aeration and risk mitigation for flood discharge tunnel in
TRANSCRIPT
IN DEGREE PROJECT THE BUILT ENVIRONMENT,SECOND CYCLE, 30 CREDITS
, STOCKHOLM SWEDEN 2020
Aeration and Risk Mitigation for Flood Discharge Tunnel in Zipingpu Water Conservancy Project
JORGE CONTRERAS MORENO
KIBRET DAWIT GHEBREIGZIABHER
KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT
Aeration and Risk Mitigation
for Flood Discharge Tunnel in
Zipingpu Water Conservancy
Project
JORGE CONTRERAS MORENO
KIBRET DAWIT GHEBREIGZIABHER
Master of Science Thesis
Stockholm, Sweden 2020
TRITA-ABE-MBT- 20191
ISBN: 978-91-7873-574-7
KTH School of ABE
SE-100 44 Stockholm
SWEDEN
© Jorge Contreras Moreno & Kibret Dawit Ghebreigziabher, 2020
Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering
Division of Concrete Structures
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Abstract
The importance of hydraulic structures has become an essential mitigating mean for floods
that occur more often due to climate change. Thus, the importance and safety of flood
discharge tunnels has promoted further studies and experiments on the topic to mitigate
damages, such as cavitation that arise because of high speed flows.
After an experimental study on a physical model was carried out on the flood discharge tunnel
in Zipingpu Water Conservancy project, a CFD model was designed and simulated in the
commercial software ANSYS Fluent. The simulations aimed to evaluate and examine the risk
for cavitation in the tunnel, examine the design problems of the structure and analyse the
installed aerators for the mitigation of cavitation. Moreover, using CFD models as a
complementary form to physical models was analyzed.
A three dimensional geometry of the discharge tunnel was built in ANSYS Spaceclaim and the
mesh conducted with ANSYS mesh generator. The known boundary condition such as the
design flow conditions, velocity inlet, pressure inlets and pressure outlet were set. For the
model a multiphase VOF scheme with RANS approach, k-ϵ turbulence model and a standard
wall function was set.
The results from the initial simulations showed that the discharge tunnel was under cavitation
risk, since the recorded cavitation index in the tunnel was below 1.8. After having revised the
layout of the aerators in order to mitigate cavitation risk, the results from the simulations with
added aerators were sufficient to mitigate the risk as the cavitation index was still below 1.8.
The results for the cavitation index remained unchanged even in the simulated models with a
different solver setup that were used in the comparison with the experimental data in order to
validate them.
As a conclusion, it was recommended that the tunnel design has to be revised and improved
by adding more aerators and air vents to mitigate the cavitation risk. Furthermore, more studies
on the discharge tunnel or similar tunnels with similar conditions should be carried out in order
to validate the results of this study and determine if numerical models are preferable to physical
models.
Keywords: Flood discharge tunnel, cavitation risk, cavitation index, aerators, CFD model.
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Sammanfattning
Betydelsen av hydrauliska strukturer har blivit ett väsentligt förebyggande medel för
översvämningar som förekommer oftare på grund av klimatförändringar. Således har vikten
och säkerheten för översvämning tunnlar främjat ytterligare studier och experiment om ämnet
för att förebygga skador, såsom kavitation som uppstår på grund av hög hastighets flöden.
Efter att en experimentell studie av en fysisk modell genomfördes på avrinningstunneln i
Zipingpu Water Conservancy projekt, genomfördes en CFD-modell i den kommersiella
programvaran ANSYS Fluent. Simuleringarna syftade till att utvärdera och undersöka risken
för kavitation i tunneln, undersöka strukturens konstruktion problem och analysera de
installerade luftnngsmekanismer för att minska kavitation. Dessutom analyserades
användning av CFD-modeller som komplement till fysiska modeller.
En tredimensionell geometri för avrinningsstunneln byggdes i ANSYS Spaceclaim och nätet
genomfördes med ANSYS nätgenerator. Det kända gränstillståndet såsom
designflödesbetingelserna, hastighetsinloppet, tryckinloppen och tryckutloppet inställdes. För
modellen sattes ett flerfasigt VOF-schema med RANS-tillvägagångssätt, k-ϵ turbulensmodell
och en standard vägg funktion.
Resultaten från de initiala simuleringarna visade att urladdningstunneln var under
kavitationsrisk, eftersom det registrerade kavitationsindexet i tunneln var under 1,8. Därefter
redigerades beläggningen av luftningsmekanismerna för att minska risken för kavitation och
resultaten från simuleringarna med tillagda luftningsmekanismer var inte tillräckliga för att
förebygga kavitationsrisken eftersom kavitationsindexet fortfarande låg under 1,8. Resultaten
för kavitationsindexet förblev oförändrade även i de simulerade modellerna med en annan
lösningsuppsättning som användes i jämförelsen med experimentella data för validerings
skull.
Som en slutsats rekommenderades att tunnel designen måste revideras och förbättras genom
att lägga till flera luftningsmekanismer och luftventiler för att minska kavitationsrisken. Vidare
bör fler studier på urladdningstunneln eller liknande tunnlar med liknande förhållanden
genomföras för att validera resultaten av denna studie och bestämma om numeriska modeller
är att föredra framför fysiska modeller.
Nyckelord: Avrinningstunnel, kavitationsrisk, kavitationsindex, luftningsmekanism, CFD-
modell.
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Preface
The diploma project reported in this Master Degree thesis was carried out during the first half
of 2020. Due to the Covid-19, we were unable to travel to China as originally planned. Instead,
the work was performed at Royal Institute of Technology (KTH) and still in close cooperation
with Tsinghua University, Beijing. In the light of the situation, necessary adjustments and
arrangements were made in terms of project topic, layout, supervision, means of
communications etc.
We would like to thank Prof. Yongliang Zhang, Tsinghua University, and Prof. James Yang,
Vattenfall and KTH, and for the supervision, access to data, advice and discussions. We would
also like to devote our thanks to Ph.D. student Shicheng Li and Dr. Penghua Teng at
Department of Civil and Architectural Engineering, KTH, for all the help with numerical
simulations including program learning during the performance of the project.
We are grateful to our examiner Anders Ansell at the Division of Concrete Structures, Royal
Institute of Technology for the coordination. Jorge Contreras Moreno would like to give special
thanks and gratitude to the National Council of Science and Technology of Mexico (CONACyT)
for providing financial support for living and studying at the Royal Institute of Technology in
Stockholm.
The diploma project is funded by Energiforsk AB within the frame of dam safety
(www.energiforsk.se), with James Yang as coordinator. The project has been going since
2004, with more than 115 university students who have done their diploma work in different
universities of technology in China.
Stockholm, June 2020
Jorge Contreras Moreno
Kibret Dawit Ghebreigziabher
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Contents
Abstract ...................................................................................................................................... i
Sammanfattning ...................................................................................................................... iii
Preface ....................................................................................................................................... v
List of acronyms ...................................................................................................................... xi
1 Introduction ...................................................................................................................... 1
Aims and objectives ................................................................................................ 1
Limitations ............................................................................................................... 2
2 Background ...................................................................................................................... 3
Water scarcity and water conservancy projects....................................................... 3
Flood discharge tunnel ............................................................................................ 4
General description of Cavitation ............................................................................ 6
Cavitation damages ................................................................................................. 7
Air entrainment ........................................................................................................ 9
Spillway tunnel aerators ........................................................................................ 10
Numerical method ................................................................................................. 12
3 Theory ............................................................................................................................. 13
Mathematical model .............................................................................................. 13
Discretization method (Finite Volume Method) ................................................... 14
3.2.1 Finite Volume Method ............................................................................. 14
3.2.2 Discretization methods ............................................................................. 14
3.2.3 Pressure based solver ............................................................................... 16
3.2.4 Reynolds Averaged Navier-Stokes .......................................................... 17
Multiphase flow model .......................................................................................... 18
Turbulence model .................................................................................................. 19
3.4.1 𝒌 − 𝝐 Turbulent model ............................................................................. 19
Boundary conditions .............................................................................................. 20
viii
3.5.1 Inlet and outlet .......................................................................................... 20
3.5.2 Wall function boundary ............................................................................ 20
Meshing ................................................................................................................. 21
3.6.1 Choice of mesh ......................................................................................... 21
3.6.2 Estimation of discretization error ............................................................. 22
4 Methodology ................................................................................................................... 25
Geometry ............................................................................................................... 25
Mesh generation .................................................................................................... 29
Numerical model setup .......................................................................................... 29
4.3.1 Boundary conditions ................................................................................ 30
4.3.2 Choice of solver ....................................................................................... 30
Numerical convergency ......................................................................................... 30
Grid independence check ...................................................................................... 31
Post-processing ...................................................................................................... 32
Model validation .................................................................................................... 32
Evaluation of different scenarios ........................................................................... 32
Aerator layout ........................................................................................................ 33
5 Results ............................................................................................................................. 35
Tunnel spillway geometry ..................................................................................... 35
Mesh ...................................................................................................................... 36
Numerical simulation ............................................................................................ 38
Grid independence ................................................................................................. 44
Data post-processing ............................................................................................. 45
Validation .............................................................................................................. 47
Results of different scenarios ................................................................................ 48
Aerator layout ........................................................................................................ 51
6 Conclusions and discussions ......................................................................................... 53
Geometry, mesh and grid independence ............................................................... 53
Comparison of models ........................................................................................... 54
Evaluation of flow scenarios ................................................................................. 54
Aerator layout behaviour ....................................................................................... 55
Source of errors ..................................................................................................... 55
7 Recommendations .......................................................................................................... 57
8 Reference list .................................................................................................................. 59
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9 Appendix ......................................................................................................................... 61
9.1 Experimental data ...................................................................................................... 61
9.2 Parametric analysis .................................................................................................... 64
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List of acronyms
CFD Computational Fluid Dynamics
CFRD Concrete Faced Rockfill Dam
CV Control Volume
GCI Grid Convergence Index
FVM Finite Volume Method
RANS Reynolds Averaged Navier Stokes
VOF Volume of Fluid
1
1 Introduction
With the occurrence of changing climatic and temporal characteristics taking place on earth due
to environmental degradation and pollution, heavy rains and flooding are becoming more
common. China is a country hit by the effects of climate change, which apart from flooding
have also created water scarcity and drought problems ((Ministry of water resources, 2006)).
To tackle these problems, the country has built numerous water conservancy projects that serve
as flood control, water supply and hydropower generation.
In this study the discharge tunnel at the Zipingpu Water Conservancy project, which is used for
flood control, has been analysed. A numerical model will be established using a computational
fluid dynamics (CFD) software called ANSYS. The model will then be validated in respect to
the results procured from the physical scale model that has been constructed at Tsinghua
University in Beijing, China. Numerical models are preferable because they are more cost and
time effective in comparison to physical models. But, nevertheless, physical scale models are
used to validate numerical models.
Aims and objectives
To complete this study various aims and objectives were set and are presented here.
The aim of the study is to:
- Analyse if there are design problems in regard to cavitation and optimize the design of
the tunnel to minimize the risks of cavitation
- Analyse whether the shape and the installed aerators in the discharge tunnel are suitable
to mitigate cavitation risks
- Analyse whether a numerical model is preferable to a physical scale model.
To achieve this aims the following objectives had been set:
- Establish a numerical model in ANSYS Fluent for 3D complex fluid flow, to simulate
the flow characteristics such as aeration and cavitation risk mechanisms
- Calibrate the numerical model through a comparison with the results acquired from the
experimental study on the physical scale model.
2
Limitations
This study was to be conducted at Tsinghua University in Beijing where a physical scale model
was conducted, and other studies of the Zipingpu Water Conservancy project were conducted.
However due to complications that arose in connection with the Covid-19 pandemic, the study
was conducted in Stockholm instead. This meant that limitations had to be considered in the
study and especially in the results. The encountered limitations were as follows:
- The ANSYS R3 2019 version that was used had an educational license which means
that the mesh element number was limited to a maximum of 512 000 mesh elements
and therefore simulations with finer mesh elements were not able to run.
- The experimental data that was used in the post-processing section and validation to
compare the calculated results was not the data for the tunnel of this study, but instead
of a similar tunnel that is a hydraulic structure in the Zipingpu Water Conservancy
project. The experimental data was extracted from an older study that was conducted on
the other tunnel since it was said to be similar after deliberation with the supervisors
(Hamberg and Dahlin, 2019). Therefore, the comparisons that were made between the
calculated results and experimental data might deviate slightly or could have been better
if the right experimental data for the tunnel in hand was available.
- Time constraint was a limitation. The study took off behind schedule but was concluded
within the deadline that was set beforehand. This affected the process of the grid
independence study and possibility of running different simulations with different types
of interpolation and discretization methods.
3
2 Background
Water scarcity and water conservancy projects
China is home to the largest population on earth. With its ever-growing economic development,
population and living standards, must meet the needs of its citizens. One of the biggest
challenges that the country is facing is water scarcity and meeting the consumption needs
(Jiang, 2009). This crisis has given the Chinese government the incentive to establish a water
conservancy plan (Liu and Yang, 2012). However, this plan although aimed to achieve water
sustainability may also cause environmental and socio-economic repercussions if correct
assessments are not carried out beforehand.
China’s water scarcity can be connected to different causes: natural characteristics, where the
southern part of China holds most of the water resources, whereas the northern part China that
accounts for 45.2% of the population has only 19.1% of the water resources; economic
development, where the hasty industrialization resulted in a 15% consumption of world natural
resources for water; population growth, where China is home to over 13 billion people (20% of
the world population) but only has 6.5% of the world’s total freshwater resources; water
resource management, where the water conservancy plans are carried out poorly; water quality,
where poor water quality in the rivers and basins add up to the scarcity and threaten China’s
economic development, food security and life quality (Jiang, 2009). Moreover, due to the
climatic and temporal characteristics that result in an annual precipitation of 60-70% in summer
season add to the water scarcity problem (Liu et al., 2013).
China has been tackling water scarcity by raising funds for investments to water conservancy
projects and to achieve a sustainable water management. China has more than 87000 dams, the
established plan and investments are aimed to repair and sustain more than half of them and
moreover, as mentioned, to manage them sustainably. However, the introduced water
management plans set up by the government focus mainly on the quantity and not quality of
the water. The quality of water is to be taken in account, considering that almost 40% of the
rivers are polluted and 80% of the lakes suffer from eutrophication (Liu and Yang, 2012).
As stated earlier, China being one of the fastest growing and developing economies have
dedicated resources and investments to counteract the water scarcity present in the country.
Which lead to a boom of construction of hydraulic structures such as water conservancy
projects, that serve different purposes such as hydropower generation, urban water supply,
irrigation supply and flood control (Liu et al., 2013). Hence, China is home to the largest
number of dams and the largest hydropower plant, i.e. the Three Gorges Hydropower Project
(TGHP) (Jiazhu, 2002).
4
The construction of dams and reservoirs has had a positive impact in flood control measures
(Liu et al., 2013). Moreover, China has a total reservoir capacity of 932.312 billion m3
(Ministry of water resources, 2006) and an installed hydropower generation of 352 million kW
year 2018 that accounts for almost 20% of China’s electricity generation and 27% of the
world’s capacity (International Hydropower Asssociation, 2019, Jia, 2016). Furthermore, the
construction of dams and reservoirs has facilitated the possibility of farming areas that
previously was not possible due to the temporal and climatic distribution of precipitation. This
resulted in a 60.35 million ha of irrigated area by 2010, that is pathing the way to food security
for the country (Liu et al., 2013).
China has executed numerous water conservancy projects to alleviate their water scarcity. To
ensure quality, sustainability and financial support various policies and plans have been carried
out by the government. Amongst the recent plans is the “Tenth Five-Year Plan” that consist of
crucial water conservancy projects such as Baise, Linhuaigang, Shapotou, Ni’erji and Zipingpu
(MINISTRY OF COMMERCE, 2002).
The Zipingpu water conservancy project, which this study is focused on, is located in Sichuan
province that is deemed to be one of the most important bases for hydropower and water
resource hence it has an ample water resource, and therefore, many hydraulic plants are built in
this area. The Zipingpu water conservancy project started construction in 2001 and concluded
in 2006. It is mainly designed for the purpose of irrigation and urban water supply, but it also
functions as a hydropower plant with a capacity of 760 MW and flood control (Xinhua News,
2002). The Zipingpu water conservancy project consists of different hydraulic structures and is
located in the upper reaches of the Minjiang river, 60 km northwest of the capital of Sichuan
province, Chengdu, and 9 km west of Dujiangyan City (Tanchev, 2014). To name some of the
main structures: a concrete faced rockfill dam (CFRD) with a height of 156m, a spillway, a
sand blasting hole, a hydropower system and two flood discharge tunnels that act as the main
structures for flood control and sand discharge in the occurrence of a flood . One of the flood
discharge tunnels will be the focus of this study.
Flood discharge tunnel
Apart from the water scarcity that is present in China, flooding is also another problem that the
country tackles with. Due to its climatic characteristics and geographical location, the country
has been subject to the Eastern Asian monsoon that results in heavy flooding (Zhang and Liu,
2006). China's investment in water conservancy projects and the adherent policies also covers
the flood control and management in the country, and therefore, China has had achievements
in their flood control programs (Ministry of water resources, 2006).
Among flood control hydraulic structures are flood discharge tunnels. A flood discharge tunnel
is commonly structured by a sloping section, a toe curve and an approximately sloping or almost
horizontal section that joins into an energy dissipator or tailwater at its end (Khatsuria, 2005).
5
The performance and stability of discharge tunnels is dependent on the design schemes used to
tackle different problematics regarding shape, size and flow properties. Regarding the shape
and size, the toe curve is of importance for the performance of a tunnel. In theory, the flow in
a tunnel is three dimensional and therefore, given the high velocity of the flow, the pressure on
the tunnel surface would be highly positive as the flow fastens to the surface and cavitation
damage occurs. Thus, the toe radius must be designed accordingly, too large of a radius is
economically unsustainable and too small might cause unwanted and risky flow conditions,
therefore, the toe radius should be larger than the tunnel radius before the toe start (i.e. 2.5 -
10.5 times the radius of a tunnel) (Khatsuria, 2005). Another important aspect is the cross-
section shape of the tunnel. Given that it is a tunnel in question, a circular cross-section is the
obvious choice for its stability and uniformity under high pressure. However, the most
frequently used shape is in the form of a “horse shoe”, where the ceiling and walls of the tunnel
are arched and the bottom is flat to resemble a horse shoe (U.S. Army Corps of Engineers, 1980,
Dandekar and Sharma, 1979).
In order to obtain an economically sustainable dam construction, the diversion tunnel used for
deviating the river flow away from the construction can be then used as a permanent flood
discharge tunnel/tunnel spillway upon completion of the project by adding an energy dissipator
to it (Tian et al., 2009). The energy dissipator at the outlet of a discharge tunnel is commonly a
flip bucket or a stilling basin, and is critical to achieve a smooth transition from the tunnels
cross-section shape into a flat bottom to ensure the stability and integrity of the structure
(Khatsuria, 2005). Furthermore, also for economic reasons, in the design of the cross-section
the diameter of the tunnel is held to a minimum without undermining the purpose to be fulfilled
by it, hence a tunnel is never allowed to flow with full capacity in order to leave space for air
flow (Khatsuria, 2005). Therefore, it is advised to design a spillway tunnel with a ratio of ¾ or
⅞ of the full flow capacity to accomplish and allow a balanced air-water flow to avoid unsafe
flow conditions (Khatsuria, 2005). The impacts of these unsafe flow conditions can be seen due
to the flow's high velocity and low pressure, which in return can result in cavitation damages,
hence the need for aeration.
For the purpose of this study one of the discharge tunnels in the Zipingpu water conservancy
project was considered. According to the design information of the Zipingpu Water
Conservancy project, the flood discharge sluicing sediment tunnel was transformed from a
diversion tunnel into a “dragon-up” type pressurized short tunnel whose inlet is non-pressure
flow, including imported open channel, inlet tower, inclined shaft section, diversion tunnel with
slow slope section, and exit tunnel section.
This tunnel has a length of 720.55 m, the inlets bottom elevation and exit elevation are 800 m
and 745.156 m, respectively. The given discharge rates are 1530.87 m3/s and 1666.74 m3/s at
the design water level and the check level respectively, with a maximum flow velocity of 45
m/s and a flow rate of 212.9 m3/s.m at the outlet of the discharge tunnel.
6
General description of Cavitation
As mentioned above, cavitation is a severe damage that affects the stability and integrity not
only of a discharge tunnel but other hydraulic structures such as spillways and chutes
(Yazdandoost and Attari, 2004). In this chapter the cavitation phenomena will be covered and
explained.
Cavitation is when voids are formed in a liquid medium and they can be classified in two
groups: vaporous cavitation, when the void is filled with water vapor and gaseous cavitation
when the void is filled with a gaseous medium (Falvey, 1990). Often, it is compared and
examined with the aspect of boiling water at a local atmospheric pressure. The vapour pressure
increases in boiling water. When the vapour pressure of the boiling water reaches and is equal
to the local pressure, bubbles are formed due to the transformation phase of the water to vapour
at the boiling point (Khatsuria, 2005, Falvey, 1990).
The pressure obtained in the water can be seen as a governing factor for the phenomena of
cavitation. Therefore, boiling can be obtained at lower pressures consequently as the pressure
decreases even up to a level where it can be obtained at room temperature. Nevertheless,
bubbles are still formed even in this case and are defined as vaporous cavitation (Khatsuria,
2005, Falvey, 1990). Boiling and cavitation, although intertwined with each other, are not the
same thing. Both illustrate a phase change from liquid to vapour, where boiling changes in
temperature but holds the local pressure constant, and the opposite for cavitation where the
temperature is held constant while the local pressure changes (Falvey, 1990).
To describe gaseous cavitation, the example of bubbles formed in a bottle of carbonated water
is used. Before a bottle is opened, the carbonated water is still because it’s kept sealed under
high pressure. When opened, the pressure in the bottle decreases, and bubbles form as the liquid
becomes saturated and the carbon dioxide diffuses (Khatsuria, 2005, Falvey, 1990).
The damage caused by cavitation occurs when the bubbles filled with vapor pressure collapse.
When the bubbles collapse or explode along a solid structure, the high pressure due to the
collapse generates a force capable of damaging hydraulic structures such as a discharge tunnel
(Khatsuria, 2005). The collapse mechanism of a single bubble is described as a process of
phases where the bubble diameter decreases until it reaches a minimum and then increases again
or rebounds. This process is repeated several times, and in every cycle the bubble diameter
decreases eventually down to a microscopic size, as shown in Figure 2.1. A shock wave with a
velocity equivalent to the speed of sound in water is formed in the rebound phase. It is stated
that the shock wave emits a pressure 200 times the ambient pressure at the collapse site
(Khatsuria, 2005, Falvey, 1990).
7
Figure 2.1. Collapse mechanism of a single bubble near a solid surface (as shown in Khatsuria 2005 and Falvey
1990)
Cavitation damages
Cavitation damages are more prone to be found in the structures where there are surface
irregularities, such as rough spots and joints between structures (Falvey, 1990, Chanson, 1989).
Basically, in the regions where the surface irregularities are present, a flow separation might be
formed, and the pressure lowered. And in the presence of high flow velocities, it results in
bubble formation, due to the low pressure (below the local pressure), which will later on
collapse when it reaches a region of higher pressure and be liable to cavitation damages
(Chanson, 1989). To be noted is that the cavitation bubble consists of several bubbles and is
referred to as a cavitation cloud (Falvey, 1990).
The surface irregularities on spillway are directly connected to the surface roughness, of which
they are classified as singular or isolated roughness and uniformly distributed roughness.
Singular roughness may also be specified as: offset into the flow, offset away from the flow,
abrupt curvature or slope away from the flow, voids or grooves, roughened surface and
protruding joint (Khatsuria, 2005, Falvey, 1990). As a result of the unusual change in the flow
at the irregularities, cavitation occurs due to the turbulence in the shear region in all the listed
cases of singular roughness. As cavitation is formed, cavitation damage will occur as the
bubbles collapse either close to the flow boundary or in the flow itself depending on the
roughness shape (Falvey, 1990).
In difference to singular roughness, the uniformly distributed roughness develops cavitation in
the flow due to fluctuations that take place over a larger area. The fluctuations that arise, can
be a result of the concrete surface erosion due to abrasions or to poor lining of the concrete
surface finish (Khatsuria, 2005).
Hydraulic structures always run a risk of high probability for cavitation damages when they are
in contact with high velocity flows. Therefore, factors have been established in order to know
if a surface will be damaged or not (Falvey, 1990), and are as follows:
8
- Determining the cause of cavitation
- Determining the location of the damage
- Determining the intensity of the cavitation
- Determining the flow velocity
- Determining the air concentration in the water
- Determining the surface resistance to damage
- Determining the exposure time of the surface
The cavitation index is also used to assess whether a flow is prone to cavitate. The index is
dimensionless and is calculated according to the following equation:
𝜎𝑖 =𝑃𝑜 − 𝑃𝑣
12 𝜌𝑉𝑜
2
where ꝍi is the cavitation index, P0 (Pa) is the static fluid pressure, Pv (Pa) is the vapour pressure,
𝜌 (kg/m3)is the density of the fluid and V0 (m/s) the velocity of the flow (Khatsuria, 2005). For
the cavitation index ꝍi = 3 there is no cavitation, for ꝍi = 1.8 the cavitation is incipient, for 0.3
< ꝍi < 1.8 it’s a developed cavitation and for ꝍi < 0.3 it’s a supercavitation (Falvey, 1990).
As for the location of the cavitation damage, it always takes place downstream of the cavitation
source, which is the collapsing cavitation cloud. Near the end of the cavitation cloud, it has
been shown to be the region for the maximum damage. Moreover, with the increase of the
discharge and surface irregularities’ height, the maximum damage also results in an increase.
However, for a cylinder, the cavitation damage takes place when the cavitation cloud’s length
equals the cylinder diameter (Falvey, 1990).
Good examples of cavitation damages in spillway tunnels are the ones occurred in the spillway
tunnels of the Hoover Dam and Glen Canyon Dam, both located in the USA. For the Hoover
dam, reports show that cavitation occurred due to misalignment upstream of the damage
location. The velocity of the flow at the time of the damage was recorded to be 45 m/s and the
damage size was a hole 14 m deep, 35 m long and 9 m wide (Khatsuria, 2005). The Glen Canyon
dam also experienced the same damages as the Hoover dam after the flooding of the Colorado
river in 1983. The damage left the spillway with a hole as big as its diameter. The damage was
reported to be due to residing cavitation damages on the concrete lining in several locations
(Falvey, 1990).
9
Figure 2.1. Cavitation damage on the Hoover dam to the left and Glen Canyon dam to the right (Falvey, 1990).
To mitigate and avoid cavitation damages, aerators have been used in spillways that serve the
purpose of inserting air into the water, especially close to the surface to minimize the damage
risk (Yazdandoost and Attari, 2004, Falvey, 1990, Chanson, 1989). Moreover, studies have
shown that for velocity of 12 m/s - 20 m/s cavitation damages may be avoided by concrete-
lining, improving or eliminating irregularities in the surface and using better material for flows
between the (Ruan et al., 2007, Chanson, 1989).
Air entrainment
As mentioned previously, aerators are commonly used to prevent cavitation damages. The use
of aerators falls in the category of air-entrainment. Air entrainment is one of the best ways to
avoid cavitation damages especially when the air is put as close as possible to the flow boundary
(Khatsuria, 2005).
The phenomena of air entrainment is defined as the exchange of air contained within the
atmosphere and water. Some other synonyms for air entrainment are air bubble entrainment or
aeration. In addition, the air entrainment may occur from natural or artificial origins (Chanson,
1996). Natural air entrainment is referred to as self-aeration, which occurs when turbulence
starts in a spillway and the turbulent boundary intersects the water surface and air entrains the
bubbles in the turbulent boundary. For the artificial air entrainment is meant forced aeration by
means of modification to the design, i.e. installing an aerator. Thorough studies should be
carried out to decide the type and location of an aerator. The aerator is fed air through different
mechanisms of air supply systems, such as the commonly used air-intake conduit or a duct
10
system. The air entrainment process can be seen when the water surface appearance turns from
clear and glossy to irregular, white and bubbly (Khatsuria, 2005).
Air entrainment is also defined as the entrapment of un-dissolved air bubbles and air pockets
through the flowing liquid. These concentrations can be classified locally (local aeration) or
continuously (interfacial aeration) along the air-water flow (Chanson, 1996). The local air
entrainment is a concentration of air bubbles located at the intersection of the impinging jet and
at hydraulic jump. On the other hand, the interfacial aeration is defined as the air entrapped
along an air-water interface, for example a chute (Yazdandoost and Attari, 2004).
In tunnel spillways air entrainment mechanism has differing characteristics depending on the
flow condition, mainly partly full or pressurized. When the tunnel is partly full, the air
entrainment mechanism is considered as an open channel flow (as described earlier). Here the
main parameter to be considered in the design is the total air discharge, which consists of both
the air flowing freely on the water surface and the air added to the flow through external means
such as air vents (Khatsuria, 2005, Chanson, 1997). For pressurized or full flow tunnels air
entrainment can cause serious damages of conveyance, unstable flow conditions and damage
to the concrete lining. Moreover, depending on the design of the tunnel, the air might flow
upwards instead and create air pockets on the water surface that will need to be released through
air diffusing systems. Moreover, the downstream flow conditions in the shear layer are to be
considered since they affect the air transport within the tunnel, and therefore, plays the ratio of
length and diameter of the tunnel a role in the transport (Khatsuria, 2005).
Spillway tunnel aerators
As mentioned earlier, aerators are measures taken to induce forced aeration in order to prevent
and mitigate cavitation damages. It was also mentioned that cavitation damage can be avoided
by eliminating surface irregularities for flow with a velocity up to 20m/s, but for flow with a
velocity over 30m/s aerators must be used to avoid cavitation damages (Ruan et al., 2007,
Chanson, 1989).
There are various types of aerators, and the most basic devices are illustrated as: steps,
deflectors, grooves, offset or a combination of them (Khatsuria, 2005, Falvey, 1990, Chanson,
1989, Volkart and Rutschmann, 1984). Figure 2.3 shows the illustrated aerators and their
combinations.
11
Figure 2.3. Basic types of aerators. (Khatsuria, 2005, Falvey, 1990, Volkart and Rutschmann, 1984)
Some basic criteria to consider in design an aerator system are (Khatsuria, 2005):
- Positioning of the first aerator
- Form of the aerator (type and size)
- Quantity of entrained air by the aerator
- Form of the air supply mechanism (type and size)
- Spacing between the aerators
To establish the location of the aerator, firstly the area where cavitation is deemed to be possible
is the most likely primary parameter and is based on the cavitation index with a value of 0.2 or
smaller. Afterwards the flow characteristics, depth and velocity, at the cavitation region and the
curvature of the boundary should be taken in consideration (Khatsuria, 2005).
In tunnel spillways it’s hard to apply the above mentioned aerator types due to the provided
space in tunnels because of their shape and the depth of the tunnel. And if applied the size of
the aerators is usually limited because of the available space, but also because, in some
examples such as the Glen Canyon dam, the criteria of the trajectory not hitting the tunnel roof
to avoid blocking the flow is set (Khatsuria, 2005). Nevertheless, with careful and thorough
design they all can be used.
Grooves are usually used in tunnels since they make the air supply through air vents easier.
Grooves are commonly combined with either an offset or a deflector to make the aeration
mechanism more efficient (Volkart and Rutschmann, 1984). In this study, the discharge tunnel
12
at Zipingpu water conservancy has a shape of horseshoe and the installed aerator is as shown
in Figure 2.3.
Numerical method
In hydrodynamics the study of fluid and flow characteristics and dynamics is an important step
for the design of a hydraulic structure. To tackle the laws of physics and nature that surround
fluid dynamics different approaches have been taken. Amongst these approaches, there are the
practical and theoretical ones, where the first ones are dependent on experiments and the latter
on relations between nature and mathematical equations (Griebel et al., 1997). In recent
decades, thanks to the advancement in technology, a more efficient and cost-effective approach
has evolved, the numerical method and simulation. The numerical method is a complement to
mathematical analysis where complex equations of fluid dynamics cannot be solved. The
detailed explanations of the equations and the theory behind computational fluid dynamics
(CFD) will be discussed in chapter 3.
The numerical method is used to obtain an approximate numerical solution by discretization of
the differential equation governing fluid dynamics. The numerical method is part of the
computational fluid dynamics (CFD). CFD run methods have the capability of solving fluid
equations in 2D and 3D (Ferziger et al., 2020). For the discharge tunnel at Zipingpu Water
Conservancy project a 3D model will be simulated using CFD. The CFD software used for the
simulation of the fluid flow is a software called ANSYS Fluent developed by ANSYS.
13
3 Theory
Mathematical model
In Newtonian fluid flows, Navier-Stokes equations and the governing equations of fluid
mechanics are commonly used in mathematical modelling (Griebel et al., 1998). This is one of
the mathematical fundamentals for the Computational Fluid Dynamics modelling.
For centuries, scientists have tried to explain the mechanism of the fluids through physics and
mathematics. The Navier-Stokes equations describe the flow of Newtonian fluids accurately at
a mathematical level analysis. For very low Reynolds numbers and simple geometries, it is
possible to obtain complete explicit results (Wolfram, 2002).
These equations were derived from the conservation mass laws and momentum equations in
the 1840’s (Wolfram, 2002) to a system of nonlinear partial equations with independent
variables (Griebel et al., 1998). The equations include three different parameters described by
the following concepts (Griebel et al., 1998) :
● ��:Velocity field,
● 𝜌:pressure,
● 𝜚:density
The system of equations basically describes the advective-convective forces of the fluids
(velocity field) and the external forces (i.e. pressure and viscosity) that oppose resistance. The
term Re is the dimensionless Reynolds (Re) number, the 𝑔 term belongs to body forces like
gravity acting throughout the bulk of the fluid and Δ represents a gradient differential operator
(Griebel et al., 1998). The equations that are dimensionless can be given in their simplest form
(where 𝑑𝑖𝑣 �� is the divergence of velocity), as follows:
𝜕𝜌��
𝜕𝑡𝑑�� + (�� ∗ Δ)�� + Δ𝜌 =
1
𝑅𝑒∗ Δ𝜐 + 𝑔 → 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑑𝑖𝑣 ��: = 0 → 𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
On the other hand, the Navier-Stokes equations are not suitable for this project due to its
complexity regarding boundary conditions (model setup). Therefore, this mathematical model
has to be adapted parallelly with numerical methods in order to solve the CFD model in this
study case.
14
Discretization method (Finite Volume Method)
After choosing the mathematical model, a discretization approach is commonly used in order
to calculate a fluid flow. This process consists of approximation of the differential equations by
a system of algebraic equations (Ferziger et al., 2020). In this case study, ANSYS Fluent will
be used for CFD modelling and it is based on the Finite Volume Method (FVM) approach
(Jeong and Seong, 2014).
3.2.1 Finite Volume Method
The FVM subdivides into a finite number of control volumes (CVs) that makes it different from
the Finite Difference Method. The FVM uses the integral form of the conservation equations
as the starting point (Equation 3.3) (Ferziger et al., 2020). FVM solution depends of the integral
from with respect to the conservation equations (Chakraverty, 2019):
∫ 𝜌𝜙 ∗ 𝑛 𝑑𝑆𝑠
= ∫ ΓΔ𝜙 ∗ 𝑛 𝑑𝑆𝑠
+ ∫ 𝑞𝜙 𝑑𝑉𝑣
(𝐸𝑞. 3.3)
The surface integrals represent the convection (𝜌𝛷𝑉 ∗ 𝑛), diffusion (𝛤𝛥𝛷 ∗ 𝑛) and the flux
vector (𝑞𝜙 𝑑𝑉) in each CV face (Ferziger et al., 2020). However, the velocity and the fluid
properties are commonly known, but not the value for 𝛷 (i.e. dimensionless scalar value). In
this project, the Φ value will be calculated by using an interpolation method explained in the
next chapter. Also, in terms of transportation equations, an integration over the volume of a CV
is required (𝑞𝛷𝑑𝑉) (Ferziger et al., 2020). The converted expression (Equation 3.4) commonly
used in the FV method can be written as:
𝜕𝜌𝜙
𝜕𝑡+ ∑ 𝜌𝑓 ∗ ��𝑓 ∗ 𝜙𝑓 ∗ 𝑆�� = ∑ Γ𝜙𝑓 ∗ Δ𝜙𝑓 ∗ 𝑆��
𝑁 𝑓𝑎𝑐𝑒𝑠
𝑓
𝑁 𝑓𝑎𝑐𝑒𝑠
𝑓
+ S𝜙 ∗ 𝑉 (𝐸𝑞. 3.4)
One of the FVM approaches usually used for CFD to define CVs is by using a grid system
(meshing) and computational nodes at the centre of the CVs. On the other hand, the
computational node locations can be defined before the CVs (Ferziger et al., 2020). The choice
between both approaches depends on the geometry of the domain.
3.2.2 Discretization methods
Even though there are approximations to the integrals, an interpolation method is needed to
calculate the 𝛷 value. Usually, the commercial codes use different schemes, but also advice to
choose an appropriate method for a situation in specific. As mentioned before, ANSYS Fluent
is used to run the CFD model where the Upwind Interpolation (First order and Second Order)
15
and the Quadratic Upwind Interpolation (QUICK) methods can be used among others
(Hamberg and Dahlin, 2019).
When a first-order accuracy is required, the CV faces are determined by assuming that the nodal
center values represent a mean value throughout the CV (ANSYS, 2009). Thus, the value 𝛷f is
equal to the CV center value of 𝛷 in the upstream CV. On the other hand, in the second order
scheme the CV faces are calculated using a multidimensional linear reconstruction scheme.
Moreover, in the second order approach a higher accuracy is achieved at the CV faces through
an expansion of Taylor series of the nodal-centred solution on the nodal center (ANSYS, 2009).
Therefore, the face value 𝛷f is computed using the following expression (Eq. 3.5):
Φ𝑓 = Φ + ΔΦ ∗ 𝑟 (𝐸𝑞. 3.5)
where 𝑟 is the displacement vector from the centre of the upstream CV to the centroid of the
face. It is important to mention that the term ∇φ requires a formulation approach in order to be
solved. The formulations to determine the gradient of φ (∇φ) can be: Green-Gauss Cell-Based,
Green-Gauss Node-Based and Least Squares Cell-Based.
The QUICK scheme is usually run in quadrilateral and hexahedral meshes, where upstream and
downstream faces with unique characteristics can be identified. The QUICK method usually
tends to be more accurate on structured meshes aligned with the flow direction. Also, this
method is based on a weighted average of second-order upwind and central interpolations of
the variable (ANSYS, 2009).
For this project, the governing equations must be discretized in both space and time (ANSYS,
2009). The temporal discretization implies the integration of every differential equation over a
time step. The following expression represents the first-order discretization variable φ:
𝜙𝑛+1 − 𝜙𝑛
Δ𝑡= 𝐹(𝜙) (𝐸𝑞. 3.6)
While the second order discretization:
3𝜙𝑛+1 − 4𝜙𝑛 + 𝜙𝑛−1
2Δ𝑡= 𝐹(𝜙) (𝐸𝑞. 3.7)
Therefore, there are two approaches usually used in time discretization: Implicit and Explicit
Time Integration. Firstly, the implicit method is to test the function F in a future time step
(Equation 3.8), which can be solved by iterations at each time step before moving to the next
step.
𝜙𝑛+1 − Φ𝑛
Δ𝑡= 𝐹(𝜙𝑛+1) (𝐸𝑞. 3.8)
16
The term “implicit integration” is referred to a given cell 𝜙𝑛+1 that is related to 𝜙𝑛+1 in
neighbouring cells through F(𝜙𝑛+1) (Eq 3.9). The implicit equation is solved by performing
iterations at each time level moving to the next time step (ANSYS, 2009):
𝜙𝑛+1 = 𝜙𝑛 + Δ𝑡 ∗ 𝐹(𝜙𝑛+1) (𝐸𝑞. 3.9)
Secondly, the explicit method performs 𝐹(𝜙) at the current time level and it is available when
the model used the density-based solver. Thus, this method is not used in this project because
the usage of the pressure-based solver.
3.2.3 Pressure based solver
In order to solve the governing equations a flow solver must be used. There are two types of
solvers that can be used in ANSYS Fluent: the density or pressure-based solver. In this project
we are analysing a multiphase flow (gas-liquid), therefore the density-based solver can be
neglected due to the different densities between the fluids. Nevertheless, both approaches use a
similar discretization process (i.e. FVM), but when it comes to solve the discretized equations
the scheme is different (ANSYS, 2009).
In difference to the density-based solver, the pressure-based solver uses a solution algorithm
where the governing equations (i.e. mass conservation equations) are not linear but coupled to
one another. The process involves a series of iterations wherein the set of equations is solved
until the solution converges (ANSYS, 2009). The governing equations can be solved segregated
or coupled from the other equations.
In the segregated algorithm, each governing equation (e.g. u, v, T, 𝑘, 𝜖) is solved step by step,
one after another. Also, this method is memory-efficient, which means that the discretized
equations are stored in the memory once at a time. Therefore, the convergence solution is slower
than the coupled method (ANSYS, 2009). This algorithm solution (Figure 3.1) is illustrated in
the following diagram:
17
Figure 3.1 Conceptual model of coupled algorithm of the pressure-based solver (ANSYS,
2009)
The coupled algorithm approach solves coupled systems by integrating the momentum
equations and the pressure-based continuity equation. Besides, since the momentum equations
are solved, the rate of solution convergence shows a better performance than the segregated
approach, but its memory requirement increases by 1.5 to 2 times in comparison to the
segregated approach.
3.2.4 Reynolds Averaged Navier-Stokes
There are different numerical approaches to describe turbulent flows, but the Reynolds-
averaged Navier-Stokes (RANS) is a practical way which does not require as much
computational calculations as some others, such as the Large Eddies Simulations (LES). Firstly,
the RANS method was developed by Osborne Reynolds over a century where the governing
equations were averaged over the spatial volume, not time. Usually, the RANS is averaged over
the time (Ferziger et al., 2020).
18
In RANS, the starting point is the Reynolds decomposition of flow variables into mean and
fluctuating parts. Then, the Reynold decomposed variables are inserted in the Navier Stokes
equations, followed by an averaging of the equations involved (Alfonsi, 2009). For the velocity
components (ANSYS, 2009):
𝜐𝑖 = ��𝑖 + 𝜐′𝑖 (𝐸𝑞. 3.10)
where ��𝑖 and 𝜐′𝑖 are the mean and fluctuating velocity components. The same approach is used
for the scalar quantities:
𝜙𝑖 = ��𝑖 + 𝜙′𝑖 (𝐸𝑞. 3.11)
where 𝜙 denotes a scalar variable like pressure, energy, or concentration of species. Thus, after
substituting the expressions (number of the equations) into the continuity and momentum
equations (General Navier Stokes equations) and taking a time average yields the RANS
equations (ANSYS, 2009). They can be written as:
𝜕
𝜕𝑡𝜌 +
𝜕
𝜕𝑥𝑖
(𝜌𝜐𝑖) = 0 (𝐸𝑞. 3.12)
𝜕
𝜕𝑡(𝜌��𝑖) +
𝜕
𝜕𝑥𝑗(𝜌𝜐𝑖𝜐𝑗)
= −𝜕𝑝
𝜕𝑥𝑖+
𝜕
𝜕𝑥𝑗[𝜇 (
𝜕𝜐𝑖
𝜕𝑥𝑗+
𝜕𝜐𝑖
𝜕𝑥𝑖−
2
3𝛿𝑖𝑗
𝜕𝜐𝑖
𝜕𝑥𝑖)] +
𝜕𝜐𝑖
𝜕𝑥𝑗(−𝜐′
𝑖𝜐′𝑗
) (𝐸𝑞. 3.13)
Multiphase flow model
Often in engineering, the multiphase flow models are used for different applications such as
combustion systems. Multiphase flows with a phase change could be used to assess different
parameters such as cavitation, melting, solidification, boiling and condensation (Ferziger et al.,
2020). In the present model, a multiphase flow approach has to be chosen due to the water and
air being the main fluxes to test cavitation within the hydraulic structure (tunnel).
In order to solve any multiphase flow, as the present project, it is important to choose an
appropriate model. There are two numerical schemes for multiphase flow: the Euler-Lagrange
method and Euler-Euler method, but in this case, the second one (Euler-Euler method) was
chosen due to different reasons that are explained in this chapter (ANSYS, 2009).
In ANSYS Fluent, there are different Eulerian-Eulerian methods such as Volume of Fluids
(VOF) model, Mixture model, Eulerian Model. The VOF model was chosen mainly due to its
suitability for free-surface flows which is the case of this simulation. Moreover, VOF can only
be performed with the pressure-based solver.
19
Turbulence model
To solve and close the equations (RANS) we must include a turbulence model. Most of the
studied flows in engineering are turbulent and therefore require a different numerical approach
compared to laminar flows. This project has a turbulent flow containing velocity fluctuations
that mix transport properties such as momentum, energy, and species of concentration, but also
cause the transported properties to fluctuate (ANSYS, 2009).
For many years, the studies on turbulence models were merely experimental, but the costs and
time required for this approach were high in comparison with numerical simulations. Currently,
the most accurate method uses the direct numerical simulation (DNS) in which the Navier-
Stokes equations are solved to calculate the motions in a turbulent flow (Ferziger, 2020).
The DNS is also the simplest approach from the conceptual point of view. The obtained
information of a DNS contains detailed characteristics of the flow. In other words, the results
are like the experimental approaches and can be used to create statistical information or a
“numerical flow visualization”. This data can be used to acquire deeper knowledge of the
physics of the flow or to build a quantitative model.
3.4.1 𝒌 − 𝝐 Turbulent model
According to ANSYS (2009), the simplest turbulence simulations are given by the two-equation
models, where the solution can be independently solved in terms of the turbulent velocity and
length scales. The 𝑘 − 𝜖 builds up the length and time scales from the turbulent kinetic energy
𝑘 and the turbulent dissipation rate 𝜖 (Alfonsi, 2009). This model is suitable for 3D modelling.
The CFD software (ANSYS Fluent) used for this project includes three different turbulence
models: Standard, RNG and Realizable. The three models are similar, but there are some
differences explained herein in terms of:
• Calculating turbulent viscosity
• The Prandtl number that governs the turbulent diffusion
• The generation and destruction in the 𝜖 equation (ANSYS, 2009)
The Standard 𝑘 − 𝜖 method is the simplest one of the three mentioned and it is the one used for
this CFD model. This model assumes that the flow is fully turbulent, and the effects of
molecular viscosity are negligible (ANSYS, 2009). The components (𝑘, 𝜖) in the Standard
scheme are obtained from the following equations:
𝜕
𝜕𝑡(𝜌𝑘) +
𝜕
𝜕𝑥𝑖(𝜌𝑘𝜐𝑖) =
𝜕
𝜕𝑥𝑗[(𝜇 +
𝜇𝑡
𝜎𝑘)
𝜕𝑘
𝜕𝑥𝑗] + 𝐺𝑘 + 𝐺𝑏 − 𝜌𝜖 − 𝑌𝑀 + 𝑆𝑘 (𝐸𝑞. 3.14)
20
and
𝜕
𝜕𝑡(𝜌𝜖) +
𝜕
𝜕𝑥𝑖(𝜌𝜖𝜐𝑖) =
𝜕
𝜕𝑥𝑗[(𝜇 +
𝜇𝑡
𝜎𝜖)
𝜕𝜖
𝜕𝑥𝑗] + 𝐺1𝜖
𝜖
𝜅(𝐺𝑘 + 𝐺3𝜖𝐺𝑏) − 𝐶2𝜖𝜌
𝜖2
𝑘+ 𝑆𝜖(𝐸𝑞. 3.15)
where 𝐺𝑘 represents the generation of turbulent kinetic energy due to the mean velocity
gradients, Gb is the generation of turbulence due to buoyancy and 𝑌𝑀 represents the contribution
of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. The 𝐺𝜖’s
components are constants and the 𝜎𝜖and 𝜎𝑘 are the Pradntl number for 𝑘 and 𝝐, respectively.
Finally, 𝑆𝑘 and 𝑆𝜖 are user-defined terms (ANSYS, 2020).
In parallel, the turbulent viscosity 𝜇𝑡, is computed by the following expression:
𝜇𝑡 = 𝜌𝐶𝜇
𝑘2
𝜖 (𝐸𝑞. 3.16)
where 𝐶𝜇 is a constant (ANSYS, 2020).
Boundary conditions
3.5.1 Inlet and outlet
For FVM analysis, the boundary conditions require that the fluxes involved must be known in
terms of quantities and nodal values. CFD models have many boundary conditions that allows
the fluid(s) to enter and exit the flow domain. There are different boundary conditions that are
used for different CFD models, but for this project the following list shows the specific ones
for the case of the discharge tunnel of Zipingpu Water Conservancy project:
● Velocity inlet value is used to define the velocity and scalar properties of the flow at
inlet boundaries.
● Pressure inlet value is used to define the total pressure and other scalar quantities at flow
inlets.
● Outflow boundary values, used to model flow exits where the details of the flow velocity
and pressure are not known prior to solution of the flow problem (ANSYS, 2009).
3.5.2 Wall function boundary
In CFD models, the wall boundaries are used to bound fluid and solid regions. In ANSYS
Fluent, the no-slip function is given at walls by default, which was also the one chosen for this
project. Moreover, it indicates that the fluid sticks to the wall and moves with the same velocity
as the wall, only if it is moving.
21
According to ANSYS (2009), the following information is required for a wall boundary:
● Wall motion conditions (for moving or rotating walls)
● Shear conditions (only for slip walls, optional)
● Wall roughness (for turbulent flows, optional)
● Thermal boundary conditions (for heat transfer calculations)
● Discrete phase boundary conditions
● Wall adhesion contact angle
For this project, a stationary wall with No-slip shear condition are chosen because both comply
with the real conditions of the hydraulic structure (tunnel).
Meshing
As mentioned earlier the Navier-Stokes equation is hard to solve analytically due to its
complexity and therefore a numerical method will be used in this study. In solving a numerical
method, the geometry of the studied structure plays a crucial role in the choice of methods,
computation time and cost. Therefore, for the geometry a proper mesh/grid has to be applied in
order to smooth out the model computation and get accurate results.
3.6.1 Choice of mesh
The geometry is divided into subdomains in order to carry out the calculation and these
subdomains represent the mesh or grid that is chosen. Some meshing options are structured,
block-structured, and unstructured. Moreover, the beforehand chosen discretization method is
deciding for the shape of the mesh, e.g. if the algorithm in the discretization method is set to
simulate orthogonal grids then non-orthogonal grids cannot be used; or if the CV is needed to
be hexahedral or quadrilateral then triangles or tetrahedral cannot be used. To be noted is that
the accuracy and quality of the mesh is higher if the CV is hexahedral in 3D and quadrilateral
in 2D (Ferziger et al., 2020).
Depending on the geometry in hand, the mesh can be carried out differently. For instance, when
the geometry is simple it is relatively easy to choose a mesh type. But when the geometry
becomes more complicated different approaches can be carried out to ensure the accuracy and
quality of the mesh. Structured or block-structured meshing is compatible with simple
geometries, the problem with these techniques is that the mesh accuracy deteriorates near the
wall and therefore an overlapping scheme is needed. Overlapping grids give a better accuracy
and mesh quality in simulations. Moreover, the accuracy of the mesh can be obtained by having
smaller mesh sizes where the geometry and the fluid characteristics are complex. Nevertheless,
it is not advised to have small mesh sizes throughout the entire geometry domain, given the fact
22
that the smaller the mesh size the longer the computational time will be and as a result will
affect the cost (Ferziger et al., 2020).
When computed profiles of a certain variable are presented, it is recommended that numerical
uncertainty be indicated by error bars on the profile, analogous to the experimental uncertainty.
It is further recommended that this be done using the GCI in conjunction with an average value
of 𝑝 is equal to pave as a measure of the global order of accuracy.
3.6.2 Estimation of discretization error
As mentioned earlier a discretization method is needed in order to solve the differential equation
through approximations. These approximations commonly result in errors that are defined as
the difference between the solution of the discrete approximation and the solution of the
governing equation prior to the discretization (Ferziger et al., 2020).
The mesh size plays an important role in the weight of the obtained error in discretization. The
recommended method in estimating the discretization error is the GCI (Grid Convergence
Index) that is based on the RE method (Richardson Extrapolation) (Celik et al., 2008). After
having set the size of the different meshes, i.e. coarse-, medium- and fine mesh (h1, h2, h3), the
GCI is calculated according to the equations below. The apparent order p is calculated using
Equation 3.17, as follows:
𝑝 =1
ln(𝑟21)|𝑙𝑛 |
휀32
휀21|| + 𝑞(𝑝) (𝐸𝑞. 3.17.1)
𝑞(𝑝) = ln (𝑟21
𝑃 − 𝑠
𝑟32𝑃 − 𝑠
) (𝐸𝑞. 3.17.2)
𝑠 = 1 ∗ sgn (휀32
휀21) (𝐸𝑞. 3.17.3)
where r32 = h3 /h2, r21 = h2 /h1, r is the mesh ratio and it is preferred if the mesh refinement is
structured even though the mesh itself is unstructured, i.e. same decrease or increase between
the different mesh sizes. 휀32 = 𝜙3 - 𝜙2, 휀21 = 𝜙2 - 𝜙1 and 𝜙 is the dimensionless scalar value
obtained from the interpolation method, as mentioned earlier. Thereafter the relative error is
estimated using Equation 3.18 and the GCI is calculated using Equation 3.20.
𝑒𝑎21 = |
𝜙1 − 𝜙2
𝜙1| (𝐸𝑞. 3.18)
When the used mesh refinement ratio is 1 and it is constant other equations prevail. Therefore
instead of using Equation 3.17.2, a simplified version of the equation can be used to calculate
the order p for a three-grid solution (Eq. 3.19.1), even the equation for the relative error changes
(Eq. 3.19.2) (Wagner et al., 2002, Roache, 1998). These are calculated using the following:
23
𝑞(𝑝) =ln (
𝑓3 − 𝑓2
𝑓2 − 𝑓1)
ln(𝑟)(𝐸𝑞. 3.19.1)
𝑒𝑎21 = |
𝑓1 − 𝑓2
𝑓1| (𝐸𝑞. 3.19.2)
where f1, f2 and f3 are the different grid solution. The GCI for fine mesh is then obtained by
Equation 3.20,
𝐺𝐶𝐼𝑓𝑖𝑛𝑒21 =
1.25𝑟𝑎21
𝑟21𝑃 − 1
(𝐸𝑞. 3.20)
where 1.25 is a safety factor used for three-grid solution, but the value of the safety factor can
go up to 3, which is considered moderate and mostly recommended for two-grid solutions
(Roache, 1998; Wagner et al., 2000). The obtained GCI should be low in order to achieve a grid
independent result.
24
25
4 Methodology
In this chapter, the undertaken steps in the methodology in order to conclude the study are
represented. The steps will cover the building of the geometry, meshing scheme, model setup,
simulation, post-processing of the results, validation of the model and lastly evaluating different
scenarios of aerators and discharge rate dependent on the achieved results from the initial
simulations.
The ANSYS version 2019 R3 workbench offers different tools to perform the CFD model, from
the geometry construction to the post-processing stage. The used softwares in the process were
as follows:
- ANSYS Spaceclaim was used in the building of the geometry
- ANSYS mesh generator was used for meshing the geometry
- ANSYS Fluent was used for the simulation
- Excel was used to analyze the results from the simulations in order to post-process and
validate the results.
Geometry
The geometry was built as close as possible to the provided design drawings of the discharge
tunnel. Some details were neglected in order to simplify the meshing and simulation process, a
detailed illustration will follow. The design drawings of the tunnel are illustrated below, where
Figure 4.1 represents the side view of the entire tunnel excluding the tunnel inlet, Figure 4.2
represents the side view of the tunnel inlet and Figure 4.3 the plan view of the tunnel inlet.
In Figure 4.1, all the measurements for the elevations and length distances are given in meters.
Since in this figure the inlet is excluded the tunnel length starts at 0+28.793 m and ends at
0+583.00 m. After the end point of the tunnel, an extension of the tunnel is included as it is
seen in the figure. The side view of the inlet in Figure 4.2 has its measurements given in cm
and the elevations are given in m. As it is the start of the tunnel the length starts from 0+0.000
m and the connection to the rest of the tunnel through a floodgate at 0+028.793 m. The floodgate
is marked with a red circle since it was not included in the simulated geometry. The plan view
of the tunnel inlet which is represented in Figure 4.3 has its length measurements given in cm
and the red circle marks an extension that was excluded from the simulated geometry.
26
Figure 4.1. Side view of the entire tunnel.
27
Figure 4.2. Tunnel inlet side view
Figure 4.3. Tunnel inlet top view.
As mentioned earlier the simulated geometry was designed as identical to the provided design
drawings (Figure 4.1-4.3). Some detailed characteristics of the structure have been excluded,
such as the above mentioned floodgate in Figure 4.2 and the tunnel inlet extension in Figure
28
4.3 (marked with a red circle). Moreover, two aerators located at point 0+389.37 m and
0+545.97 m have been simplified by only considering the deflectors at the bottom and
neglecting those on the sidewalls. However, the gates located within the tunnel stretch at point
0+128.793 m and 8.5m to the right of 0+192.695 m have been included. Furthermore, the outlet
extension starting at point 0+583.00 m has been neglected in this study. The rest of the
specifications such as measurements and shapes were kept similar to the design drawings.
In designing the geometry ANSYS Spaceclaim was used and it is a program within the
workbench of ANSYS. The ANSYS Spaceclaim is a relatively simple software with an intuitive
and simple layout. To start with the build of the geometry a coordinate system, i.e. x,y and z,
was set on the plane. The x-coordinate was set to point the flow direction, i.e. horizontal plane;
the z-coordinate was set coming out of the plane and the y-coordinate along the vertical plane.
The straight and uniform shaped part of the tunnel, starting from point 0+241.27 m up to the
end of the tunnel, was the initial part of the geometry design. Hence this part has the same
shape, the horseshoe shaped cross section (3-3 in Figure 4.1) was drawn in 2D on the z-y plane.
The used software allows 2D drawn objects to become solids per automatic, therefore, by
changing to a 3D plane and using the “pull” function, the cross section was pulled according to
the length presented in the design drawings and a 3D geometry of the tunnel section was
created. In order to design the ramps for the aerators, the shape of the ramp was drawn in 2D
on the x-y plane and then pulled across the width of the tunnel to create a cut in the geometry.
Lastly, the tunnel section was inclined by using the rotate function to the right slope in
accordance with the design drawings.
For the next section of the tunnel, between 0+192.695 m and 0+241.27 m, in point 0+192.695
m a cross section (2-2 in Figure 4.1) was drawn in 2D on the z-y plane and then merged with
the first drawn tunnel section at point 0+241.27 m by using the “blend” function. The step for
the aerator was drawn the same way as described earlier. Afterwards, at point 0+156.264 m, the
next cross section (1-1 in Figure 4.1) was drawn and then merged to the rest of the tunnel. The
latter cross section (1-1) was then pulled towards point 0+99.256 m and the ramp and step for
the aerator were drawn as described earlier.
In the next section of the tunnel, between 0+28.793 m and 0+99.256 m, a sketch of the arching
on the top of the tunnel and of the bottom line of the tunnel was created in 2D on the x-y plane
in order to create a reference path to follow for the cross section when pulling it. When this
section of the tunnel was completed, the tunnel inlet was drawn following the same processes
that have been mentioned.
As a final step, the geometry was thoroughly checked for inaccuracies to avoid complications
when creating the mesh and simulations. Moreover, since the tunnel was drawn in different
sections where several CVs were created, it was checked that the tunnel sections were
connected to each other so that the whole tunnel behaves as a single section when running
simulations.
29
Mesh generation
In order to set mesh grids to the structure the mesh generator in ANSYS workbench was used.
Aspects such as the element number and mesh quality were the key points that were focused in
creating the mesh in order to succeed in the simulations. As illustrated in the previous chapter,
the geometry was executed in 3D, and therefore, the meshing of the geometry was carried out
in 3D also.
Since the geometry was divided in different sections, the CVs (control volume) were easily
meshed and specific mesh editing was possible per CV to achieve high mesh quality. The faces
of the CVs were not edited or meshed on their own since they followed the same meshing
scheme of the CV they belonged to. In accordance with the facts in chapter 3.6, hexahedral and
tetrahedral meshing schemes were used throughout the whole tunnel geometry. The parts of the
tunnel such as the downstream horizontal section of the tunnel and the CV for the upstream part
of the tunnel inlet were meshed with a hexahedral scheme. The inclined section of the tunnel
and the curve connecting the inclined and horizontal sections were meshed with a tetrahedral
scheme. The mixture of these meshing schemes was crucial to avoid high element numbers in
order to achieve an efficient simulation and to create a more realistic representation of the
geometry.
When the mesh quality such as skewness, orthogonal quality and element quality were checked,
the inlet for water and air pressure, outlet, walls and flow domain were specified using the name
selection function to facilitate the process of the boundary conditions setup in the numerical
model setup (Chapter 4.3). The first step and the deflectors at the DS (downstream horizontal
section) for the aerators were named as pressure inlets, the steps at points 0+128.793 m and
0+184.195 m were named as wall, the boundaries of the tunnel as walls and the internal part of
the tunnel as the flow domain, the first face as water inlet and the last one as outlet.
The grid size for both the hexahedral and tetrahedral meshes was kept the same in order to
maintain a node connectivity between the different CVs. Although, some mesh refinement was
conducted on the faces of the aerators and on the tunnel ceiling. The refinement on the ceiling
was done due to its curvature shape in terms of computing a uniform pressure scalar for the free
surface.
Numerical model setup
The numerical model setup was conducted using ANSYS Fluent model setup. The carried out
operations to set the model included the boundary conditions, governing equations, pressure
solver and turbulence model.
30
4.3.1 Boundary conditions
In setting up the boundary condition an idea of how the tunnel would behave had to be foreseen
in order to set them right. The initial boundary was set at the first face, which is the inlet, and
it was set to be the velocity inlet for the water, hence the initial flow conditions of the tunnel
were known. As mentioned earlier in chapter 4.2, named selections were set on the geometry
in the meshing process. The names that were set were then used as a guideline to set the right
boundary conditions for the pressure intakes. Therefore, the faces of the aerators and the
gateway for aeration right after the inlet were set as pressure inlets with the default atmospheric
pressure which was known beforehand. However, the faces for the aerators at points 0+128.793
m and 0+184.195 m were set as walls, and the pressure inlet was instead set on the side of the
step according to the dimensions of the vents in the design drawings. Moreover, the aerator at
point 0+28.793 m had a pressure inlet both at the face of the step and side air vents. The bottom,
side walls and ceiling were then set as walls. To be noted is that, it was given that the tunnel
does not run full of water, therefore there is a free surface of air in the ceiling. But to avoid
misconception in the boundary condition, the ceiling was not set as a pressure inlet, instead a
mesh refinement was used to facilitate that as mentioned in chapter 4.2. Since there was no data
provided of the flow at the outlet, the outlet face was set as a pressure outlet. The flow domain
and the internal faces of the CVs were set as internal since it was set beforehand that all the
CVs behave as a whole tunnel (see chapter 4.2).
4.3.2 Choice of solver
To initialize the simulations the settings regarding the governing equations were set to the
simplest ones that do not require high computational time and to be used as a starting point for
the different simulations to be conducted.
In the tunnel of this study the interaction between two phases is considered, air and water.
Therefore, a multiphase model was chosen with a VOF scheme and RANS approach. Since the
VOF model was used, the solver was set to be pressure-based and the formulation was chosen
to be explicit. For the turbulence model the standard k- 𝝐 model was used with the standard wall
function.
The solvers setup changed slightly in the different simulations that were conducted, e.g. for the
validation of the model.
Numerical convergency
After having set up the solver and the boundary conditions, the discretization methods to solve
the governing equations were set. As mentioned earlier, even in this step, the simplest
discretization methods that do not require a high computational time were used. Therefore, for
31
the solutions the gradient was set to be Green Gauss Cell-based and for the discretization of the
solutions the first order upwind scheme was chosen.
Afterwards a hybrid method was chosen for the initialization method. Moreover, it was set that
the initialization would start from the velocity inlet which was then patched in order to be
considered full of water, since ANSYS Fluent considers the heaviest phase as the secondary
phase which in this case was the water and therefore the tunnel is initially empty with only air
pressure in it which is considered the primary phase.
From the initial flow conditions that were procured beforehand, the inlet velocity was calculated
to be 12.86 m/s. This value was calculated by the ratio of the known discharge 1530.87 m3/s
for the design water level of 871.2 m and the inlet area of 119.05 m2 which was obtained through
the design drawings.
It was noticed that the model reached steady state after 3500s in the early stages of simulations
and therefore it was chosen to run the rest of the simulations with a time step of 4500s, with 15
iterations per time step and a time step size of 0.01. Some simulations were also run with time
step sizes of 1 and 0.001, but satisfactory results in regard to convergence and computational
time were not reached and therefore disregarded in this study.
The time of simulation varied from 12 to 15 hours depending on the type of turbulence model.
In some models the simulation was stopped since the reporting monitor of the water flow
presented a steady state without variations. At the same time, the monitors of the interactions
were monitored to assure the convergence of the numerical process.
Grid independence check
Results obtained from CFD simulations cannot be fully trusted, therefore, a grid independence
study is needed in order to evaluate if the results vary or depend on the mesh size. Normally
this is done through running simulations using different mesh sizes, e.g. coarse, medium and
fine, and analyzing the results which are then verified by calculating the GCI (see chapter 3.6.2)
to estimate which mesh size is more appropriate for the simulations. However, for this study,
due to the lack of time, the grid independence study was conducted by using a parametric
analysis method. In this method, only one simulation was run with a mesh size of 1.5 m and
then the parametric analysis function present in ANSYS workbench was used to set the
parameters in which their solutions output with different mesh sizes were compared. The chosen
parameters for the output solutions were the total pressure at the outlet, average velocity at the
outlet, the maximum velocity at the outlet and the relative flux difference (see Appendix 9.2).
32
Post-processing
In accordance to the set aims and objectives, the results that were compared to the provided
experimental data (see Appendix 9.1) were the static pressure and cavitation index in the post-
processing stage. The results procured from the simulation were exported to Excel and the
comparison carried out. The cavitation index was calculated according to the formula in chapter
2.4, where for the velocity value the velocities near the bottom boundary were exported from
the simulation and the vapor pressure was set to 2.3388 kPa for a water temperature of 20˚C.
The comparison of the results aimed to evaluate how the model was performing in comparison
to the experimental data, and moreover, to evaluate where changes in the geometry or model
setup are needed.
Model validation
After having chosen an appropriate mesh size from the grid independence study and post-
processed the results, a validation of the model was conducted by changing the solver setup and
comparing the results with the experimental data. In the validation of the model, the aim was
to find the best numerical model to use that fits the experimental data as close as possible. To
conduct this process, the turbulence model and wall function were the solver setups that were
changed. Therefore, the three k-ϵ models were combined with different wall functions to create
three model setups that were used for the validation. The three models are illustrated in Table
4.1:
Simulated model k-ϵ model Wall function
Model 1 Standard Standard
Model 2 RNG Standard
Model 3 Realizable Non-equilibrium
Table 4.1. Model setups for validation.
Evaluation of different scenarios
After having compared the results in the post-processing stage and having validated the results
using different models according to Table 4.1, the model setup that was deemed the best was
chosen to run a simulation with added pressure inlets on the inclined section. This process was
carried out in order to evaluate and compare how the tunnel behaves when different amounts
33
of air were put in it. The added pressure inlets were conducted using the boundary condition
function and were set on the faces of the aerators at points 0+128.793 m and 0+184.195 m,
which were set as walls before since the air was flowing through air vents instead.
Furthermore, another simulation was conducted by choosing the discharge rate of the check
level of 1666.74 m3/s. This discharge yielded a velocity at the inlet of 14.83 m/s which was
changed in the setup for the numerical solution. This simulation was aimed to evaluate how the
aerators behaved and if they were sufficient to handle high discharge rates in critical conditions
of strong floods.
To be noted is that both of these simulations were run with the same model setup as Model 1
(see Table 4.1), with a standard k-ϵ turbulence model and standard wall function. Table 4.2
illustrates the settings of the two scenarios.
Simulated
scenarios k-ϵ model Wall function Discharge rate Aerators
Model 4 Standard Standard 1530.87 m3/s Added
Model 5 Standard Standard 1666.74 m3/s Added
Table 4.2. Model setups for scenarios.
Aerator layout
In simulating the above mentioned models two aerator layouts were used. The layouts aimed to
facilitate a comparison of the fluids (gas-liquid) behaviour in the tunnel and to evaluate which
aerator layout yields better results that can mitigate cavitation. The amount of airflow produced
by the different aerators within the two layouts was also checked to evaluate the performance
of the aerators. Layout 1 consisted of a pressure inlet set on the aerators according to the design
drawings. This meant that the first aerator at point 0+28.793 m had both air vents and step
aerator set as pressure inlets, at points 0+128.793 m and 0+184.195 m it was only the air vents
that were set as pressure inlets and on the horizontal section of the tunnel it was the faces of the
deflectors that were set as pressure inlets. In layout 2 at points 0+128.793 m and 0+184.195 m
aerators were added by setting the steps as pressure inlets, while the other aerators remained
unchanged. Furthermore, layout 2 was analysed for the model with the higher discharge rate
(Model 5). In Table 4.3 the layouts and corresponding models are illustrated.
34
Layout Model 1 Model 2 Model 3 Model 4 Model 5
Layout 1 Yes Yes Yes No No
Layout 2 No No No Yes Yes
Table 4.3. Aerator layout and corresponding models.
35
5 Results
Tunnel spillway geometry
The final geometry of the tunnel that was used in this study is illustrated in Figure 5.1 and 5.2.
As mentioned earlier, the geometry followed the design drawings with negligible deviations.
As it can be seen in the figures below, the side wall aerators on the horizontal section of the
tunnel were excluded in this project.
The height of the tunnel at the point of connection between the curve and the horizontal section
was increased by 1m to achieve the desired and requested mesh quality by the program, such
as element quality, orthogonal quality, high aspect ratio and skewness. Moreover, the nodes of
the mesh in that point were not connecting properly since the grid cells from both control
volumes had different adjacent faces. To be noted is that the change in height at the connection
point between these CVs does not affect the flow properties in the tunnel. The rest of the
measurements of the tunnel remained unchanged to those in the design drawings.
Figure 5.1. The simulated geometry with the different CVs. View from the pressure outlet.
36
Figure 5.2. The simulated geometry with the different CVs. View from the water inlet.
Mesh
The mesh used for the simulations was mostly hexahedral in the horizontal section of the tunnel
and tetrahedral in the inclined section of the tunnel. The CV for the inlet was hexahedral. As
mentioned earlier a refinement on the ceiling was conducted with a min mesh size of 0.375m
whilst the rest of the geometry had a mesh size of 1.6 m. For this combination the total number
of elements resulted in 231675 (Appendix 9.2). The number of mesh elements in comparison
to the mesh size was high due to the scheme chosen for the tetrahedral mesh, which in this case
was a patch independent scheme. This scheme was implemented to achieve the needed mesh
quality in the flow domain of the inclined section of the tunnel. It is important to mention that,
there were models run without the refined mesh in the curvatures and proximities, but those
simulations were neglected since the results led to a tunnel totally full of water due to a poor
numerical interaction between both flows. The mesh distribution throughout the whole tunnel
is represented in Figure 5.3-5.5.
Figure 5.3 illustrates the meshing schemes of the inlet CV and the inclined section of the tunnel.
As it can be seen the inlet CV is hexahedral with rectangular and structured meshes, and the
inclined section is tetrahedral with triangular and unstructured meshes. The refinement on the
ceiling and the edges is also noticeable, moreover, the darker zones with fine refinement are the
positions of the air vents.
In previous simulations, the mesh refinement was not used in order to get less mesh elements,
but the final result showed a fluid domain full of water and a lack of air inflow through the
aerators which represented a source of error between the mesh and the discretization process.
Thus, as mentioned in the previous paragraph, after a deliberation with the supervisors and
37
according to Ferziger et al. (2020), an Adaptive Refinement Method (AMR) was used in
curvatures, proximities and air vents.
Figure 5.3. Inlet and inclined section meshing.
Figure 5.4 illustrates the hexahedral mesh of the horizontal section of the tunnel. However, the
CV connecting to the inclined section was made tetrahedral because of the complexity of the
geometry. Moreover, as mentioned earlier, the darker zone in the tetrahedral mesh is the
position of the air vents.
Figure 5.4. First half of the horizontal section meshing.
Figure 5.5 also represents the hexahedral mesh of the horizontal section. It can be seen that the
two small CVs right after the deflectors for the aeration are tetrahedral meshes because
inaccuracies and errors arose when connecting the mesh nodes of those CVs to the CVs holding
the deflectors for the aeration. Therefore, in order to facilitate a smooth transition and
connection between the CVs it was chosen to have tetrahedral meshes in the latter mentioned
CVs.
38
Figure 5.5. Second half of the horizontal section meshing.
Numerical simulation
By using CFD-Post tools, a graphical representation of three different parameters resulted after
running the simulation. These plots contain the final results from the numerical simulation. The
first two figures (5.6 and 5.7) represent the two fluxes, in this case, air and water with number
0 and 1 respectively. As it is noticeable, the air flow is working properly in the air vents within
the structure. Thus, there it is assumed that the air entrainment is occurring between the air and
water.
39
Figure 5.6. Volume of fraction water/air. Inlet and inclined section.
Figure 5.7. Volume of fraction water/air. Horizontal section.
40
The following plots (Figure 5.8- and 5.9) correspond to a centered plane (Axis z) along the x
axis representing the static pressure. It can easily be seen that most of the pressure is located at
the bottom of the tunnel. Therefore, the results reported in chapter 5.6 were taken from the
bottom zone.
Figure 5.8. Static pressure. Side view of the inlet and inclined section.
41
Figure 5.9. Static pressure. Side view of the horizontal section.
The following images (Figure 5.10 and Figure 5.11) represent a top view of the chute with the
distribution of the static pressure. The values below 31276.109 Pa show the area of the water
plunge occurring at the zone of aerators.
Figure 5.10. Static pressure. Bottom view of the inlet and inclined section.
42
Figure 5.11. Static pressure. Bottom view of the horizontal section.
The Figures 5.12 and 5.13 illustrate the behaviour of the velocity. The tunnel shows gradual
increment of velocity occurring along the shaft that can reach up to 45 m/s that represents a
high-speed flow, according to literature velocities above 20 m/s create cavitation problems
(Teng, 2019).
43
Figure 5.12. Velocity. Inlet and inclined section.
Figure 5.13. Velocity. Horizontal section.
44
Grid independence
As mentioned in chapter 4.5, the grid independence study was conducted using a parametric
analysis function. The parameters that were considered here are the average velocity, total
velocity and total pressure, all at the outlet, and the relative flux difference. The results are
illustrated in Figure 5.14-5.17 with a mesh size range of 1.5-3m. A finer mesh size, lower than
1.5m, was not possible to conduct in this study due to an exceeding mesh element number for
the finer meshes because of the patch independent scheme that was chosen for the tetrahedral
meshing method.
According to the graphs below the results can be both dependent and independent in regard to
which parameter is considered. It can be seen that for the average velocity the solutions are
mesh dependent while for the maximum velocity the solutions are mesh independent after a
mesh size of 2.2 m. The total pressure, Figure 5.16, does not show much variation and therefore
the solutions for it are mesh independent. However, to evaluate the other simulations that were
carried out in this study the chosen mesh size was 1.6 m. This choice was based on the results
from the relative flux difference in Figure 5.17, as it can be seen the mesh size of 1.6 m had the
smallest relative flux difference of -1%. A detailed table of the parametric analysis is provided
in Appendix 9.2.
Figure 5.14. Parametric analysis. Average velocity at the outlet.
Figure 5.15. Parametric analysis. Maximum velocity at the outlet.
45
Figure 5.16. Parametric analysis. Total pressure at the outlet.
Figure 5.17. Parametric analysis. Relative flux difference.
Data post-processing
When the grid independence study was concluded and a suitable mesh size chosen for the
simulation run with the simplest model setup, the results of the static pressure and cavitation
index of the simulation were compared with the experimental data. The compared results are
illustrated in Figure 5.18 and 5.19. From the figures below it is noticeable that the results from
the simulation although they follow a somewhat similar pattern to the experimental results, it
is hard to say that the results match. This difference is mostly noticeable in Figure 5.18 where
the result from the calculated data is quite linear in the horizontal section of the tunnel except
for the peaks that were recorded at the position of the aerators.
As for the cavitation index in Figure 5.19 the calculated data corresponds moderately to the
experimental data, except for some peak and low points that were recorded in the horizontal
section of the tunnel and the inclined section that has lower cavitation index values. From the
literature study it was learned that the lower the cavitation index the higher the risk of cavitation
damage becomes (Khatsuria, 2005). Therefore, it was assumed that a modification of the
aerators was needed.
46
Figure 5.18. Comparison of the static pressure between the experimental data and the calculated data.
Figure 5.19. Comparison of the cavitation index between the experimental data and the calculated
data.
47
Validation
With the solver setups illustrated in Table 4.1, three models were simulated in order to validate
them. As it is illustrated in Figure 5.20, for the static pressure the results from the different
models are similar to each other and it is hard to evaluate which model corresponds the most to
the experimental data. Even for the results of the cavitation index, Figure 5.21, all models
behave similarly and yield similar results. In comparison to the experimental data it can be seen
that some of the sections of the tunnel have closer values to the models and some differ by small
margins. Therefore, even in this case it was hard to evaluate which model corresponds best with
the results of the experimental data.
After having run the validation, since the results of the models were similar to each other, it
was concluded that the best way to run the simulations would be using the numerical model
setup of Model 1. This conclusion was drawn due to the fact that the setup with the standard k-
ϵ and standard wall function is the least expensive model compared to the other two model
setups.
Figure 5.20. Comparison of the static pressure between the experimental data and the three models.
48
Figure 5.21. Comparison of the cavitation index between the experimental data and the three models.
Results of different scenarios
According to Table 4.2 a model with a turbulence model of standard k-ϵ and standard wall
function with the two added aerators was compared to Model 1. The results are illustrated in
Figure 5.22-5.23 and show that the results for the static pressure and cavitation index for both
Model 1 and Model 4 overlap each other. Therefore, the scenario with the added aerators does
not change the fluids behaviour in the tunnel.
49
Figure 5.22. Comparison of the static pressure between Model 1 and Model 4.
Figure 5.23. Comparison of the cavitation index between Model 1 and Model 4.
50
Figure 5.24 and 5.25 illustrate the static pressure and cavitation index respectively of the
comparison conducted between Model 4 and Model 5 (see Table 4.2). Model 5 with the higher
discharge rate results in a slightly higher static pressure than Model 4 but lower cavitation index
due to a higher water velocity.
Figure 5.24. Comparison of the static pressure between Model 4 and Model 5.
Figure 5.25. Comparison of the cavitation index between Model 4 and Model 5.
51
Aerator layout
In Figure 5.26-5.28 are illustrated the aerator layouts that were used to run all five models. As
mentioned before at points 0+128.793 m and 0+184.195 m aerators were added by setting the
face of the step of the aerator as a pressure inlet, Figure 5.27 shows the positioning of those
aerators, while Figure 5.26 illustrates only the air vents at those points. Figure 5.28 illustrates
the aerator at point 0+28.793 m with both the air vents and the step aerator, and the layout of
the two similar aerators on the horizontal section of the tunnel. The red surfaces in the figures
represent the pressure inlets.
Figure 5.26. Aerator layout with air vents.
Figure 5.27. Aerator layout with the added aerators.
Figure 5.28. Aerator layout for the first aerator and the aerators on the horizontal section of the tunnel.
52
The airflow of the aerators for both layouts were analysed and are illustrated in Table 5.1 and
5.2 for Model 1 and Model 4 respectively (see Table 4.3). Furthermore, the ratio of airflow and
water flow that represents the effect of the aerator layout was analysed. It can be seen that both
layouts yield the same results, moreover, it is noticeable that the ratio is higher in aerator 3 and
4 where there is a higher flow velocity. Table 5.3 illustrates the airflow and the ratio of airflow
and water flow for the simulation run with the higher flow discharge (Model 5). The Qw that
was used to calculate the ratio in Table 5.1 and 5.2 was the known discharge rate of 1530.87
m3/s for the design water level of 871.2 m and in Table 5.3 the discharge rate of the check level
of 1666.74 m3/s.
Layout 1 Aerator 1 Aerator 2 Aerator 3 Aerator 4 Aerator 5 Total airflow
Airflow (Qa) 0.1757 0.20356 0.33194 0.31478 0.30441 1.33039
Qa/Qw (%) 0.0115% 0.0133% 0.0217% 0.0206% 0.0199% 0.0870%
Table 5.1. Summary of air flow and Qa/Qw for layout 1.
Layout 2 Aerator 1 Aerator 2 Aerator 3 Aerator 4 Aerator 5 Total airflow
Airflow (Qa) 0.17533 0.20109 0.34236 0.314775 0.30456 1.338115
Qa/Qw (%) 0.0115% 0.0131% 0.0224% 0.0206% 0.0199% 0.0875%
Table 5.2. Summary of air flow and Qa/Qw for layout 2.
Layout 2 Aerator 1 Aerator 2 Aerator 3 Aerator 4 Aerator 5 Total airflow
Airflow (Qa) 0.18858 0.20503 0.3720 0.33750 0.33216 1.43528
Qa/Qw (%) 0.0123% 0.0123% 0.0223% 0.0202% 0.0199% 0.0861%
Table 5.3. Summary of air flow and Qa/Qw for layout 2 with the model with higher flow discharge
(Model 5).
53
6 Conclusions and discussions
Geometry, mesh and grid independence
One of the aims of this study was to analyse whether a numerical model is preferable to a
physical model. After having established a numerical model and ran all the needed simulations
with their respective evaluations and taken in consideration the computational time of the
numerical model, it was concluded that a numerical study would indeed be preferable. This
conclusion was based on the fact that scale models have to be structured and built, which means
that the need for an experimental space, materials and resources were deemed costly and time
demanding. Another aspect that was considered is the flexibility of a scale model in
experimenting and evaluating a variety of flow conditions and boundary conditions. Even
though the above mentioned cases were all made effortless with a numerical model, a scale
model was still useful in having comparative data in order to validate the numerical model.
Nevertheless, even if numerical models were deemed preferable in this study due to the
flexibility of running simulation with a variety of conditions, yet they present their own
complications. The complications that were noticed in this study were concerning the
complexity of the geometry and the simplifications that were needed to be done in order to
obtain an acceptable and functioning mesh to run the simulations. The dependency of the
meshing schemes on the geometry is an aspect that was managed carefully. As a result, it was
observed that the computational time and accuracy of the numerical model was dependent on
the used meshing methods and the quality it yielded.
As it can be seen in chapter 5.1 the geometry was modelled similar to the design drawings with
some simplifications that were mentioned. The tunnel has complex geometry with changing
slopes and different cross section shapes that resulted in varying curvatures on the ceiling. As
mentioned earlier, since the geometry would affect the accuracy of the desired mesh, a mesh
refinement was added to the curvatures, proximities and air vents to achieve better quality. The
patch independent scheme that was used to achieve the refinement and good quality in the flow
domain gave a very high number of elements in proportion to the mesh size of 1.6 m that was
used for the simulations. As a result, a finer mesh could not be conducted.
The latter affected the grid independence check as it was limited to a range of mesh sizes of
1.5-3 m (see Appendix 9.2). The grid independence check conducted through a parametric
analysis gave varying results depending on which parameter was considered. Therefore, to
evaluate which mesh size was most suitable for the study only the parameter for relative flux
difference was used (Figure 5.17). In comparison to the other parameters that were used,
average velocity, maximum velocity and total pressure, the relative flux difference was deemed
more certain and suitable for the purpose of conducting a grid independence check.
54
Comparison of models
From Figures 5.18, the VOF model 1 showed a similar pattern in comparison to the
experimental data. The highest pressures were reached along the inclined shaft where the water
was flowing at high velocities. It was assumed that there was a source of error since the
experimental information were only 56 data points obtained along the tunnel (Appendix 9.1),
while, in the CFD post processing the data was obtained from more data points (730) since it
was set a centered polyline along the tunnel. This could affect to a certain extent the recognition
of patterns or more clear similarities on the graphs obtained from the computational model
compared to the experimental data.
According to Falvey (1990), for cavitation index below 1.8 structures develop cavitation or
super-cavitation. As it could be seen from the data, for both cases (experimental and
computational data) the numbers have shown that the structure is within the range of present
cavitation problems with a water discharge of 1530 m3/s of the design level 871.2 m.
The three different turbulence models yielded similar results. It means that there are not
significant variations among the models. The results from cavitation index and static pressure
presented values with small differences amongst the three models (Model 1-3). Although
ANSYS Fluent software recommended through an interactive dialog to use RNG model, the
validation had shown that the difference between extracted data would not make any difference
when it comes to the cavitation index analysis. Therefore, the 𝑘 − 𝜖 standard model with
standard wall function was used for further simulations since it was the least computational
expensive.
Evaluation of flow scenarios
The scenario of water discharges of 1530 m/s and 1667 m/s were analysed by locating patterns
and analysing the cavitation index, but also other parameters such as the velocity and the
volume of fraction. It was noticed that in the scenario of model 5 (1667 m/s) the water plunge
was higher, and this can be seen on Figure 5.24 at the longitudinal direction of 86 m. Thus, it
meant that with higher discharges the concentration of static pressure will be higher due to the
plunge and the air inflow through the air vents or aerators.
The static pressure showed increases in other zones, but this is due to the higher velocity that
lead to a higher sucking effect in the air vents and aerators. However, although the static
pressure is a product of the air entrainment through the water that acts like a shield to avoid
cavitation of the spillway (Falvey, 1990), the cavitation index, in both cases, indicated numbers
below 1.8. Thus, there will be cavitation problems for both scenarios even with the added
aerators (Model 4 and 5, Table 4.2).
55
Aerator layout behaviour
According to Falvey (1990), one of the recommendations to improve the cavitation number is
to change the aerator configuration (layout) to increase the intrusion of air in the water. The air
concentration in Table 5.1 (Qa/Qw) did not even achieve 1%, which means that it is much lower
than the recommended values according to Teng (2019), where it was mentioned that 1.5% -
2.5% of air concentration would reduce significantly cavitation damages and that 7% - 8%
would be suitable to not have cavitation.
According to Yang et al. (2019) aerator air flow conditions could be affected by the air-vent
layouts. Therefore, by adding aerators, it was expected to see substantial changes in the air
inflow, but almost nule changes were obtained (see Table 5.2 and 5.3). Thus, this represented
a justification to assume that there was a geometrical problem according to the design
improvement recommendations (Falvey, 1990).
Source of errors
There are different facts that could have affected the accuracy of the results. The mesh quality,
as mentioned before, could have been perfected. It was preferable to have inflated cells along
the bottom of the tunnel to avoid inaccuracies or low orthogonal quality, but the mesh generator
could not record that command order during this project. Therefore, there could be a range of
errors within the tetrahedral zones of the fluid domain.
Another important fact is the velocity extracted from ANSYS CFD-post. When it comes to
turbulence flows the velocity becomes irregular due to the eddies and flow changes. The spatial
resolution of the nodes can affect the accuracy of the velocity since it is a result of an ensemble
averaged method.
56
57
7 Recommendations
From the results obtained in this study it was verified that there are design problems regarding
cavitation. As it was recorded, the results for the cavitation index in the tunnel for this study
were low and therefore a high cavitation risk. Therefore, in order to optimize the discharge
tunnel or other tunnels of similar conditions it is advised, according to this study, that design
improvements are needed to minimize the risk of cavitation. The modification can consist of
adding more aerator devices to the structure, such as deflectors with or without ramps on the
walls and air vents under the deflectors.
In addition, another solution would be to use a special concrete to avoid cavitation problems,
this concrete should have high resistance to shear forces (500 – 600 kg/cm2) and density fillers
in order to mitigate cavitation incidents (Arutyunov and Gomolko, 1967).
Furthermore, further studies on the discharge tunnel are recommended to validate the reliability
of the results that were obtained in this study. The further studies should include finer mesh
sizes, different initial conditions, boundary conditions, implement a discretization method of a
higher degree such as the second order upwind interpolation. Moreover, various combinations
of model setup, beyond those used in this study (see Table 4.1), should also be evaluated in
order to determine which analytical setups are most suitable to this or similar case studies.
58
59
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61
9 Appendix
9.1 Experimental data
The experimental data was used in this study was obtained from a similar thesis by Hamberg
M. and Dahlin S. (2019). The data is illustrated in Table 9.1.
Table 9.1. Static pressure and cavitation index measured in the tunnel for the design level of
871.2 m.
Longitudinal direction (x) Static pressure (Pa) Cavitation index
0 415460 1,69
4,186 337580 1,31
8,358 273010 0,95
12,572 194420 0,59
16,786 117010 0,39
20,986 57130 0,26
23,193 105960 0,22
89,765 105960 0,35
94,171 81680 0,41
100,752 60230 0,33
105,502 71940 0,31
110,675 64000 0,27
120,335 38260 0,23
129,284 42000 0,24
132,791 -4950 0,23
147,618 124240 0,18
62
150,802 124330 0,18
153,987 108470 0,28
157,53 85850 0,28
164,221 146540 0,25
174,023 157750 0,22
183,64 159860 0,29
193,434 123880 0,3
231,66 141850 0,3
232,64 133870 0,26
234,6 95240 0,19
235,58 87050 0,34
236,56 73030 0,54
237,54 60530 0,38
238,52 52220 0,29
239,5 59850 0,2
240,48 52180 0,29
241,26 46440 0,19
242,04 58200 0,32
243,44 50530 0,2
246,24 10480 0,24
251,14 16080 0,27
256,04 53640 0,25
280,54 5680 0,37
305,824 66530 0,3
329,54 9120 0,3
354,068 28290 0,26
63
378,54 44800 0,22
416,77 -1490 0,29
427,54 67210 0,24
430,34 59820 0,31
433,14 50930 0,44
435,94 52680 0,24
438,74 50050 0,41
452,04 35680 0,33
476,54 16260 0,34
501,04 49090 0,35
525,54 24530 0,32
550,124 55330 0,32
574,624 91790 0,29
599,152 11670 0,31
64
9.2 Parametric analysis
Table 9.2. The parameters and results obtained from the parametric analysis to conduct the grid
independence study.
#
Mesh
elem
ent
size
[m]
Patch
indepen
dent -
Max
element
size [m]
Patch
indepen
dent -
min size
limit
[m]
Mes
h
nod
es
Mesh
eleme
nts
Avera
ge
velocit
y [m
s^-1]
Total
pressu
re [Pa]
Max.
velocit
y [m
s^-1]
Water
flow -
out
[kg s^-
1]
Water
flow -
in [kg
s^-1]
Relativ
e flux
differe
nce
1 1,5 1,5 0,375 611
93
23177
3
31,97
133
26196
7,4
39,932
098
-
139273
5,6
144553
2,2 -4%
2 1,6 1,6 0,375 601
54
23167
5
32,08
650
23012
0,8
40,073
059
-
143814
4,4
144553
2,2 -1%
3 1,7 1,7 0,375 550
24
22300
4
31,21
826
21442
3,1
39,446
579
-
132302
9,3
144553
2,2 -9%
4 1,8 1,8 0,375 533
13
22002
6
30,09
764
21925
8,0
39,508
545
-
169381
9,0
144553
2,2 15%
5 1,9 1,9 0,375 524
34
22022
7
30,25
664
22750
2,7
39,568
584
-
171925
8,5
144553
2,2 16%
6 2 2 0,375 517
63
21875
0
30,59
330
21689
0,3
39,236
908
-
152925
4,4
144553
2,2 5%
7 2,1 2,1 0,375 507
67
21784
2
30,67
390
25440
9,6
39,493
282
-
166843
1,3
144553
2,2 13%
8 2,2 2,2 0,375 505
53
21728
2
30,67
681
21651
0,6
39,371
639
-
161983
0,7
144553
2,2 11%
9 2,3 2,3 0,375 505
22
21702
2
29,80
775
21178
1
39,541
626
-
165346
1,2
144553
2,2 13%
1
0 2,4 2,4 0,375
501
43
21724
6
30,53
720
19231
3,2
39,566
483
-
159888
2,9
144553
2,2 10%
1
1 2,5 2,5 0,375
500
10
21705
3
30,19
709
21173
1,7
39,476
551
-
180348
9,9
144553
2,2 20%
65
1
2 2,6 2,6 0,375
494
49
21561
4
30,31
785
19104
0,5
39,445
286
-
175262
4,8
144553
2,2 18%
1
3 2,7 2,7 0,375
500
87
21733
0
30,11
772
19598
1,5
39,461
445
-
166598
6,4
144553
2,2 13%
1
4 2,8 2,8 0,375
493
65
21569
6
30,02
991
18341
0,8
39,461
578
-
164905
3,2
144553
2,2 12%
1
5 2,9 2,9 0,375
441
49
18992
0
31,51
261
23692
9,5
39,450
481
-
174129
1,1
144553
2,2 17%
1
6 3 3 0,375
443
55
19088
5
30,05
74
20175
8,5
39,428
215
-
174016
6,4
144553
2,2 17%
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