossible within the space available and usually is expensive when need to adjust a geometrical parameter of the model.
Advances in computing power in the last three decades have made the mathematical analysis of three- dimensional time-dependent flow a practical and reliable approach for the design of hydraulic structures.
These analytical techniques, known as Computational Fluid Dynamics (CFD), have become establishes tools in the design of hydraulic structures and are now a very important tool in the assessment of the optimal methods for increasing spill capacity.
CFD analysis involves the solution of the governing equations for fluid flow at discrete points on a computational grid, giving the analyst a one, two, or three dimensional representation of the fluid flow domain. This numerical simulation capability gives engineers the power to quickly, reliably and inexpensively explore different aspects of the water regime during a wide range of hydraulic conditions for assessing the effectiveness of various alternatives for spill enhancement.
1.5 Aim of this Study and Approach to the Solution
Although several PKW projects have been completed and many researchers have published their own studies on PKW discharge efficiency, there is not yet an empirical method that considered all of the published experimental data. Their hydraulic design procedures were usually based on similarity laws to scale up from physical model to field-size situation. One of the challenges related to PKW design is the great number of geometric parameters involved (Bieri et al., 2009; Laugier, 2007; Laugier et al., 2009). Hence, there was a strong need in geometrical studies on PKW to improve the understanding of
10
the flow and the impact of its main geometric parameters on the discharge capacity and to set up an optimal design.
In a first step, the study aims at improving the comprehension of the flow behavior on PKW. To achieve this goal, a numerical model has been exploited in detail, allowing to visualize the flow along the structure and to characterize it in terms of discharge capacity, streamlines. The study results highlight the contribution of flow over each crests in terms of discharge capacity.
In a second step, the influence of overhangs crest lengths has been extensively studied
1.6 Constitution of the Thesis
This thesis consists of five chapters, two of which describe original researches. Three chapters supplement the former chapters and make them consistent with the purpose of the thesis.
The Introduction chapter explains the need for improving the discharge capacity of spillway of some old dams, the need for using numerical modeling for the research and the goals of this thesis.
Chapter 2 reviews the literatures on hydraulic behavior of PKW.
Chapter 3 presents a three-dimensional finite volume modeling for comparison of the discharge capacity between PKW and rectangular labyrinth. Discharge flows over every crest of weirs is of great interest in this research.
Chapter 4 describes the influence of overhangs crest length on the discharge capacity of PKW using a three-dimensional finite volume modeling. The model is verified by comparing with results from laboratory study.
Chapter 5 summarizes this study in a conclusion and gives an opinion regarding further studies.
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12 2.1 Introduction
Global warming is predicted to amplify the rainfall intensity due to the increases in evaporation from soil and storage capacity of moisture in atmosphere. Though the total annual precipitation has not changed significantly for several decades, the intensity of storms is likely to be more severe (Trenberth, 2011). According to Mei and Xie (2016), typhoons that struck East and Southeast Asia have intensified by 12 to 15 percent over the past 37 years. It is predicted that the ocean warming suggests that typhoons striking Japan, as well as eastern China, Taiwan, and the Korean Peninsula, will intensify further in the subsequent years. This might cause an embankment collapse of outdated small reservoirs, such as irrigation ponds due to insufficient discharge capacity of spillways. For example, in Southwestern Japan, torrential rain with return period more than 200 years in July 2018 caused a severe damage to several irrigation ponds, some of which were completely destroyed. With the different available numerical climate models and the development of new methodologies for analysis of climatic and hydrologic data, as well as higher requirements of the communities on safety management standard, a large number of existing dams necessitate improvement or rehabilitation of spillways in order to remove the excess flow from hydro system and prevent overtopping. To give in to these demands, Piano Key weir (PKW) was developed. This type of spillway, which is a further development of labyrinth weir, was first initiated by Hydrocoop in collaboration with Biskra University (Algeria), the Hydraulic Laboratory of Elictricité de France (France) and Roorkee University (India) (Lempérière and Ouamane, 2003). It uses cantilevered apexes to restrict its basis length (Blanc and Lempérière, 2001) and permits its installation on top of existing structures such as concrete gravity dams and arch dams (Barcouda et al., 2006) and internal ramps in the cycles, thus reducing the forces exerting on the lateral walls and hence the structural cost (Paxson et al., 2013). The recent studies have shown that this weir can provide more than four times as much discharge capacity as a traditional ogee-crested weir can at a defined hydraulic head and a constant width occupied by the structure on the dam (Ouamane and Lempérière, 2006).
13
Table 2.1 The World register of PKWs
Dam name Country Completion year
Bakkhada Algeria 1938
Beni Bahdel Algeria 1940
Goulours France 2006
Saint-Marc France 2008
Etroit France 2009
Gloriettes France 2010
Rattling Lake Canada 2011
Escouloubre France 2011
Gouillet France 2011
Malarce France 2012
Beaufort France 2013
Black Esk United Kingdom 2013
Dak Mi 4B Vietnam 2013
Dak Rong 3 Vietnam 2013
Giritale Sri Lanka 2013
Loombah Australia 2013
Sawra Kuddu India 2013
Emma Switzerland 2013
Campauleil France 2014
Charmines France 2015
Rambawa Tank Sri Lanka 2015
Rassisse France 2015
Raviege France 2015
Van Phong Vietnam 2015
Da Dang 3 Vietnam 2016
Dak Mi 3 Vietnam 2016
Record France 2016
Xuan Minh Vietnam 2016
Gage France 2017
Hazelmere South Africa 2017
Oule France 2018
Ouljet Mellegue Algeria 2018
Lewis Creek reservoir USA 2019
14
Over the last decades, many studies have been carried out in more than 15 institutions in order to comprehend the hydraulic behaviors of PKW, the influences of several geometrical parameters on PKW discharge efficiency (Anderson and Tullis, 2012; Erpicum et al., 2010; Ho Ta Khanh et al., 2011; Laugier, 2007; Leite Ribeiro et al., 2012a). More than 30 PKWs are in operation or under construction in France (Laugier, 2007; Laugier et al., 2009), Vietnam (Ho Ta Khanh et al., 2011, 2012), Sri Lanka (Jayatillake and Perera, 2013), Switzerland (Eichenberger, 2013) and Scotland (Ackers, 2013) with more than two-thirds associated with dam rehabilitations (increase in spillway capacity). Some PKWs are multipurpose schemes with irrigation and hydro features (Dakmi 2 and Van Phong barrages, Vietnam). Van Phong barrage is so far the longest PKW (475 m), with the largest capacity (14,400 m3.s-1), in the world. On the left bank is located the intake with the irrigation channel and on the right is located the powerhouse with 2 units. The water through the 2 turbines can be collected downstream by a small weir to irrigate the lowest area near the coast (Ho Ta Khanh, 2017). The World Register of Piano Key Weirs (https://www.uee.uliege.be/cms/
c_5026433/en/world-register-of-piano-key-weirs-prototypes), which is mentioned in Table 2.1, aims at gathering all PKW projects. Two Algerian labyrinth spillways have also been added in the register as they are similar to the PKW concept. Some other projects are also being studied in the USA (Crookston et al., 2016).
Although several PKW projects have been completed and many researchers have published their own studies on PKW discharge efficiency, there is not yet an empirical method that considered all of the published experimental data. Their hydraulic design procedures were usually based on similarity laws to scale up from physical model to field-size situation. Boillat et al. (2011) highlights the need for the creation of an experimental database about PKW. One of the challenges related to PKW design is the great number of geometric parameters involved (Bieri et al., 2009; Laugier, 2007; Laugier et al., 2009). Hence, to synthesize information from various sources, most of which were published in three conference proceedings (Erpicum et al., 2011, 2013, 2017b), this paper presented a succinct review of PKW hydraulic design procedure and addressed some future prospect of research.
15 2.2 Background
The geometric specificities of the PKW include a large set of parameters. The PKW -unit can be defined as the fundamental structure of a PKW, comprised of two lateral walls, an inlet and two half outlets located on each side of the inlet (Figure 2.1). The key geometric parameters are weir height P, the number of PKW-units Nu, lateral crest length B, in- and outlet widths Wi and Wo, up- and downstream overhang lengths Bo and Bi and the wall thickness Ts (Pralong et al., 2011). i; o and s subscripts refer to the inlet key, i.e. the key that is filled with water for a reservoir level at the PK weir crest elevation; the outlet key, i.e. the dry key for the same reservoir level and the lateral wall, respectively.
The width of a unit Wu is equal to Wi + Wo + 2Ts and the total width W of the weir is equal to NuWu. The developed crest length Lu of a unit is equal to Wu + 2B and the total crest length L of the weir is equal to NuLu.
The efficiency of a PKW can be also affected by other parameters of secondary importance, as the shape of entry under the upstream overhangs (i.e., adding “noses”), the section of the crest (i.e., elevating the crest via a parapet wall), crest shape profiles (i.e., flat-topped, half-rounded, upstream quarter-rounded and downstream quarter-rounded) (Cicero and Delisle, 2013).
Basic geometry of a PKW, called type A, includes symmetrical overhangs. When the downstream or the upstream overhang is omitted, the PK weir is of type B or C, respectively. A PKW without overhangs, i.e. a rectangular labyrinth weir with ramped floors, is called type D (Figure 2.2) (Hien et al., 2006; Lempérière, 2011; Machiels et al., 2014).
16
Figure 2.1 PKW unit and main geometric notation
Figure 2.2 Different types of PKW
2.3 PKW Flow Description
Machiels et al. (2011) showed that the flow over a PKW is the cumulative of three different types of overflow: flow over the inlet key downstream crest, flow over the outlet key upstream crest and flow over the lateral wall. Indeed, for low heads (H/P ≤ 0.2), where H is the water head, P is height of weir,
17
the streamlines of the three-dimensional flow field of a PKW indicates that the flow approaching the inlet key remains straight as it comes into the key and then is forced upward by the ramped floor. Flow approaching the outlet key, by contrast, diverges to both sides at the centerline of the outlet key toward the inlet keys. Thus, the downstream crest of the inlet is always supplied by the current along the slopes.
Meanwhile, only the surface current comes into the outlet. On the lateral crest, because of flow inertia downstream, the discharge over lateral crest in its downstream part is contributed by the partial flow in front of the inlet key and in its upstream portion by the current coming front of the outlet, under the crest level (Machiels et al., 2011) (Figure 2.3.a). For low heads, the whole weir crest flow is of free outflow, and the water nappes flowing through the adjacent lateral crests enter the outlet key without interfering each other, which will help evacuate water out of the outlet key. Therefore the volume flow rate of the PK weir is relatively high.
Figure 2.3 3-D streamlines for low and high heads of (a) H/P = 0.08 and (b) H/P = 0.25
For high heads (H/P ≥ 0.2), as highlighted by Machiels et al. (2011) and Denys et al. (2017), the streamlines over the PKW are less homogeneously distributed. The downstream crest which resembles a sharp-crested inclined weir (pivot weir) is supplied by both the bottom current and the front inlet current. The upstream crest is still supplied by the surface current, in the meanwhile the lateral crest is secondarily supplied by the front outlet current, flowing under the crest level (Figure 2.3.b).
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19
proportional relationship between L/W and discharge coefficient Cd. Indeed, there are differences of about 50 percent in discharge capacity between tests with L/W = 7 and L/W = 3 for a constant width occupied by the structure on the dam W and low ratios H/P (Leite Ribeiro et al., 2011). However, higher values of L/W than 7 do not have an effect on Cd (Kabiri-Samani and Javaheri, 2012). This finding is in agreement with the previous study of Ouamane and Lempérière (2006). Figure 2.4 indicates that most ratio lies between L/W = 4 and 6. Laugier et al. (2017) recommended that value of 6 to 7 is the most efficient ratio from the perspective of hydraulic efficiency and economic feasibility.
In the case of a small upstream head (H/P < 0.2), the flow rate of the PKW is comparatively large and the discharge enhancement ratio between PKW and the traditional linear weir (r in equation 2.3 below) is higher than 3, which indicates an obvious increase in the discharge efficiency. The PKW discharge efficiency losses occur when increasing the hydraulic head. When H/P > 1.2, the PKW is only 1.2–1.3 times more efficient than the linear weir (Guo et al., 2019). The efficiency losses can be elucidated by the formation of local submergence just downstream of the upstream overhang of the weir and the interference of the lateral nappes overflowing the adjacent weir walls of the inlet key. As a result, many projects were constructed in accordance with the range of H/P of 0.2–0.8 (Figure 2.4).
As a direct result from research of Ouamane and Lempérière (2006), the discharge capacity of PKW with Wi/Wo > 1 is higher than that of the symmetric configuration (Wi/Wo = 1) and that with Wi/Wo < 1.
Machiels (2012) described the flow in the inlet key as the “engine” of PK weir flow, whereas flow in the outlet key played a role as the “brake”. With larger inlet keys the area of the inlet flow and the wetted perimeter increase, leading to reduced inlet velocities, contraction area and head loss. This refers to the fact that a wider inlet will advance flow into the weir. However, a wider inlet key coincides with a narrow outlet key which then restricts flow out of the weir and allows local submergence occurrence inside. A suitable balance must thus be struck between these two opposing effects. Lempérière and Jun (2005) and Hien et al. (2006) recommended Wi/Wo = 1.2 as an optimal value producing a maximum discharge coefficient Cd. This finding is consistent with mean value of Wi/Wo close to 1.25 in Figure 2.4.
Regarding the overhangs lengths ratio Bi/Bo, the highest hydraulic efficiency of a “high” PKW (P/Wu
20
> 0.5) is reached for a value of 3. For a “low” PKW (P/Wu < 0.5), there is not rather difference as Bi/Bo
ratio ranges from 1 to 3 (Machiels et al., 2014).
2.5 Analytical Equations of PKW Discharge Capacity
As summarized by Schleiss (2011), because of the complexity of a PKW geometry, three equations may be used to derive its discharge capacity.
The first equation uses the standard free flow weir equation involving a total head H on the weir versus the transverse width W as
2 2 3
PKW 3 dW
Q C W gH (2.1)
with QPKW as discharge, H as total head, W as total width of the weir, g as the gravitational acceleration.
The effect of weir is thus dependent on the discharge coefficient CdW (Ouamane and Lempérière, 2006).
Completing determination of CdW is very difficult due to the complexity of the several factors influencing it (Lakshmana Rao S, 1975).
Relating to this approach, Kabiri-Samani and Javaheri (2012) have presented a discharge coefficient related to the width of the weir CdW for sharp-crested PKWs with limitations:
0.1 ≤ H/P ≤ 0.6; 2.5 ≤ L/W ≤7; 1 ≤ B/P ≤ 2.5; 0.33 ≤ Wi/Wo ≤ 1.22; 0 ≤ Bi/B ≤ 0.26; 0 ≤ Bo/B ≤ 0.26;
Hd/H ≤ 0.6 as:
0.426
0.675 0.377 0.306 1.504 0.093
0.212 o i 0.606
B B
B B
dW i
o
W
H L B
C e
P W W P
(2.2)
The statistical analysis indicated that the coefficient of determination between the model of Kabiri- Samani and Javaheri (2012) and equation (1) combined with (2) is reported as R2 = 0.986.
The second equation performed by comparison with the theoretical discharge capacity Qs of a linear sharp-crested weir of same width, considering the discharge enhancement ratio r (Leite Ribeiro et al., 2012b):
2 3
PKW PKW
S S
Q Q
r Q C W gH (2.3)
21
with QPKW is the PKW discharge. The discharge coefficient of the linear sharp crested weir Cs can be assumed as constant with Cs = 0.42 (Hager and Schleiss, 2009).
The values of r were given as a function of primary and secondary parameters. The primary parameters with a dominant effect on the discharge capacity of a PKW are the developed length L, the total transverse width W, the height of the inlet entrance measured from the PKW crest (including possible parapet walls) Pi, and the total head H. The secondary parameters have a relatively small influence on discharge capacity, including the ratio of inlet to outlet key width Wi/Wo, the ratio of inlet to outlet height Pi/Po, the relative overhang length (Bi + Bo)/B, and the relative parapet wall height Ro/Po. Here, parapet walls are vertical extensions placed over the crest of a PKW.
The measured r collapsed with a trend line of normalized with δ as:
1 0.24
r wpba
(2.4)This equation was validated for half rounded crests with the limitations:
0.1 ≤ H/P ≤ 2.8; 3.0 ≤ L/W ≤7.0; 1.5 ≤ B/P ≤ 4.6; 0.50 ≤ Wi/Wo ≤ 2.0; 0.4 ≤ (Bo + Bi)/B ≤ 0.8;
0.72 ≤ Po/Pi ≤ 1.38; 0 ≤ Ro/Po ≤ 0.22 where:
L W P
i 0.9 WH
(2.5)
and 𝑤, 𝑝, 𝑏, 𝑎 were found to be individual correction factors, expressing the influence of the secondary parameters:
0.05 i o
w W W
(2.6)
0.25 i o
p P P
(2.7)
0.5
0.3 B Bo i
b B
(2.8)
2
1 o
o
a R
P
(2.9)
Here, r (δ = 0) = 1 (when L = W or small Pi combined with great H). All tests considered in Leite Ribeiro et al., (2012b) included a range of 1.2 ≤ r ≤ 5.3 and the coefficient of determination between
22 the measured data and equation (4) is R2 = 0.976.
The third equation was derived from a detailed analytical formulation. Machiels et al. (2013) estimated the specific discharge (q) for flat topped PK weirs as the sum of the specific discharges on the downstream (qd), the upstream (qu) and the lateral crests (qs)
PKW o i 2
u d s
u u u
Q W W B
q q q q
W W W W
(2.10)
The three specific discharges can be estimated by using standard weir equations:
2
1 3
0.374 1 1 0.5 2
1000 1.6
u
T
q H gH
H H P
(2.11)
2
1 3
0.445 1 1 0.5 2
1000 1.6
d H
q gH
H H P
(2.12)
2
1 0.833 3
0.41 1 1 0.5 2
833 1.6 0.833 0.833 i o
s e W W
e e
P
q H K K gH
H H P H P
(2.13)
where PT = P + Pd, with dam height below the PKW Pd, the mean side wall height Pe:
1 2
o o
e B T B P
P P
B B
(2.14)
and, 𝛼 and 𝛽 are parameters that characterize the influence of the inlet key slope Si on the side crest discharge efficiency:
0.7 3.58 7.552
i i
S S
(2.15)
1.446
0.029e Si
(2.16)
Furthermore, KWi describes the influence of the inlet key width Wi on side crest discharge efficiency with γ is an empirical expression:
1 2 Wi
i
K W
(2.17)
0.0038 i 0.0055
o
W
W (2.18)
and KWo takes into account the side crest length decrease induced by the outlet key flow and the side nappe interference. It depends on H/Wo:
23
3 2 2
2 1 2 1 2 2 1
3 3 3 3
2 1 2 1 2 1 2 1
1
3 6 3
2
0
Wo
o o o
H H H
K W W W
(2.19)
The values of the two thresholds 𝛿1 and 𝛿2 are directly related to the outlet key slope (So):
1 0.788S01.88 5
(2.20)
2 0.236S01.94 5
(2.21)
2.6 Comparison
Pfister and Schleiss (2013) presented a review of all three hydraulic design approaches for a symmetrical A-type PKW prototype. The resulting discharge-head relationship curves are shown in Figure 2.5. In addition, the rating curve of a linear sharp-crested weir with same width and height is expressed, computing in respect of Vischer and Hager (1998). It was found that three studies basically estimate similar PKW discharge for a certain head, but not identical results. In general, the second approach predicts the highest discharge capacity, and the analytical equation in the third approach provides the lowest values.
Figure 2.5 Comparison of rating curves for PKW and straight sharp crested weir
24
This difference is partially a result of the crest shape of the PKW. In approaches conducted by Leite Ribeiro et al.(2012a) and Kabiri-Samani and Javaheri (2012), half-rounded crest and sharp-crested PKW were used, respectively, while the analytical equation in Machiels et al.(2013) study used flat- topped shape of weir. Performance of flat-topped crest in terms of discharge capacity is still inferior to half-rounded crest and sharp-crested weir. The differences may also allude to the problem of not guaranteeing the suitable performance of the equations for predicting the PKW discharge capacity outside its parameter limitations. According to Pfister et al., (2012), small discrepancies in this parameter range may lead to apparently incredible errors in the PKW discharge calculation. Comparison of the analytical formulation with experimental data from different physical modeling studies, as well as real projects PKWs configurations, shows a 15 percent accuracy in unit discharge prediction on a wide range of geometric parameters values (Machiels et al., 2013). Consequently, for a design process, the most appropriate PKW hydraulic design formula shall be considered on the application limits of the capacity equations (Anderson and Tullis, 2012; Kabiri-Samani and Javaheri, 2012; Leite Ribeiro et al.
2012a, 2012b; Machiels et al., 2014). If possible, Erpicum et al., (2017a) recommended to consider applying different formulae in order to ensure a better predictive power.
2.7 Scale Effects and Discharge Capacity Prediction Using Mathematical Models 2.7.1 Scale Effects
Scale effect occurs when a prototype hydraulic process is simulated at a laboratory scale due to dissatisfaction of similarity laws. It is suggested that scale effect in a PKW physical model test could be efficiently reduced if both the geometric, kinematic and dynamic similitudes are satisfied simultaneously. This is often unachievable when a common fluid (e.g., water) is used for both the model and prototype. The viscosity and surface tension can affect the formation of free surface in experimental models. To avoid these effects, Erpicum et al. (2016) conclude that model-heads smaller than 0.03 m underestimate the discharge capacity at PKWs, and that some 0.06 m are required to correctly reproduce the flow features in terms of nappe formation and jet geometry. However, according to Tullis et al.
25
(2020), minimum heads to avoid scale effects varied with both model scale and actual weir size. Simply limiting the model geometric scale ratio is not sufficient when considering scale effects for PKW model.
The actual sizes of the prototype and model are indeed important consideration. The minimum dimensionless head (H/P) below which scale effects occur generally decreases with increasing model size.
2.7.2 Mathematical Models
Although specific physical models are considered to be an effective way of investigating fluid flows, simply conducting physical model tests prevents a comprehensive understanding of the hydraulic and discharge characteristics of PKWs. Moreover, due to high cost of experiments, in recent years, researchers are encouraged to use mathematical methods for investigating the hydraulic characteristics of PKWs. Mathematical modeling consists of computational fluid dynamic (CFD) methods and soft computing techniques. In the context of CFD the Navier-Stokes equations coupled with turbulence models using numerical methods are solved based on a multigrid algorithm. Recently, for using the CFD techniques number of available commercial software such as Fluent and Flow-3D and free open codes such as OpenFOAM have been employed. Using the CFD technique for simulating the flow over PK weirs was reported by many investigators (Crookston et al., 2018; Hu et al., 2018; Li et al., 2019;
Oertel, 2016; Paxson and Savage, 2006). Soft computing approach, along with the CFD modeling, has become popular and has drawn research interest from investigators. Soft computing is not a single method, but it is a combination of several methods such as artificial neural networks (ANN’s), Genetic Programming, Support Vector machine, Group Method of Data Handling, Adaptive Neuro Fuzzy Inference System (ANFIS). Predictions of the hydraulic characteristics of PKW have been confirmed by Zounemat-Kermani and Mahdavi-Meymand (2019). General results indicated that all mathematical methods can simulate the PKW discharge more accurately than empirical relations. However, it is fair to say that mathematical method has not yet emerged as a reliable means for PKW design. The experts recognize the need to verify and validate the code. The errors of the mathematical method can be minimized by comparing the results to those of experiments which are always the final proof.
26 2.8 Conclusions
PKW is a complex structure involving a large number of geometric parameters and an interaction of flows over up/downstream crest and lateral crest of the keys. This review paper is a result of a demand for a comprehensive coverage on the hydraulic behavior of PKW. The flow over a PKW is the cumulative of three different types of overflow: flow over the inlet key downstream crest, flow over the outlet key upstream crest and flow over the lateral wall. The downstream crest of the inlet is always supplied by the current in front of the key. Meanwhile, only the surface current comes into the outlet.
The lateral discharge capacity decreases considerably when upstream head increases. The flow characteristics over a PKW are dependent on primary and secondary geometrical parameters, which increase difficulty level of discharge capacity prediction. Three general design equations were proposed in recent years. Summary of analysis results illustrated that there is a fine agreement between three equations, however discrepancies still exist. Besides that, researchers are encouraged to use mathematical models to calculate the PKW discharge. Although, recent published studies have proved the accuracy of the numerical solution, this approach still need to be further investigated in the future.
+<'5$8/,&&203$5,621%(7:((13,$12.(<:(,5$1' 5(&7$1*8/$5/$%<5,17+:(,5
28 3.1 Introduction
The availability of hydrological and meteorological data coupled with new dam-safety guidelines have increased the Probable Maximum Flood (PMF), the Spillway Evaluation Flood (SEF), or the Inflow Design Flood (IDF) that a dam is required to pass (Felder et al., 2017). One of the most common problems for obsolete dams is spillways that are no longer sufficient to handle updated flood flow due to climate change. If water cannot escape quickly enough through spillways, it could flow over the top of a dam, which would increase the likelihood of extensive erosion that can cause it to collapse (Lempérière, 2017).
Increased discharge capacity of an existing spillway can be achieved by increasing either the spillway crest length or discharge coefficient or operating head, or any combinations (Xlyang and Cederström, 2007). The operating head for a given spillway can be increased by either lowering the spillway crest and installing gates, or raising the dam crest to permit higher reservoir levels. However, adopting these approaches will lead to a great rise in costs of investment and operations management. A modest increase in the coefficient of discharge can generally be realized by reshaping the crest and by channel improvement, but at great cost. A more common modification to existing dam to accommodate larger floods is the enlargement of the spillway crest length without an associated increase in structure width, which is always limited by the layout of the discharge structures or site conditions.
Labyrinth weir is an especially suitable method which is employed to alleviate the problem of restricted spillway widths. Labyrinth spillways are polygonal overflow weirs folded in plan-view to provide a longer total crest length for a given overall spillway width. Although there are many geometric configurations of labyrinth weirs, three of them are widely used: triangular, trapezoidal and rectangular.
Due to their polygonal shape, labyrinth weirs provide higher discharge capacity than linear overflow weirs for the same width and upstream energy head (Anderson, 2011). The best example is the labyrinth weir of the BeniBahdel dam built in Algeria in 1938. Its approximate discharge capacity is 1200 m3.s-1 at a head of 0.5 m with the total crest length of 1200 m which is shrunk into a channel of 80 m width.
At the same head (0.5 m), a traditional sharp-crested weir would discharge only 95 m3.s-1 (Lempérière
29
and Vigny, 2011). Although labyrinth weirs may be very cost effective, they require a specific topography; this may explain their limited success (Bilhan et al., 2018; Blanc and Lempérière, 2001;
Lempérière and Ouamane, 2003).
Piano key weirs (PKWs) are a modified type of labyrinth weir that has rectangular cycles with overhangs and ramps in each cycle (Blanc and Lempérière, 2001; Lempérière and Ouamane, 2003).
The use of overhangs decreases the footprint of the structure (Figure 3.1) and permits its installation on top of the existing structures such as gravity dams and embankment dams. Compared to a labyrinth weir, the inclined bottoms in the cycles of the PKW help to reduce the lateral forces exerting on the side walls and hence the structural cost (Eslinger and Crookston, 2020).
Figure 3.1 PKW (left) with geometrical parameters; RL weir (right)
A significant amount of research has been carried out during the last years to investigate the hydraulic behavior of labyrinth and PKWs with three landmark international conferences (Crookston et al., 2018, 2019; Erpicum et al., 2013; Erpicum, Laugier, et al., 2017; Erpicum et al., 2011; Lempérière and Ouamane, 2003; Machiels et al., 2014) but relatively few investigations on the topic of rectangular labyrinth (RL) weirs (side leg angel α = 0o). Tullis et al. (Tullis et al., 1995) developed a comprehensive design procedure for trapezoidal labyrinth spillways estimating the discharge capacity with side leg
30
angles of the weir varying from 6o to 35o. Machiels et al. (Machiels et al., 2011) tested a large scale model to enhance the understanding of the flows over the PKWs. Karimi et al. (Karimi et al., 2018) studied the hydraulic characteristics of Piano Key side weirs and emphasized the significant advantages of Piano Key side weirs and rectangular labyrinth side weirs in terms of discharge capacity compared with conventional linear side weirs.
However, aforementioned studies used only scaled models based on similitude theory. Although specific physical models are considered to be an effective way of investigating fluid flows, simply conducting physical model tests is insufficient to fully comprehend the hydraulic and discharge characteristics of non-linear weirs. In recent years, advances in computing power and computational fluid dynamics (CFD) algorithms have been used extensively to investigate complex flow physics instead of relying on reduced scale models, and evaluate the design and operation of the non-linear weirs (Crookston et al., 2018, 2016, 2012; Ghanbari and Heidarnejad, 2020; Hu et al., 2018; Li et al., 2019; Paxson and Savage, 2006; Safarzadeh and Noroozi, 2017). Although some of those are preliminary studies without validation by physical models, their results showed that CFD approach has a promising future of weir investigations. For the more specific case of nonlinear weirs, there have been numerous hydraulic studies regarding discharge capacity but relatively few investigations on the topic of comparison between PKW and RL weir.
The objectives of this study are:
1) to validate and verify a numerical model for determining the discharge capacity over the PKW and RL weir;
2) to compare the discharge efficiency of the PKW and RL weir;
3) to carry out a qualitative comparison between PKW and RL weir regarding to the discharge per unit length;
4) to develop a better understanding of the flow dynamics passing over the PKW and RL weir.
