UPTEC ES 20032

Examensarbete 30 hpAugusti 2020

Computational fluid dynamics calculations of a spillway’s energy dissipation

Robin Lindstens

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Computational fluid dynamics calculations of aspillway’s energy dissipation

Robin Lindstens

To make sure that a dam is safe it is important to have good knowledge about the energy dissipation in the spillway. Physical hydraulic model tests are reliable when investigating how the water flow behaves on its way through the spillway. The problem with physical model testing is that it is both expensive and time consuming, therefore computational fluid dynamics, CFD, is a more feasible option. This projects focuses on a spillway located in Sweden that Vattenfall R&D built a physical model of to simulate the water discharge and evaluate the energy dissipation in order to rebuild the actual spillway. The main purpose of this project is to evaluate if the physical hydraulic test results can be reproduced by using CFD, and obtain detailed results about the flow that could not be obtained by physical testing.

There are several steps that need to be completed to create a CFD-model. The first step is to create a geometry, then the geometry needs to be meshed. After the meshing the boundary conditions need to be set and the different models, multiphase model and the viscous model, need to be defined. Next step is to set the operating conditions and decide which solution method that will be used. Then the simulation can be run and the results can get extracted. In this project two CFD simulations were performed. The first simulation was to be compared with the physical hydraulic model test results and the second CFD simulation was of the rebuilt spillway. The results proved that the physical model test results could be recreated by using CFD. It also gave a better understanding of how the energy dissipation was in the spillway and indicates that the reconstruction of the actual spillway was successful since the new spillway both had a higher water discharge capacity and better energy dissipation.

Tryckt av: UppsalaISSN: 1650-8300, UPTEC ES** ***Examinator: Petra JönssonÄmnesgranskare: Per NorrlundHandledare: James Yang

Popularvetenskaplig sammanfattning

Nar ett vattenkraftverk ska byggas ar det manga aspekter som maste beaktas.Forutom sjalva dammen maste aven utskovet undersokas for att sakerstallaatt vattenkraftverket ar sakert. Om utskovet inte kan slappa ut ett tillrackligtstort vattenflode riskerar dammen att skadas over vid stora vattenfloden, tillexempel vid varfloden. Darfor ar det viktigt att standigt undersoka om dedammar som byggdes under 1900-talet kan hantera stora vattenfloden pa ettsakert satt. Skulle ett utskov behova byggas ut ar det aven nodvandigt attsakerstalla att energiomvandlingen i utskovskanalen fungerar som den ska.Om vattenflodet har for hog energi, i form av hastighet och turbulens, arrisken stor att bade utskovskanalen och floden nedanfor kanalen eroderar.Detta kan i sin tur leda till att dammen blir instabil.

Ar 2013 borjade Vattenfall R&D undersoka en damm och dess utskov foratt sakerstalla att dammen kan hantera oversvamningar. Genom att byggafysiska modeller av dammen kunde de genomfora matningar for att ta redapa vad som behovde andras for att dammen skulle vara saker. De kom framtill att det vore fordelaktigt att bygga ut utskovet for att kunna slappa mervatten men aven gora forandringar i utskovet for att forbattra energiomvan-dlingen. De forandringar som genomfordes var att bygga en energiomvand-lare i form av en bassang precis efter utskovsluckorna samt bygga atta ribbori utskovskanalen.

Syftet med det har projektet var att undersoka om de fysiska testresultatenkunde aterskapas med hjalp av computational fluid dynamics, CFD. Projek-tet skulle aven forsoka fa ut mer detaljerad information om vattenflodet anvad som var mojligt vid de fysiska testerna. Forst stalldes en geometri uppover utskovet, sedan delades geometrin in i sma delar som ar nodvandiga vidberakningen av flodet. Darefter stalldes randvillkoren upp och till sist kundesimuleringen starta. Det kordes tva simuleringar, en med energiomvandlare,ribbor och samma bredd pa utskovet som det var innan det byggdes om ochen med energiomvandlare, ribbor och breddat utskov.

Resultatet visade att CFD-simuleringarna kunde aterskapa resultaten frande fysiska matningarna pa ett trovardigt satt. Resultatet fran simuleringenmed breddat utskov indikerade att utslappskapaciteten okat samt att ener-giomvandlingen fungerade battre an tidigare.

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Executive summary

It is vital to have a good understanding of the flow and energy dissipation ina spillway. To increase the knowledge about a specific spillway it is possibleto perform physical hydraulic model tests. These test have proven to besuccessful and reliable. The only problem is that physical testing is timeconsuming and expensive since a physical model of the spillway has to bebuilt and the model often needs to be modified for different tests. Anotherpossibility is to perform the tests with computational fluid dynamics, CFD.When using CFD it is possible to make more detailed simulations of the flowthrough the spillway without having to build a physical model.

This project focuses on a spillway in Sweden that Vattenfall R&D tested bybuilding a physical hydraulic model at Alvkarleby in 2013. The test resultsresulted in a reconstruction of the actual spillway by adding eight ribs and astilling basin for better energy dissipation. The spillway was also widened by1.5 m. The purpose of this project was to evaluate if the physical hydraulictest results could be reproduced by using CFD and investigate if CFD couldproduce more detailed results concerning the energy dissipation and waterflow.

The results shows that CFD could correctly reproduce the physical tests andthat the energy dissipation got better after the reconstruction of the spillway.The water velocities leaving the spillway channel got reduced and thereforethe erosion of the downstream river is reduced. To further investigate thespillways energy dissipation it would be recommended to perform CFD simu-lations that simulates the flow with one spillway gate opened at a time. Thiswould give a better understanding of how the different spillway gates affectthe water flow. It is also recommended to investigate how the flow behavesafter it leaves the spillway channel. This would give a better understandingof the width of the erosion downstream the channel.

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Acknowledgements

This master thesis marks the end of my five year long education in energysystem at Uppsala University and Sveriges lantbruksuniversitet. The projecthas been performed in cooperation with Vattenfall R&D at Alvkarleby.

I would like to thank my supervisor, James Yang, for providing me with thisthesis work and for his technical guidance through the work. I would alsolike to thank my other supervisor, Bengt Hemstrom, for his support in thesetup of a computer and guiding me through the CFD process. Finally Iwould like to thank my subject reviewer Per Norrlund.

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Contents

1 Introduction 1

1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Problem description . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Purpose and goals . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theory 8

2.1 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Stilling basin and a surface-roughened spillway chute . 8

2.2 CFD theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Reynolds-Averaged Navier-Stokes (RANS) . . . . . . . 12

2.2.2 Boussinesq approximation . . . . . . . . . . . . . . . . 12

2.2.3 Turbulence models . . . . . . . . . . . . . . . . . . . . 13

2.2.4 Standard k-ε model . . . . . . . . . . . . . . . . . . . . 14

2.2.5 Realizable k-ε model . . . . . . . . . . . . . . . . . . . 15

2.2.6 RNG k-ε model . . . . . . . . . . . . . . . . . . . . . . 15

2.2.7 Boundary conditions . . . . . . . . . . . . . . . . . . . 16

2.2.8 Multiphase flow . . . . . . . . . . . . . . . . . . . . . . 17

2.2.9 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Methodology 19

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3.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Generation of the mesh . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 CFD-Post . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Validation of the CFD-model 26

4.1 Volume flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Flow behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 Water velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Water level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Results and discussion 35

5.1 Comparison of volume flow . . . . . . . . . . . . . . . . . . . . 35

5.2 Comparison of water velocities . . . . . . . . . . . . . . . . . . 36

5.3 Comparison of water level . . . . . . . . . . . . . . . . . . . . 39

5.4 Comparison of the flow behaviour . . . . . . . . . . . . . . . . 41

5.5 Sources of error . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Conclusions 43

Referenser 43

v

List of Acronyms

CAD - Computer aided design

CFD - Computational fluid dynamics

DNS - Direct numerical simulation

m.a.s.l - Meters above sea level

RANS - Reynolds-Averaged Navier-Stokes

RNG - Re-Normalization Group

RSM - Reynolds stress model

SST - Shear stress transport

VOF - Volume of fluid

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1 Introduction

1.1 Terminology

To be able to understand this report it is important to be familiar with thehydropower terminology. In figure 1 it is possible to see the spillway gates,stilling basin, spillway chute and the ribs. Another thing that is good toknow is that when referring to left and right when talking about hydropowerit is from the up stream perspective.

Figure 1: The physical hydraulic model which has been improved by addingeight ribs in the spillway chute [3].

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1.2 Background

Hydropower has been an important part of the Swedish energy system sincethe beginning of the 20th century. The construction of new hydropowerplants reached its peak during the years 1950-1960 and today hydropowerstands for about 40 % of Sweden’s total electricity generation. During theseyears the restrictions and laws concerning the safety of the dams was notas strict as today and the ones constructing the dams were considered re-sponsible. To determine the size of the design flood, the highest historicalflood was multiplied with a safety factor. Since then it has been proven thatthese estimations of design floods were incorrect and in the 1980’s severeflooding further proved the design flood to be inadequate. After the floodinga committee called Flodeskommitten was established to investigate the de-sign floods and produce new guidelines. In 1990 the guidelines were releasedand the hydropower companies in Sweden agreed to follow the guidelines andhave since then been working with improving the spillways to make sure thatthe dams can handle flooding in a safe manner.[1]

One of the best ways to evaluate if a dam and its spillway are fulfilling theguidelines is by performing physical hydraulic model tests. Vattenfall R&Dat Alvkarleby has evaluated over 40 dams since the new guidelines werereleased [2]. To further evaluate the dams and spillways, it is possible to usecomputational fluid dynamics, CFD, which simulates the flow of both waterand air in the spillway. By comparing the results from the hydraulic modelsand the results given by CFD it is possible to get a better understandingof the flow. The statistics from the physical hydraulic tests shows that thenew design floods are 20-50 % higher than the dams originally were designedfor [2]. To solve this, the spillways either must be modified or in somecases rebuilt. When rebuilding the spillway, it is important to consider thestructural stability. By increasing the spillway capacity the mass flow willget larger and the the risk for erosion might increase which could endangerthe dam. Therefore it is important to have knowledge about the flow throughthe spillway [2].

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1.3 Problem description

A couple of years ago, a physical hydraulic model of the dam in questionwas built at Vattenfall R&D, Alvkarleby. The purpose of the physical modelwas to simulate the water discharge in the spillway and evaluate the energydissipation to investigate how to rebuild the actual spillway. The physicalhydraulic model tests showed that the energy dissipation in the spillway chutehad an uneven flow which caused a standing cross wave and a large zone offlow circulation. The main reason for this uneven flow is the asymmetry ofthe seven spillway gates. The asymmetry of the spillway gates is illustratedin figure 2. As can be seen in figure 2, spillway gates number 1 and 2 areplaced at the side of the spillway since there is limited space. The flow thatcomes through gate 1 and 2 will take a sharp turn which causes the flow tobecome more turbulent and it will affect the rest of the flow. This puts ahigher demand in the energy dissipation of the spillway. [3]

Figure 2: The seven spillway gates before reconstruction [3].

The standing cross wave and the large zone of circulation can be seen infigure 3. The standing cross wave can be seen in the spillway chute and thezone of circulation is on the left side of the stilling basin.

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Figure 3: The physical hydraulic model with a standing cross wave markedwith red and a large zone of circulation marked by the black circle [3].

The simulations showed that a good way to improve the energy dissipationin the spillway was by installing a stilling basin and eight ribs in the spillwaychute. In figure 4 it is possible to see how the ribs affected the flow. [3]

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Figure 4: The physical hydraulic model which has been improved by addingeight ribs in the spillway chute [3].

The tests that were performed focused mainly on the discharge capacity fordifferent water level, the water velocities in different sections of the spillwayand the water level along both sides of the spillway chute. The reason forevaluating the water level along the walls of the spillway was mainly to makesure that the stilling basin would be long enough to contain the hydraulicjump that occurs in a stilling basin. By investigating the water velocitiesin different sections of the spillway chute it is possible to see if the water isslowing down before reaching the end of the chute. If the water reaches theend of the chute with too high velocity it might cause erosion in the riverdownstream of the spillway. By adding eight ribs in the spillway chute theflow got more evenly distributed and the standing cross wave disappeared butthe large zone of circulation remained. After the physical hydraulic modeltests had been performed the real spillway was reconstructed by wideningspillway gate 7, which can be seen in figure 2, from 5 m to 6.5 m. The other6 spillway gates kept their width at 5 m. The stilling basin and spillwaychute were also widened by 1.5 m to fit with the spillway gates.

After the decision to increase the width of spillway gate number 7 a physicalhydraulic model of the dam was built and some tests were performed. Thesetests focused mainly on the discharge capacity and comparing the differentgates by closing six of the seven gates and measuring the discharge capacity

5

for different water levels for all of the gates individually opened. Figure 5shows how the physical hydraulic model looked like after spillway gate 7 andthe wall to the right had been widened by 1.5 m. [3]

Figure 5: The physical model with the reconstructed spillway gate and wallto the right [3].

One of the disadvantages with hydraulic model tests is that it is hard to ob-tain detailed information about the flow, therefore there is a need to simulatethe flow by using CFD. [3]

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1.4 Purpose and goals

The main purpose of this project is to evaluate if the physical hydraulic testresults can be reproduced by using computational fluid dynamics, CFD, andobtain detailed results of the flow that could not be obtained in the hydraulicmodel tests. The software used to simulate the flow in this project wasANSYS Fluent 2019r2. By modeling the flow with CFD it is possible to geta better understanding of the flow pattern, flow circulation and outflow fromthe stilling basin. This information will be helpful when trying to understandthe safety aspects in release of extreme spillway floods. This purpose can bedivided into several part goals

• Model the flow behaviours through the spillway openings, in the thestilling basin and in the spillway channel.

• Analyze the CFD results.

• Perform a qualitative comparison with the hydraulic model test results.

1.5 Limitations

To be able to finish the project within the time frame it is important to havesome limitations. This project will not model the flow downstream of thespillway chute. The project will only include how the flow behaves from thepoint where it enters the spillway gates until it leaves the spillway chute. Theproject will also not test all the different gates separately since the calculationwhen using CFD often takes many days to complete. Therefore the focuswill be on testing the flow discharge, water levels and water velocities in thespillway for both the case with seven gates that are all 5 m wide and for thecase where spillway gate 7 is 6.5 m wide and the rest of the gates are 5 mwide. All gates will be fully opened in the simulations.

Another limit was the amount of cells in the mesh. This limit was set to1000 000 and only one turbulence model was used during the simulationssince the time limit is to short to simulate with more turbulence models.

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2 Theory

This section will explain the theory that is necessary to understand both theenergy dissipation and how CFD calculates the different characteristics ofthe flow.

2.1 Energy dissipation

It is crucial to have a good energy dissipation in a spillway to avoid erosionof the downstream river and the spillway chute. If the spillway gets damagedand needs to be shut down it might have dire consequences for the dam. Ifthe downstream river erodes it might affect the stability of both the spillwayand the dam. There are many different options to choose between whenintegrating energy dissipation in a spillway. The dam in this project hasboth a stilling basin and a surface roughened spillway chute in the form ofeight ribs.

2.1.1 Stilling basin and a surface-roughened spillway chute

Stilling basins can be used for both flows in pipes and open channel flows.Since this project focuses on an open channel the theory for flows in pipeswill not be described. A stilling basin dissipates energy to decrease the risk oferosion of the spillway. The water flows down a slope and reaches the stillingbasin. The water entering the stilling basin will have a higher velocity thanthe water in the basin and this will cause a hydraulic jump to occur [4].Figure 6 illustrates how a hydraulic jump is formed when the fast flowingwater reaches the basin.

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Figure 6: Spillway with a hydraulic jump [5].

A hydraulic jump will give an increase of the water depth and will thereforetransform some of the kinetic energy into potential energy. The flow rapidlygoes from a supercritical to a subcritical flow which forces the flow to createa wave, a hydraulic jump. For the stilling basin to be safe the basin needs tohave the right dimensions, it has to be large enough to contain the hydraulicjump. By adding a roughened bed at the bottom of the stilling basin itis possible to stabilize the hydraulic jump on a shorter distance. Wheninvestigating a hydraulic jump there are some parameters that are importantto consider. A empirical formula for the sequent depth ratio is given by (1).[6]

y2y1

= 0.832Fr1 + 1.998B − 1.250r

y1+ 0.432 (1)

In (1), y2 represents the sequent depth of the hydraulic jump, y1 representsthe inflow depth of the hydraulic jump, Fr1 is the Froude number of theinflow, B is the divergence ratio and is given by B = b1/b2 where b1 is thewidth of the stilling basin upstreams the hydraulic jump and b2 is the widthof the stilling basin downstream the hydraulic jump and r is the height of theroughness elements in the bottom of the stilling basin. The Froude numberof the inflow can be calculated with (2).

Fr1 =v1√gy1

(2)

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In (2) v1 is the local flow velocity and g is the gravitational acceleration. Itis important to know the relative length of the hydraulic jump in order toknow which dimensions the stilling basin should have. How to calculate therelative length of the hydraulic jump is given in (3). [6]

Ljy1

= 8.924Fr1 + 11.473B − 12.390r

y1− 21.541 (3)

In (3), Lj is the length of the hydraulic jump. The relative loss of energy isgiven in (4).

ELE1

= 0.250log(Fr1)− 0.024B2 − 0.023B + 0.026r

y1+ 0.244 (4)

In (4), EL is the specific energy loss in the hydraulic jump and E1 is thespecific energy upstream the hydraulic jump [6].

The case in this project is unique since the flow that enters the stilling basinis unevenly divided because of the placement of the spillway gates. It isnot necessary to have a surface roughened bottom in the stilling basin sincethe basin is long enough to contain the hydraulic jump. But it is a usefulsolution to add some ribs in the spillway channel. These ribs acts as a surfaceroughened bottom in the spillway chute. This will cause a drag to occur inthe spillway channel that forces the flow to get more evenly divided which willreduce the risk of erosion on both the spillway channel and the downstreamriver.

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2.2 CFD theory

The two fluids that are used in this project are water and air. Both of theseare Newtonian fluids and will follow Newtons law of viscosity, which can beseen in (5). [7]

τ = µdu

dy′(5)

In (5), τ [N/m2] stands for the shear stress, µ [Ns/m2] is the dynamic viscosityand du/dy′ [1/s] represents the shear rate. Since the fluids are Newtonianfluids the flow can be represented by Navier-Stokes equations. The Navierstokes equations will include two time dependent conservative equations, thecontinuity equation, seen in equation 6, and the momentum equation, seenin (7). [7]

∂ρ

∂t+∇(ρu) = 0 (6)

ρdu

dt= −∇p+ ρg + µ∇2u+ (µ+ b)∇(∇ ∗ u) (7)

In (6) and (7), ρ [kg/m3] represents the density of the fluid, u [m/s] thevelocity, p [N/m2] the pressure, g [m/s2] the gravity acceleration, t [s] thetime, b [Ns/m2] a coefficient of the viscosity and µ [Ns/m2] the dynamicviscosity.

When investigating the Navier-Stokes equation it is important to know if thefluid is considered to be compressible or not. A good way to be able to knowif the flow can be approximated as incompressible or not is to calculate thefluids Mach number. A Mach number can be described as the ratio betweenthe speed of the flow and the speed of the sound. If this ratio is less than 0.3the flow can be considered to be an incompressible flow. The flow will, in thiscase, have a Mach number less than 0.3 and can therefore be approximatedas an incompressible flow [8]. In reality a flow can never be incompressiblebut when the difference between the speed of the sound and the speed ofthe fluid is big enough it is possible to make the approximation that thefluid is incompressible. This makes the Navier-Stokes equations simpler tosolve since the divergence of the velocity will be zero. The effect of thisapproximation is that (7) can be rewritten as seen in (8). [7]

ρdu

dt= −∇p+ ρg + µ∇2u+ (µ+ b)∇(∇ ∗ u) = −∇p+ ρg + µ∇2u (8)

It is important to remember that (8) is only true for incompressible fluids.Another important factor to consider is that the flow will be turbulent. This

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makes the calculations a lot more complicated than if the flow would havebeen laminar. In reality most of the flows are turbulent since small dis-turbances will cause the laminar flow to turn into a turbulent flow if theReynolds number is high enough. [7]

The flow in this project is considered to be a turbulent flow with two Newto-nian fluids that both are incompressible. The two fluids in the flow are waterand air.

2.2.1 Reynolds-Averaged Navier-Stokes (RANS)

It is possible to solve the Navier-Stokes equation by using a direct numericalsimulation, DNS. The problem is that DNS requires large amounts of compu-tational power and will give additional results that in the most cases are notnecessary. A solution that is more computationally efficient is the Reynolds-Averaged Navier-Stokes, RANS, model. In the RANS model the complicatedNavier-Stokes equation is simplified by decomposing them into time-averagedfluctuating quantities. This will give the general RANS equation, which canbe seen in (9). [9]

∂Ui∂t

+ Uj∂Ui∂xj

= −1

ρ

∂p

∂xi+ ν

∂2Ui∂x2j

− ∂uiuj∂xj

(9)

In (9), Ui represents the mean velocity over a time period, t the time, x theposition, p the pressure, and uiuj the mean stress tensor. It is possible torewrite (9) as (10).

∂Ui∂t

+ Uj∂Ui∂xj

= −1

ρ

∂

∂xj

[pδij + µ

(∂Ui∂xj

+∂Uj∂xi

)− ρuiuj

](10)

The last term in (10), −ρuiuj, is the definition of the Reynolds stress tensor.The term δij is the Kronecker delta and this will be 1 if i = j and 0 otherwise.U represents the velocity and x the position. [9]

2.2.2 Boussinesq approximation

To be able to use (10) there is a need for further approximations. TheBoussinesq approximation states that the Reynolds stress tensor will be pro-

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portionate to the mean velocity gradients. This gives (11).

∂Ui∂t

+ Uj∂Ui∂xj

= −1

ρ

∂p

∂xi− 2

3

∂k

∂xi+

∂

∂xj

[(ν + νT )

(∂Ui∂xj

+∂Uj∂xi

)](11)

In (11), ν+νT represents the effective viscosity. The problem with Boussinesqapproximation is that the models based on it only can predict isotropic flowsin local equilibrium. The Boussinesq approximation is still used in mostof the turbulence models since it is simple and does not require as muchcomputational power as turbulence models based on more complex equations.[10]

2.2.3 Turbulence models

A turbulent flow is characterized by its randomness, it is not possible topredict what the velocity will be in a specific point in the domain at a certaintime. The flow will be irregular, chaotic and therefore diffusive. When theflow goes from laminar to turbulent the Reynolds number will get larger,which means that the flow gets less stable. When the Reynolds number getslarge enough all fluids will have a turbulent flow. In (12) it is possible to seehow the Reynolds number can be calculated.

Re =ρV L

µ(12)

The Reynolds number is represented by Re, V [m/s] is the average flow speedand L [m] is the characteristic linear dimension. By making experimentalstudies it has been possible to find out when the transition from laminar toturbulent flow occurs. In this project the flow will be along surfaces whichmeans that the transition will occur when the Reynolds number gets largerthan 500 000. This value is not set in stone and the transition from laminar toturbulent flow will be affected by other factors, such as the surface roughness,as well.

To simulate a flow with high Reynolds number might require huge amountsof computational power and would be very time consuming. Therefore engi-neers have developed many different turbulence models that makes differentkinds of approximations to be able to focus on the big picture. The mod-els represents modified versions of the original equations and these modifiedequations will require much less computational power to solve. When usingone of these turbulence models it is important to have detailed information

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about the flow that needs to be simulated and also know which model thatwill give the most accurate solution. It is also necessary to know how muchcomputational power that is available since the most accurate model mightrequire more computational power and will therefore be more time consum-ing. In some cases it might be preferred to select a less accurate solution tobe able to finish the simulation in time.

There are many turbulence models and one way to classify these are bydividing them into groups based on how many partial differential equationsthe models uses, other than the RANS and continuity equations. The mostcommon models are zero-, one- and two-equation models and this project willuse a two equation model since that is the most common model used for flowsimulations.. The most common two-equation models are called Standardk-ε, RNG k-ε, Realizable k-ε, k-ω model, SST model and Reynolds stressmodels, RSMs. [9]

2.2.4 Standard k-ε model

The standard k-ε model is one of the most common models used in the indus-try where k stands for the turbulence kinetic energy and ε is the turbulentkinetic energy dissipation rate. The reason for the popularity of this modelis that ε affects how the turbulence will be interpreted. For the turbulentkinetic energy, k, the exact transport equation is not possible to solve with-out making approximations. The transport equation for k when using thestandard k-ε model is given by (13). [9]

∂k

∂t+ Uj

∂k

∂xj= vT

[(∂Ui∂xj

+∂Uj∂xi

)∂Ui∂xj

]− ε+

∂

∂xj

[(v +

vTσk

)∂k

∂xj

](13)

In (13), σk represents the Prandtl-Schmidt number and vT represents theturbulent viscosity. The transport equation for ε when using the standardk − ε model is given in (14).

∂ε

∂t+Uj

∂ε

∂xj= Cε1vT

ε

k

[(∂Ui∂xj

+∂Uj∂xi

)∂Ui∂xj

]−Cε2

ε2

k+

∂

∂xj

[(v +

vTσε

)∂ε

∂xj

](14)

The turbulent viscosity is calculated by using (15).

vT = Cµk2

ε(15)

14

In (13), (14) and (15) there are five coefficients, Cµ, Cε1, Cε2,σk and σε. Thesecoefficients are considered to be constants with values that can be seen intable 1. [9]

Constant Value

Cµ 0.09Cε1 1.44Cε2 1.92σk 1.00σε 1.30

Table 1: Values of the coefficients in the standard k-ε model.

Even though the standard k-ε model is common and well validated it hassome flaws. The model is not suited for modelling swirls, round jets or flowswith sudden acceleration. It is also not suited for regions with low Reynoldsnumbers. Therefore models with some modifications have been developed.[9]

2.2.5 Realizable k-ε model

The realizable k-ε model is one of the modified models of the standard k-εmodel. The difference between these models is that the normal stress can benegative in the standard model when the mean strains are large enough. Thiscan cause some problems since the normal stress is positive by definition. Tosolve this problem the realizable model uses a Cµ that is variable. Cµ willvary to stop the normal stress from becoming negative. The realizable modelalso has an extra term in the ε-equation that calculates the production ofturbulent energy dissipation. This implies that the realizable model is moresuited for flows with large strain rate, in example flows with rotating shearflows and flows with axisymmetric jets. [9]

2.2.6 RNG k-ε model

Another modified version of the standard k-ε is the Re-Normalisation Groupk-ε model, more commonly known as the RNG k-ε model. The modificationthat has been made is that one extra term has been added to the original

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equation. This can be seen in (16) in comparison to (14).

∂ε

∂t+Uj

∂ε

∂xj= Cε1vT

ε

k

[(∂Ui∂xj

+∂Uj∂xi

)∂Ui∂xj

]−Cε2

ε2

k+

∂

∂xj

[(v +

vTσε

)∂ε

∂xj

]−Sε

(16)The Sε term is calculated using (17)

Sε =Cµη

3(1− η/η0)ε2

(1 + βη3)k(17)

Then it is possible to calculate η by using (18) and η0 = 4.38 and β = 0.012.

η =k

ε

√2SijSij (18)

In (18), Sij is the strain-rate tensor.

The RNG k-ε model is mainly better for simulating jets but also for simu-lating flows with swirls. [9]

2.2.7 Boundary conditions

When setting up a CFD model it is important to set the right boundaryconditions. If the boundary conditions are wrong the model will produce aresult that cannot be trusted. There are many different boundary conditionsthat can be modified, for example flow inlet, outlet, wall, internal and faceboundaries.

When setting up the boundary conditions for a flow inlet there are a coupleof different methods that can be used. The most used method is the velocityinlet boundary condition. This is used to describe the velocity of the fluid atthe inlet. It is also possible to use pressure as the inlet boundary condition.This describes the total pressure of the flow at the inlet. A third methodis to use the mass flow inlet boundary condition but this method is onlyrecommended for compressible flows. Using this method for incompressibleflows is unnecessary since the density is constant and therefore the velocityinlet boundary conditions will adjust the mass flow.

The pressure outlet boundary condition can be used to describe the staticpressure at the outlet. Another method would be to use the outflow boundarycondition. This method is a good choice when there are unknown detailsabout the flow velocity and the pressure but will not be recommended for

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compressible flows. The pressure outlet boundary condition will, in mostcases, give a better convergence rate towards a stationary position comparedto using the outflow boundary condition. [13]

Another significant factor is the boundary conditions close to the walls. Thedefault setting is a no-slip condition which makes sure that the flow that isin contact with the walls does not move. This can be changed by specifyingthe shear conditions. It is also possible to set the wall roughness, whichis recommended for turbulent flows, and the wall adhesion contact angle.Beyond these conditions it is also possible to set many other wall boundaryconditions, for example wall motion, thermal and chemical reaction boundaryconditions, but these boundary conditions are not relevant for the model inthis project. [14]

2.2.8 Multiphase flow

When a flow consists of more than one phase it is called a multiphase flow.Most of the flows in nature and in the industries are multiphase flows andneeds to be treated as such when modelled. There are a couple of models thatcan be used and the most common one is called the volume of fluid, VOF,model. The VOF model uses (19) to calculate how the phases interact.

∂F

∂t+ u ∗ ∇F = 0 (19)

In (19), F represents the volume fraction and ranges from 0 to 1. If F = 1the element is fully occupied by one of the phases and F = 0 implies thatthe element is fully occupied by the other phase. [11]

2.2.9 Meshing

To be able to model a flow with CFD it is necessary to divide the domain intocells, this is known as meshing. The meshing can be done in several differentways and all different meshing methods have different positive and negativeaspects. By dividing the mesh into a finer mesh the solution will be moreaccurate but it requires more computational power. It is possible to createthe mesh to fit the case assuming that there is enough of computationalpower available. For example if the flow is turbulent in specific locationsit is possible to make the mesh finer in these locations to get a precise so-lution without using more computational power than necessary. There are

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several different meshing methods that can be used, for 3D meshing the mostcommon methods are tetrahedron, prism, pyramid and arbitrary polyhedron.[12]

It is vital to make sure that the mesh quality is high enough. A mesh with lowquality will lead to problems with both the accuracy and the computationof the solution. When checking the mesh quality there are different criteriathat need to be controlled depending on which meshing method that has beenused. For tetrahedral and triangular meshes the skewness is an importantfactor to evaluate. Skewness refers to how similar the cells shape is comparedto a cell that is equilateral and with the same volume. In (20) it is possibleto see how the skewness is calculated.

Skewness =Optimalcellsize− cellsize

Optimalcellsize(20)

The equation in (20) only applies to triangles and tetrahedral cells. Sincethis project uses a 3D model the optimal cell size and cell size are the volumeof the cells. The quadrilateral mesh should have angles close to 90 degreesand triangular meshes should have angles close to 60 degrees. The skew-ness should never exceed 0.95. If the skewness exceeds 0.95 there will occurproblems during the calculation of the model in the form of convergencedifficulties.

An alternative method for controlling the mesh quality is to look at thesquish index. This index controls the face area vectors and compares themto a vector that goes from the centroid of the cell to each of the cells faces.If the squish index is close to 1 the mesh quality is poor and if it is closeto 0 the mesh quality is high. The squish index can be used for all types ofmeshes and as long as the maximum squish index is below 0.99 for all cellsthe mesh can be considered to be of satisfying quality. [15]

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3 Methodology

There are many tools that needs to be used in this project. The methodologysection explains how the geometry, mesh, boundary conditions, numericalsolution and post-processing were set up.

3.1 Geometry

The geometry was created using a model, created in CAD, Computer AidedDesign, that illustrated how the reconstructed spillway looked. The CADmodel was created by Vattenfall R&D before this project started as a partof the construction of the real spillway. Before the model could be used ithad to be adjusted. Figure 7 shows how the CAD model looked before it wasadjusted.

Figure 7: The CAD-model illustrating the reconstructed spillway.

Two programs were used to make the necessary adjustments, ANSYS FluentSpaceclaim 2019r2 and ANSYS Fluent DesignModeler 2019r2.

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The adjustments that had to be made was creating a volume that the fluidcould flow through, removing edges that were not supposed to be there,creating a volume above the spillway and finally removing the chamfers atthe ribs. The removement of the chamfers was necessary to be able to createa mesh with high enough quality. It was necessary to produce two differentgeometries, one that could be compared with the physical test results and onethat represented the real case. The spillway gates in the geometry that wascompared to the physical tests was 5 m wide and in the real case spillwaygate number 7 was 6.5 m and the other six spillway gates was 5 m wide.When increasing the width of spillway gate number 7 the rest of the spillwayalso had to be widened with 1.5 m. The two final geometries can be seen infigure 8 and 9.

Figure 8: The geometry for case 1 that will be compared to the physical testresults.

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Figure 9: The geometry that represents the real spillway, referred to as case2. The blue area represents the added volume compared to case 1 and thearrows marks the inlet and outlet.

The blue rectangles in figure 9 represents the added volume. When the finalgeometries were completed the different sections of the spillway was definedby naming them. In figure 9 the inlet and outlet are marked by arrows.The top side of the whole spillway is a pressure inlet that allows air into thespillway and the rest of the areas are walls. It is also possible to see whereorigin is located and in which direction x, y and z are in figure 8. The casewith the geometry in figure 8 will be referred to as case 1 and the case withthe geometry in figure 9 will be referred to as case 2 during the rest of thisreport.

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To be able to have a stable water level upstream of the spillway a regionwas created. The region can be seen as the red part of figure 10 After thesolution initializing was completed it was possible to patch the solution andfill the region with water. The purpose of the region was to make sure thatthe water level stays constant above the spillway. In reality the water levelwould not stay constant, unless the inflow from the upstream river is thesame as the outflow in the spillway. But in the physical hydraulic tests thewater level above the spillway was constant during the measurements.

Figure 10: The spillway with the created region in red.

3.2 Generation of the mesh

Before the mesh was created some demands were set. The mesh was notallowed to have more then 1000 000 elements and had to have a maximalskewness below 0.75. The reason that the mesh had to consist of less then1000 000 elements was that it would require to much computational powerto calculate the model if the number of elements got too high. The skewnessgoal was set to make sure that there would not be any problem with the meshduring the calculation of the solution. In the beginning a mesh consistingof mostly hexahedral cells was created but this mesh had extremely highskewness, 0.98. To solve this problem there had to be a shift from hexahedralto tetrahedral cells since tetrahedral cells works best when dealing with acomplicated geometry. The maximum element size, which is the maximumlength of the edges an element has, was set to 0.65 m and high smoothingwas used. The target skewness was set to 0.75. This resulted in a mesh thatcan be seen in figure 11 and figure 12 shows the area marked by the redrectangle.

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Figure 11: The meshing of the spillway for case 2.

Figure 12: A closer view of the mesh, the mesh seen is the mesh marked bya red rectangle in figure 11

The mesh for case 2 can be seen in figure 11 and 12 and it consists of 651643elements and has a maximum skewness at 0.71. The mesh for case 1 consistsof 629293 elements and has a maximum skewness at 0.75.

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3.3 Setup

In the setup all the boundary conditions have to be defined. The solver wasset to be transient, meaning that the solution is time dependent, and pressurebased, a pressure based solver uses a pressure equation to get the constraintfor the continuity of the velocity field. Then the gravity had to be includedand it was defined as -9.81 m/s2 in the z-direction.

Then the different models had to be defined. The multiphase model was set tovolume of fluid, VOF, with open channel flow and implicit body force, whichenhances the convergence of the solution by taking the partial equilibriumof the body forces and pressure gradient into consideration when solving themomentum equations. There are two phases, the air phase, which is theprimary phase, and the water phase, which is the secondary phase. Thevalues for both air and water were received from the Fluent database. Thenext model to be defined was the turbulence model. This was set to be theRNG k − ε model with standard wall functions since this turbulence modelis the recommended model for flows with swirls.

The next step was to define the boundary conditions. The inlet was definedas a pressure inlet with open channel flow that had a free surface at 3.67 min the z-direction and a bottom level at -1.9 m in the z-direction. The choiceof a pressure inlet instead of velocity inlet was simply because the water levelwas known but not the water velocity. The velocity magnitude was set to be0.9034 m/s in the x-direction, this value is an estimated value based on themeasured water flow from the physical hydraulic tests.

Then the operating conditions was defined by setting the reference pressurelocations in x-, y- and z-directions. The specified operating density was setto 1.225 kg/m3.

The solution method was at first a non iterative time advancement but thissolution method failed during the calculations and the SIMPLE scheme wasselected instead. The pressure was set to be body force weighted, the mo-mentum was second order upwind, the volume fraction was compressive, theturbulent kinetic energy was second order upwind, the gradient was leastsquares cell based and the turbulent dissipation rate was second order up-wind.

To verify that the simulation was correct a monitor was set to follow thevolume flow at the inlet. If all steps in the setup are correct the volume

24

flow will stabilize around 310 m3/s, which was the result achieved from thephysical hydraulic model tests for the same water level as defined at the inletpressure after scaling it up to real size.

The last step before running the calculation is to check the case to makesure no obvious mistakes has been made. The selected time stepping methodwas variable and with 10−5 s as the smallest time step. The total simulatedtime was set to 1000 s. The calculation of case 1 used 32 cores and thecalculation of case 2 used 40 cores. Both calculations required about 3 daysof calculation time per case.

3.4 CFD-Post

CFD-Post 2019r2 is a software in ANSYS 2019r2 that was used to extractthe relevant results from the simulation. Water velocities and water level atspecific points in the model were extracted to be compared with the physicalhydraulic test results and also to compare case 1 and 2. The the water veloc-ity directions was extracted to compare case 1 and 2 and to get informationabout the spillway that could not be received by physical model tests.

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4 Validation of the CFD-model

During the physical hydraulic model tests most of the measurements weremade on a model that had seven spillway gates that all were 5 m wide. There-fore the results from case 1 is compared to the results from these physicalmodel tests.

4.1 Volume flow

In figure 13 it is possible to see how the volume flow through the inlet sta-bilized at a value of 309.5 m3/s. The corresponding value from the physicalhydraulic model test was 310 m3/s.

Figure 13: Volume flow through the inlet from the CFD-simulation of case 1.

The first step in validation the CFD-model is by investigating the volumeflow. The volume flow measured in the physical tests were at 310 m3/swhich is really close to the volume flow in the CFD-model, 309.5 m3/s. Thisindicates that the CFD-model might be correct but to validate the modelthere is a need for further investigation.

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4.2 Flow behaviour

In figure 14 it is possible to see the directions of the water velocities. Thisgives an overview of how the flow behaves and can be compared to figure15 to investigate if the CFD-model recreates the physical hydraulic modeltest results. As can be seen in figure 14 there is a zone of circulation on theleft side of the stilling basin. This zone of circulation was observed in thephysical tests as well which can be seen in the red circle in figure 15. It isalso possible to see that the water flow have more chaotic directions in thestilling basin and when it enters the spillway channel it calms down and flowsin a more uniform direction. The flow behaviour that can be seen in bothfigure 14 and 15 follows the same pattern. It starts with high energy and azone of circulation and when it leaves the stilling basin the water flow seemsto be less turbulent.

Figure 14: The directions of all the water velocities in the spillway for case 1.

27

Figure 15: Water flow in the spillway channel during the physical hydraulicmodel tests, the red circle marks the zone of circulation [3].

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4.3 Water velocities

The water velocities at three different sections, A,B and C, of the spillway,received by the CFD-simulations of case 1, can be seen in figure 16. Thesethree sections are the same as in the physical hydraulic model tests.

Figure 16: Water velocities at section A, B and C for case 1

In figure 17, 18 and 19 the water velocities in section A, section B and sectionC received from the CFD-simulation of case 1 are compared to the water ve-locities from the physical hydraulic tests. The velocities in the physical testswere extracted at unknown depths but it was stated that these velocities werethe maximum velocities. Therefore the velocities from the CFD simulationwere extracted at various depths to be able to find the maximum velocities.

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Figure 17: Comparison of the water velocities at section A.

Figure 18: Comparison of the water velocities at section B.

30

Figure 19: Comparison of the water velocities at section C.

By comparing the measured water velocities at the three different sections,A, B and C, which can be seen in figure 17, 18 and 19 the CFD-modelcould be further validated. As can be seen in these three figures the watervelocities from the physical tests are similar to the water velocities from theCFD simulation. There are some values that differ but that does not haveto indicate that the CFD-model is incorrect. As can be seen in figure 16the highest velocities are placed at different depth depending on the distancefrom the left wall. In the physical tests the values were measured by hand andit is not mentioned at which depth the measurements were made. Generallythe measurement of water velocities is made close to the surface since thatis were the velocity theoretically would be the highest. As can be seen infigure 16 the water velocities in both section A and B are higher close tothe bottom. Another difference is that the results from the CFD-model onlyconsiders water velocity and neglects the air velocity. In the physical testsit would not be possible to only measure the water velocity, if there is anyair with velocities in the measured point it might have affected the result.The first value in figure 19 differs a lot. This is since the value from theCFD-simulation is close to 0 m from the left wall. In the physical modeltest it would not be possible to measure that close to the left wall. Thiswould explain why the difference between these two values are as big as it is.Overall the water velocities in section A, B and C follows the same patternwhich indicates that the CFD model represents the physical hydraulic modelwell.

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4.4 Water level

The water depth along both sides of the spillway channel from the CFD-simulations can be seen in figure 20 and 21.

Figure 20: The water depth along the right side of the spillway channel incase 1.

Figure 21: The water depth along the left side of the spillway channel incase 1.

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A comparison between the water level received from the CFD-simulation ofcase 1 and the water level from the physical hydraulic tests can be seen infigure 22 and 23.

Figure 22: The water level along the left side of the spillway channel, waterlevel 0 m is at the inlets water level.

Figure 23: The water level along the right side of the spillway channel, waterlevel 0 m is at the inlets water level.

33

To further evaluate the CFD-model it is possible to compare the water levelsalong both the left and the right side spillway channel. As can be seen infigure 22 and 23 both follow the same trend. The water level reaches a peakwhen the water leaves the stilling basin and then the water level gets lowerand lower as it travels down the spillway channel. There are some valuesthat differs with almost 1 m which might seem strange. The highest valuesfrom the CFD simulation in figure 22 and 23 were measured exactly wherethe stilling basin ended and this could explain why the values are that high.The values from the CFD simulation measured after the stilling basin endsare not as smooth as the values from the physical tests. The reason for thisis that some of the measurements were close to the ribs while some werebetween two ribs. The measurements close to, or on, a rib will then give ahigher water level since the ribs cause small waves to occur. What can beseen in both figures are that the results from the CFD simulation follows thesame down going trend as the results from the physical tests. The last valuein figure 22 and 23 could not be retracted from the CFD simulation sincethat point is located outside the CFD-model.

By comparing the physical hydraulic test results with the results from theCFD simulation it is possible to validate the CFD-model. The flow behaviourstrongly implies that the model is a good representation of the physical hy-draulic model. Most of the other results are similar and follows the sametrends as the physical tests results. Therefore the model can be applied tocase 2 since that model is exactly the same except for the width of gate 7and the width of the spillway channel.

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5 Results and discussion

The results are presented as graphs that compares the results from the CFD-simulations of case 1 and 2. By comparing these two cases it is possibleto determine if the energy dissipation is improved when one of the spillwaygates and the rest of the spillway are widened by 1.5 m. The results receivedfrom the CFD-simulations that provides a better understanding of the flowpattern, flow circulation and outflow from the stilling basin than the physicaltests will then be analysed. All the following results are based on a constantwater level at the spillway gates. The last part of this section goes throughdifferent sources of error that is important to be aware of and also possiblefuture work to further investigate the spillway.

5.1 Comparison of volume flow

In figure 24 it is possible to see how the volume flow through the inlet incase 2 stabilizes at 325 m3/s after about 200 s. This does not indicate thatthe last 800 s of the simulation is unnecessary, it only shows that the inflowhas stabilized, not if the outflow and flow through the spillway has stabilized.In case 1 the volume flow stabilizes at 309.5 m3/s. This increase is reasonablesince the area of the spillway gates has been increased while the water levelabove the spillway gates remains the same.

Figure 24: The volume flow through the inlet from the CFD-simulation ofcase 2.

35

5.2 Comparison of water velocities

In case 2 the water velocities at section A, B and C were measured in thesame way as for case 1. Figure 25 shows the water velocities at the threesections for case 2.

https://www.overleaf.com/project/5e3ac835313467000110d028

Figure 25: Water velocities at section A, B and C for case 2.

The figures 26, 27 and 28 compares the water velocities at section A,B andC for case 1 and case 2.

36

Figure 26: Comparison of the water velocities at section A.

Figure 27: Comparison of the water velocities at section B.

37

Figure 28: Comparison of the water velocities at section C.

It is possible to compare the water velocities in case 1 and 2 by looking atfigure 26, 27 and 28. At section A the water velocities follow the same trendbut in case 2 the water velocities are lower close to the right wall comparedto case 1. This is since both the spillway gate and the spillway channel hasbeen widened and therefore more water can flow through the spillway withlower velocity. At section B the water velocities are similar in both cases.There is one value at 21 m from the left wall that differs more than theother values. The reason for this value is that the velocities were measuredat different depths which explains why the values differ as much as they do.At section C the water velocities from case 2 are more stable and lower thanin case 1. This makes it possible to assume that case 2 has a better energydissipation than case 1. It is desired to have a water flow with low velocitythat exits the spillway channel to reduce the erosion of the river below thespillway.

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5.3 Comparison of water level

Figure 29 and 30 shows the difference in water level between case 1 andcase 2.

Figure 29: Comparison of the water level along the left side of the spillway,water level 0 m is at the inlets water level.

Figure 30: Comparison of the water level along the right side of the spillway,water level 0 m is at the inlets water level.

39

The water levels in the spillway for the two cases can be investigated bystudying figure 29 and 30. On the left side of the spillway the water levelsare almost the same for both cases. Case 1 has some points were the waterlevels are higher than in case 2 but the overall trend is the same. On theright side the water levels are a bit lower in the beginning of the stilling basinthan for case 1. But during the rest of the spillway channel the water levelfollows the same pattern. The water levels are lower in the beginning of thestilling basin since the basin has been widened with 1.5 m on the right side.The peaks on both the left and the right sides are when the flow reaches theend of the stilling basin. On the right side of the spillway the peak is higherfor case 2 and on the left side the peak is higher for case 1. This is most likelybecause the volume flow is higher on the right side for case 2. Before theend of the stilling basin the water level is higher on the right side but afterthe stilling basin the water levels are almost the same on both sides of thespillway channel. Since there is no distinct difference between the two casesit is hard to determine if the energy dissipation is better in one of the casesbased on water level. It is reasonable that the water levels follows the samepattern and are at the same level for both cases since the spillway channelhas been widened with the same width as the spillway gate.

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5.4 Comparison of the flow behaviour

The figures 14 and 31 shows the direction and magnitude of all the watervelocities in the spillway for case 1 and 2. This cannot be retrieved byphysical test, only by using CFD. In physical hydraulic testing it is possibleto see some of the flow patterns just by looking at the model but it is notpossible to see the magnitude of the water velocities.

Figure 31: The directions of all the water velocities in the spillway for case 2.

By comparing figure 31 and 14 it is possible to get a better understandingof how the water flow is affected by the widening of the spillway gate andspillway channel. In both cases the water velocities are at their highest whenentering the spillway gates 7, 6, 5 and 4. In case 2 the water directionsstabilises faster in the stilling basin compared to case 1. When the waterflow then enters the spillway channel the flow seems to behave more calm.One problem that is noted in case 2 is that there are water directions on thediagonal of the spillway chute but these have low velocities compared to thevelocities of the water that flows directly in the channels direction. In case 1there is no sign of a standing wave in the spillway channel but it seems to bemore water flow with directions in the wrong way which indicates that thereis more randomness in the velocity directions in case 1 than case 2.

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5.5 Sources of error

When using CFD modeling there are a couple sources of error that always arerelevant to discuss. One of the most common errors is that the quality of thegrid is to low. It is possible to perform a mesh independence study to makesure that the quality of the mesh is high enough but unfortunately the timelimit of the project was to short to make a mesh independence study. Themesh could have been made even finer if the time limit of the project had beenlonger since having high mesh quality often leads to longer simulation time.Another factor is the choice of turbulence model. The model used in thisproject, the RNG k−ε, is recommended for cases with free surface simulationssince it has been proved to be the most successful model. To further test howreliable the result was it would have been advantageously to run a simulationwith another turbulence model, for example the realizable k − ε model.

It is vital that the geometry corresponds well to the real case when simulatingwith CFD. If the geometry differs the results will not be accurate. In thiscase the geometry was built from a CAD model of the reconstructed spillwayand can be trusted to be very similar to the real spillway. The only volumethat might differ a bit from the real case is upstream the spillway gates. Thepart that differs is the angle on the bottom on the right side. The reasonfor this is that the angle there was unknown and had to be estimated incooperation with Vattenfall employees. This difference should not affect theresults since the water level above the spillway gates was set to a constant.

Another factor that might have affected the result are the measurement un-certainties during the physical tests. These uncertainties are unknown.

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6 Conclusions

The main purpose of the project was to evaluate if the physical hydraulic testresults could be reproduced by using computational fluid dynamics, CFD,and obtain detailed results of the flow that could not be obtained in thehydraulic model tests.

The CFD results achieved in this project corresponds well with the resultsfrom the physical hydraulic model tests. Therefore it is possible to deter-mine that the CFD calculation was successful in reproducing the hydraulicmodel tests. The results from case 2 indicates that the decision to widenspillway gate 7 and the rest of the spillway with 1.5 m was correct. Thedischarge capacity increased and at the same time the energy dissipation gotbetter. This led to the water flow leaving the spillway having a lower velocitywhich is good for the downstream river since the risk for erosion is decreased.Using CFD can be considered to be a useful complement to physical testssince it can both validate the physical experiments and give more detailedinformation about the flow.

For future work it would be recommended to run calculations with one gateopened at a time and compare these results with the results from the physicalhydraulic test. This was not possible in this project mainly because of thelimited time, CFD calculations are very time consuming. It would also beinteresting to investigate how a change of the water level above the spillwaygates would affect the energy dissipation in the spillway. It would also bepossible to retract more detailed information about how the water flow be-haves in the spillway. Another factor that would be interesting to investigateis how the flow behaves when it leaves the spillway. This would give a betterunderstanding of how big the risk for erosion of the downstream river is.

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