UPTEC ES14 007
Examensarbete 30 hp April 2014
Numerical Study on Jet Flow
Characteristics of High Head and Large Discharge Spillways
Lisa Gerdin & Mira Rosengren Keijser
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
Numerical study on jet flow characteristics of high head and large discharge spillways
Lisa Gerdin & Mira Rosengren Keijser
Today scale models are used to design spillway structures for hydropower stations. These are expensive and time-consuming to build and alter.
This study investigates the possibilities of using numerical simulations in order to facilitate the spillway design process. It would be possible to save time and resources by altering the spillway parameters in the numerical model and thus find an optimal design, which can be further investigated with a scale model.
However, it is complicated to simulate turbulent flows. Therefore the simulated flows in this study are compared to experimental measurements in order to investigate the accuracy of the numerical model. Ansys software Fluent uses Computational Fluid Dynamics (CFD) to calculate turbulent flows and is used as the simulation tool in this study.
The simulations were performed on the spillway system of Shuibuya hydropower station. There are five spillway channels with flip bucket terminals and high head. In order to investigate the risk of erosion during large flows the jet throw distance was examined in experiments on a scale model. The same parameter was investigated in this simulation study.
The acceptable error margin was set 30 % for the comparison between simulated and experimental measurements. All performed simulations met this criterion. It was therefore concluded that Fluent could be used as a sufficiently good approximation tool when it comes to turbulent flows in spillways.
Keywords: Spillway design, energy dissipation, CFD, Fluent, jet nappe, erosion.
Sponsor: Elforsk, Vattenfall AB, Uppsala Universitet ISSN: 1650-8300, UPTEC ES14 007
Examinator: Petra Jönsson Ämnesgranskare: Urban Lundin Handledare: Yongliang Zhang
Summary'in'Swedish'
Detta%examensarbete%syftar%till%att%undersöka%möjligheterna%att%använda%Ansys%program%
Fluent%som%hjälpmedel%vid%utformning%av%utskov.%I%dagsläget%används%fysiska%
skalmodeller%för%att%undersöka%flödeskaraktäristik%och%utvärderar%utskovsdesignen.%
Dessa%är%tidskrävande%och%dyra%att%bygga%upp,%framförallt%på%grund%av%att%det%är%svårt%att%
göra%ändringar%på%en%fysisk%modell.%Om%det%är%möjligt%att%använda%Fluent%för%att%simulera%
utskovsflöden%numeriskt%kan%det%underlätta%framtagandet%av%en%optimal%utskovsdesign,%
eftersom%man%då%lättare%kan%ändra%utskovsparametrar.%%
Studien%utfördes%utifrån%vattenkraftverket%Shuibuyas%utskovssystem.%Två%olika%flöden%i%
tre%av%utskovkanalerna%simulerades%i%Fluent%och%jämfördes%med%experimentella%resultat%
från%en%skalmodell.%På%så%sätt%utvärderades%Fluents%potential%att%användas%som%
hjälpmedel%vid%utskovsdesign.%Den%accepterade%felmarginalen%sattes%till%30%%,%vilket%är%
standard%för%denna%typ%av%studie.%Simuleringsresultaten%jämfördes%med%beräkningar%
från%en%empirisk%ekvation%för%att%vidare%validera%simuleringsmodellen.%För%att%utvärdera%
inverkan%från%olika%parametrar%utfördes%tre%känslighetsanalyser.%%
Då%utskoven%vid%Shuibuyas%vattenkraftverk%är%utformade%som%en%backhoppningsbacke%
och%slutar%på%hög%fallhöjd%kommer%en%jetstråle%att%produceras%(vid%höga%flöden).%
Resultaten%tolkades%från%jetstrålens%nedslagsområde,%då%detta%är%direkt%kopplat%till%
risken%för%erosion.%Det%är%fördelaktigt%med%ett%stort%nedslagsområde%eftersom%
energiomvandlingen%i%nedslaget%inte%blir%lika%koncentrerad%till%en%punkt,%vilket%minskar%
risken%för%erosion%och%i%förlängningen%dammhaveri.%
Gambit%användes%för%att%bygga%upp%simuleringsmodellerna.%Eftersom%datorkapaciteten%
var%begränsad%delades%simuleringsvolymen%upp%i%två%mindre%volymer;%en%del%för%kanalen%
och%en%del%för%jetstrålen.%Därigenom%kunde%totala%antalet%beräkningsceller%ökas.%
Resultatet%från%alla%sex%simuleringarna%hamnade%inom%felmarginalen%vid%jämförelse%med%
experimentresultaten.%Slutsatsen%av%studien%blev%därmed%att%Fluent%har%potential%att%
användas%som%ett%hjälpmedel%vid%utskovsdesign.%Både%experimenten%och%simuleringarna%
visade%att%utskovskanal%1%gav%den%mest%utspridda%jetstrålen,%vilket%tyder%på%att%
erosionsrisken%är%minst%vid%detta%utskov.%Korrelationen%mellan%resultaten%från%
simuleringarna%och%experimenten%skulle%ändå%kunna%förbättras%genom%att%använda%en%
dator%med%större%beräkningskapacitet,%så%att%antalet%beräkningsceller%kan%ökas.%
Känslighetsanalyserna%indikerade%att%andra%ordningens%lösningsmetod%skulle%kunna%
förbättra%simuleringsresultaten,%men%då%skulle%också%mer%beräkningskapacitet%krävas.%
Nyckelord:+Utskovsdesign,%energiomvandling,%CFD,%Fluent,%jet%nappe,%erosion.%
'
% %
Summary'in'Chinese'
%
' '
Acknowledgements'
The%study%was%carried%out%at%Tsinghua%University,%Beijing,%China,%and%is%part%of%an%
ongoing%cooperation%between%Tsinghua%University,%Elforsk%AB%(the%R&D%organization%of%
the%Swedish%power%companies)%and%the%leading%universities%of%technology%in%Sweden.%
The%purpose%of%the%project%is%to%provide%Swedish%students%with%international%experience%
in%the%hydropower%field.%
We%would%like%to%thank%the%financial%sponsors%of%this%masters%project,%namely:%Elforsk%
AB,%Vattenfall%AB%and%the%Department%of%Engineering%Sciences,%division%of%electricity,%at%
Uppsala%University.%
We%would%like%to%thank%professor%Yongliang%Zhang%and%associate%professor%Li%Ling%for%
supervising%and%support%during%our%research%at%the%Department%of%Hydraulic%
Engineering%at%Tsinghua%University,%Beijing.%We%are%also%thankful%for%all%the%help%we%got%
from%ms%Dan%Zhang%with%our%computers,%and%for%showing%us%some%of%Wudaokou.%We%
would%also%like%to%thank%ms Ying Guan helping%us%during%our%stay%in%Beijing%and%at%
Tsinghua%Campus.%
Furthermore,%we%would%like%to%thank%dr.%James%Yang%at%Vattenfall%R&D%for%arranging%this%
exchange.%
We%would%also%like%to%thank%professor%Urban%Lundin%for%support%and%supervising%during%
our%work%with%the%report.% %
Abbreviations'
CFD%–%Computational%Fluid%Dynamics%
CFRD%–%Concrete%Faced%Rockfill%Dam%
LES%–%Large%Eddy%Simulation%
PDE%–%Partial%Differential%Equation%
RANS%–%ReynoldsbAveraged%Navier%Stokes%
RSM%–%Reynolds%Stress%Model%
SST%–%Shear%Stress%Transport%kbomega%model%
VOF%–%Volume%Of%Fluid%
' '
Nomenclature'
!%b%density%[kg/m3]%
!%b%viscosity%[mm2/s]%
!%b%gravity%[m/s2]%
!%b%area%[m2]%
!%b%velocity%[m/s]%
!"%b%Reynolds%number%[b]%
!%b%flow%rate%[m3/s]%
!%b%flow%quantity%such%as%energy,%momentum%or%mass%
!%b%dynamic%viscosity%[Pa∙s]%
!%b%turbulence%kinetic%energy%[J/kg]%
!!"!turbulent%dissipation%rate%[J/(kg∙s)]%
!!%b%external%forces%%[N]%
!%b%pressure%[Pa]%
!%b%absolute%temperature%[K]%
%!%b%thermal%conductivity%coefficient%[W/(m∙K)]%
!%b%viscous%shear%stress%tensor%[N/m2]%
!%b%total%enthalpy%[J]%
!%b%conserved%energy%[J]%
!%b%horizontal%position%of%jet%flow%[m]%
ℎ!!b%velocity%head%[m]%
!!%b%of%inclination%from%the%horizontal%plane%of%the%water%jet%at%outlet%[rad]!
!%b%vertical%position%of%jet%trajectory%[m]%
!!%b%initial%jet%velocity%[m/s]%
%
Table&of&Contents&
1.#Introduction#...#3!
1.1.#Purpose#...#3!
1.2.#Method#...#3!
1.3.#Limitations#...#3!
1.4.#Assumptions#...#4!
2.#Background#and#theory#...#5!
2.1.#Energy#dissipation#and#scour#...#5!
2.2.#Spillways#...#7!
2.3.#Shuibuya#...#9!
2.4.#Scale#model#...#10!
2.4.1.!Shuibuya!scale!model!project!...!11!
2.5.#Computational#Fluid#Dynamics#...#12!
2.5.1.!Navier;Stokes!equations!...!12!
2.5.2.!Turbulence!...!13!
2.6.#Simulating#in#Fluent#...#14#
2.6.1.!Preprocessing!...!16#
2.6.2.!Settings!in!FLUENT!...!17!
2.6.3.!Post;processing!...!22!
3.#Method#...#24!
3.1.#Settings#in#Fluent#...#25!
3.1.1.!Boundary!conditions!...!26#
3.1.2.!Solver!...!26!
3.2.#Grid#independence#study#...#26!
3.3.#PostLprocessing:#convergence#...#27!
3.3.1.!The!channel!...!27!
3.3.2.!The!jet!nappe!...!28!
3.3.3.!Interpreting!the!simulations!...!28!
3.4.#Validation#...#28!
3.4.1.!Sensitivity!analysis!...!28!
3.4.2.!Comparison!to!existing!results!...!29!
3.4.3.!Analytical!validation!...!29!
4.#Results#...#30!
4.1.#Grid#independence#study#...#30!
4.2.#Convergence#...#31!
4.3.#Jet#throw#distance#...#35!
4.4.#Analytical#results#...#38!
4.5.#Sensitivity#analysis#...#41!
5.#Discussion#...#46!
5.1.#Impact#of#air#entrainment#...#46!
5.2.#Convergence#...#48!
5.4.#Sensitivity#analyses#...#48!
6.#Sources#of#errors#...#48!
6.1.!Difficulties!when!comparing!results!...!48!
6.2.!Numerical!diffusion!...!49!
7.#Conclusion#...#49!
8.#References#...#51!
Appendix#I#...#54!
Reynolds#number#calculations#for#both#flow#rates#...#54!
Appendix#II#...#55!
Grid#independence#study#of#channel#simulations#...#55!
Grid#independence#study#of#jet#nappe#simulations#...#59!
Appendix#III#...#62!
Convergence#study#results#...#62!
Simulation!1!...!62!
Simulation!2!...!63!
Simulation!3!...!64!
Simulation!4!...!66!
Simulation!5!...!67!
Simulation!6!...!68!
Appendix#IV#...#69!
Residuals#...#69!
Simulation!1!...!69!
Simulation!2!...!71!
Simulation!3!...!72!
Simulation!4!...!73!
Simulation!5!...!74!
Simulation!6!...!75!
Sensitivity!analysis!I!...!76!
Sensitivity!analysis!II!...!77!
Sensitivity!analysis!III!...!78!
Appendix#V#...#79!
Appendix#VI#...#83!
Section#! ##Author#...#83!
!
!
1.&Introduction&
1.1.&Purpose&
The!purpose!of!this!study!was!to!evaluate!the!possibilities!of!using!Ansys!software!Fluent!when!
investigating!the!behavior!of!turbulent!flows!in!the!field!of!hydraulic!engineering,!more!specifically!
when!designing!spillway!structures.!Today!the!flow!behavior!is!examined!in!scale!models!which!are!
expensive!to!build!and!difficult!to!alter.!Computer!models!provide!the!possibility!of!easily!changing!
parameters,!for!example!the!length!of!a!spillway!channel.!This!is!a!feature!that!would!facilitate!the!
design!process!of!a!hydropower!station!significantly.!
1.2.&Method&
The!evaluation!of!Fluent!as!a!simulation!tool!was!performed!by!comparing!simulation!results!with!
experimental!measurements!carried!out!in!a!scale!model!of!the!Shuibuya!spillway!system.!In!order!to!
perform!simulations!in!Fluent!a!geometric!model!of!the!spillways!had!to!be!built!and!divided!into!
control!volumes.!This!was!done!with!Gambit!software!according!to!drawings!of!the!Shuibuya!dam!
structure.!Due!to!limited!computational!power!the!spillway!was!divided!into!two!different!
geometries;!a!channel!and!a!jet!nappe.!This!means!that!two!separate!simulations!had!to!be!
performed!for!each!spillway!simulation.!See!table!I,!section!1.3,!for!performed!simulations.!
Three!sensitivity!analyses!were!performed!in!order!to!investigate!the!accuracy!of!the!simulation!
model!and!the!impact!of!three!particular!setting!parameters.!
When!comparing!the!simulation!results!with!the!results!from!the!experimental!measurements,!the!
jet!throw!distances!from!the!two!different!methods!were!compared.!As!a!further!evaluation!the!jet!
throw!distance!was!also!calculated!with!an!empirical!equation.!The!definition!used!for!jet!throw!
distance!in!this!study!is!explained!in!section!2.1!below.!
The!working!process!will!be!described!more!thoroughly!in!chapter!3.!
1.3.&Limitations&
Since!Fluent!simulations!are!time!consuming!there!was!not!enough!time!to!simulate!for!all!five!
spillways!and!flow!rates!during!this!study.!Three!spillways!were!chosen;!the!one!with!the!longest!
channel,!the!one!with!the!shortest!channel,!and!the!one!in!the!middle.!Regarding!the!flow!rates!the!
design!flow!of!11,940!m3/s!and!a!larger!flow!rate!of!14,810!m3/s!were!chosen.!Simulations!were!
performed!for!both!flow!rates!in!three!spillways,!adding!up!to!six!different!simulations.!See!table!1!
below.!
! !
Table#1.#Overview#of#the#original#simulations#performed.#
Original!simulations! ! !
Simulation! Spillway! Flow!rate![m3/s]!
Simulation!1! Channel!1!+!jet!nappe! 14,810!
Simulation!2! Channel!1!+!jet!nappe! 11,940!
Simulation!3! Channel!3!+!jet!nappe! 14,810!
Simulation!4! Channel!3!+!jet!nappe! 11,940!
Simulation!5! Channel!5!+!jet!nappe! 14,810!
Simulation!6! Channel!5!+!jet!nappe! 11,940!
!
The!accuracy!of!the!simulation!is!strongly!dependent!on!the!number!of!cells!used!in!the!model.!In!
order!to!receive!accurate!results!one!wants!a!model!with!many!cells![1].!However,!a!large!number!of!
cells!require!large!computational!power!and!can!also!be!very!time!consuming![2].!The!number!of!
cells!in!the!simulation!model!was!therefore!limited!by!time!constraints!and!computational!power.!
1.4.&Assumptions&
When!modeling!the!Shuibuya!spillway!channels!in!Fluent!a!number!of!assumptions!were!made.!The!
flow!rate!of!the!incoming!water!was!assumed!to!be!divided!equally!over!the!five!spillway!channels!of!
the!Shuibuya!hydropower!plant.!The!flow!of!water!was!also!assumed!to!be!entering!the!spillway!
channels!horizontally,!i.e.!the!velocity!of!the!incoming!water!is!assumed!to!be!in!horizontal!direction!
only![3].!The!five!spillway!channels!can!be!modeled!one!and!one!in!Fluent!without!any!significant!
impact!on!the!results![3][4].!Furthermore,!the!approaching!channel!was!neglected!in!the!simulation!
model.!The!fact!that!the!curved!part!of!the!approaching!channel!could!be!neglected!was!shown!in!
the!report!Numerical*modeling*of*a*slotted*flip*bucket*spillway*system*[4].!However,!the!simulation!
model!used!in!this!study!also!neglected!the!straight!part!of!the!approaching!channel,!which!means!
that!the!inflow!of!water!in!the!simulation!model!takes!place!just!upstream!the!flip!bucket!terminal,!
see!figure!1!below!for!a!birdview!of!the!Shuibuya!spillway!system.!Furthermore,!it!was!assumed!that!
the!effect!of!boundary!layers!could!be!neglected!due!to!the!large!flow!rates![5].!
!
Figure#1.#Birdview#of#the#Shuibuya#hydropower#plant#[6].#
When!interpreting!the!simulation!results!a!definition!of!how!large!the!volume!fraction!of!water!shall!
be!in!order!to!count!as!water!has!to!be!established.!It!was!assumed!that!when!a!volume!contains!
more!than!or!equal!to!30!%!volume!fraction!of!water,!it!should!be!defined!as!water.!This!is!a!fairly!
common!definition!that!has!been!used!in!several!studies![7].!
2.&Background&and&theory&
One!reason!why!it!is!important!to!investigate!the!behavior!of!spillway!flows!is!its!correlation!to!scour!
and!in!worst!case,!dam!failure.!How!spillway!design!is!related!to!energy!dissipation!and!scour!is!
explained!in!the!following!sections!2.1!and!2.2.!
2.1.&Energy&dissipation&and&scour&
Large!amounts!of!energy!have!to!be!dissipated!in!high!head!dams.!The!spillway!and!downstream!
basin!design!is!therefore!important!regarding!energy!dissipation![8].!!
!
Figure#2.#This#figure#shows#where#the#five#different#phases#of#energy#dissipation#take#place#in#a#surface#spillway#with#a#
spilling#basin#[8].#Phase#4#and#5#are#not#considered#in#this#study.#
Energy!dissipation!can!be!divided!into!five!phases,!as!shown!in!figure!2!above.!
1. on!the!spillway!surface!
2. in!a!free!falling!jet!
3. at!impact!into!the!downstream!pool!
4. in!the!stilling!basin!
5. at!the!outflow!into!the!river!
Energy!dissipation!is!dependent!on!the!spillway!design,!e.g.!the!chosen!specific!discharge!or!
difference!between!upstream!and!downstream!water!levels.!The!design!of!the!basin!also!has!a!large!
impact!on!the!energy!dissipation,!however!the!focus!of!this!study!lies!on!the!dispersion!and!aeration!
of!the!free!falling!jet!and!therefore!there!will!be!no!further!discussions!regarding!the!energy!
dissipation!at!phase!4!and!5.![8]!!
The!energy!loss!in!the!spillway!depends!on!a!number!of!factors,!among!others!the!height!of!the!
spillway!crest!above!its!toe!(S![m]),!the!overfall!head!(H![m])!and!the!specific!discharge!(q![m3/sm2]).!
At!Shuibuya,!as!in!many!modern!spillway!designs,!energy!dissipation!is!increased!using!free;falling!
jets.!This!can!be!achieved!either!at!the!end!of!a!ski;jump!or!downstream!of!a!flip!bucket.!The!energy!
losses!in!phase!2!are!not!very!substantial,!unless!terminal!velocity!is!reached,!even!if!the!jet!is!
disintegrated![8].!!
An!advantage!with!free;falling!jets!is!that!large!energy!dissipation!occurs!as!the!jet!hits!the!
downstream!basin,!phase!3![8].!However,!large!amounts!of!kinetic!energy!in!a!concentrated!jet,!
plunging!into!the!basin!downstream!the!spillway!can!result!in!problems!with!scour.!Scour!might!
undermine!dam!structure!and!result!in!dam!failure![9].!In!jet!spillways!it!is!beneficial!to!have!a!
dispersed!and!intensively!aerated!jet!nappe!in!order!to!increase!the!energy!dissipation!in!the!basin!
[7].!A!dispersed!and!aerated!jet!nappe!also!results!in!a!decreased!risk!of!scour![10].!!
Large!energy!dissipation!is!characterized!by!large!jet!throw!distance.!Jet!throw!distance!is!defined!as!
the!distance!between!the!inner!and!outer!edge!of!the!jet!nappe,!see!figure!3!below.!Both!inner!and!
outer!edges!are!measured!as!the!horizontal!distance!from!the!spillway!terminal!to*respective!plunge!
edge.!
!
Figure#3.#A#jet#nappe#can#be#the#result#when#water#falls#over#a#sharp#edge,#as#in#this#figure.#The#jet#nappe#is#defined#as#a#
free#falling#water#jet.#The#jet#throw#distance#is#the#distance#between#inner#and#outer#edge#[11].#
2.2.&Spillways&
Overfall!spillways!are!commonly!built!at!high!head!dams![12].!!An!overfall!spillway!may!terminate!
with!a!free;falling!jet!or!with!a!ski!jump!slope!followed!by!a!more!or!less!concentrated!jet.!Figure!2!in!
section!2.1!shows!an!overfall!spillway!with!a!ski!jump!slope![13].!The!jet!can!plunge!into!a!pool!and!it!
is!often!desired!to!produce!a!jet!that!does!not!expose!the!plunging!pool!to!high!dynamic!pressures.!
This!is!achieved!with!a!spread!jet!with!large!air!entrainment![12].!
A!stepped!spillway!is!usually!an!overfall!spillway!followed!by!steps!on!the!spillway!face!in!order!to!
decrease!the!energy!dissipation!in!phase!1![13].!
At!earth!and!rockfill!dams,!side;channel!spillways!are!mainly!used.!In!a!side;channel!the!water!from!
the!reservoir!comes!in!from!the!side!of!the!channel,!see!figure!4.!The!channel!is!followed!by!a!chute!
or!a!tunnel.!The!side!of!the!channel,!where!the!water!comes!in!from!the!reservoir,!is!usually!designed!
as!a!normal!overfall!spillway.!Regarding!energy!dissipation!it!is!beneficial!if!the!side!of!the!channel!
facing!the!reservoir!is!deep!and!steep.!This!design!is!usually!expensive!though,!and!it!is!often!more!
beneficial!to!build!a!shallow!and!wide!channel!from!an!economical!point!of!view![13].!
!
Figure#4.#SideLchannel#spillway#seen#from#above#[13].#
A!chute!is!a!steep!channel,!which!can!be!used!to!transport!discharge!from!a!side;channel!with!low!
overfall.!In!order!to!increase!the!energy!dissipation!in!the!chute!the!outlet!can!be!widened!gradually.!
This!way!the!flow!per!unit!width!will!decrease,!resulting!in!increased!energy!dissipation![13].!
The!shaft!spillway!is!funnel;shaped,!which!means!that!a!circular!inlet!is!connected!to!a!tunnel!that!
has!a!90!degrees!bend,!see!figure!5.!The!tunnel!is!sometimes!part!of!the!bottom!outlet!or!a!turbine!
tailrace!tunnel![13].!
!
Figure#5.#A#shaft#spillway#[13].#
Siphon!spillways!are!closed!conduits!shaped!as!a!U!turned!upside;down.!The!main!advantages!of!
siphon!spillways!is!the!possibility!to!increase!the!specific!flow!rate!and!the!fact!that!it!will!contribute!
with!automatic!control!of!water!levels!within!a!small!range!of!flow!rates![13].!
To!sum!it!up,!where!and!how!the!energy!dissipation!will!occur!can!be!determined!with!different!ways!
of!design.!In!order!to!achieve!optimum!energy!dissipation,!the!spillways!are!often!tailor;made!for!
specific!projects![13].!!This!is!also!the!case!at!the!Shuibuya!hydropower!plant!with!its!ski!jump!
spillway!system!with!flip!bucket!terminals.!A!flip!bucket!terminal!is!a!narrowing!structure!at!the!end!
of!the!spillway!channel.!This!means!that!the!water!jet!will!have!a!higher!velocity!when!it!exits!the!
spillway!terminal,!and!therefore!reach!a!longer!jet!throw!distance.!!!
2.3.&Shuibuya&
The!Shuibuya!hydropower!plant!is!located!in!Qingjiang!River!in!the!Hubei!Province.!It!has!an!
underground!powerhouse!with!the!installed!capacity!1600!MW![14].!This!hydropower!station!was!at!
the!time!of!construction!the!major!power!source!for!peak!load!regulation!in!the!Central!China!Grid,!
with!a!reservoir!capacity!of!4.58!billion!m3.!It!has!a!concrete!faced!rockfill!dam!(CFRD),!which!at!the!
time!of!construction!was!the!highest!in!the!world!with!its!233!meters.!!The!discharge!tunnel!is!on!the!
right!bank!and!the!spillway!on!the!left!bank.!The!power!plant!started!operating!in!2007![15].!!
!
Figure#6.#Birdview#of#Shuibuya#hydropower#plant#[6].#
This!study!is!focused!on!the!spillway!system,!which!consists!of!five!surface!spillways.!There!is!an!open!
approach!channel!that!terminates!in!five!open!spillway!channels.!Figure!6!shows!the!hydropower!
plant!seen!from!above.!The!five!channels!all!have!identical!rectangular!geometry,!but!different!
horizontal!lengths!and!elevations!of!the!flip!bucket!terminals,!which!figure!7!below!illustrates.!The!
upper!structure!shows!a!vertical!cross!section!of!spillway!channel!5!and!the!lower!shows!all!five!
channels!from!above.!As!seen!in!figure!7!the!length!of!the!channels!varies!from!227.5!m!to!296.0!m.!
The!elevation!of!the!flip!bucket!terminal!varies!from!309!m!(channel!5)!to!252!m!(channel!1)!above!
sea!level.!!
!
Figure#7.#Schematic#display#of#the#spillway#system.#The#upper#part#of#the#figure#shows#a#vertical#crossLsection#of#the#
spillways.#The#lower#part#shows#a#horizontal#crossLsection#of#the#spillways#[6].#
The!distance!from!flip!bucket!terminal!to!downstream!water!level!depends!on!the!current!flow!since!
the!water!level!varies!with!the!flow,!see!table!2!below.!The!design!flow!is!11,940!m3/s![7].!1.0!%!flood!
frequency!means!that!the!design!flow!will!occur!once!every!100!years,!and!the!14,810!m3/s!flow!will!
occur!once!every!500!years,!which!corresponds!to!a!flood!frequency!of!0.2!%.!!
Table#2.#The#elevation#of#upstream#and#downstream#water#level#depends#on#the#flow#rate,#the#different#water#levels#are#
represented#in#this#table#[9].#
Water!level!variations!due!to!different!flow!rates!
Flood!
frequency!
Flow!rate!
[m3/s]!
Upstream!water!level!
[m]!
Downstream!water!level!
[m]!
0.2%! 14810! 400.8! 226.8!
1.0%! 11940! 398.0! 222.0!
!
2.4.&Scale&model&
Engineering!problems!involving!fluid!mechanics!are!often!solved!experimentally!using!scale!models.!
The!flow!conditions!or!the!geometry!of!the!structure!for!the!flow,!or!both,!are!often!so!unique!that!
previous!experimental!results!cannot!be!used!since!the!conditions!differ!too!much.!The!actual!
engineering!design!is!then!called!a!prototype!and!the!replica!a!model.!It!is!even!more!necessary!to!
perform!empirical!work!when!it!comes!to!turbulent!flows!since!they!are!difficult!to!predict.!
Furthermore!it!is!more!complicated!to!describe!turbulent!behavior!in!small;scale!flows.!This!means!
prototype.!Turbulent!flows!are!also!very!difficult!to!solve!using!simple!calculation!systems.!The!
complexity!of!the!flow!conditions!requires!very!large!computational!power![16].!However,!with!the!
development!of!computational!power!during!the!latter!half!of!the!20th!century!the!possibility!to!solve!
turbulent!flow!problems!with!the!help!of!computers!has!increased![17].!One!of!the!reasons!to!further!
investigate!the!possibility!to!solve!fluid!mechanical!problems!using!Computational!Fluid!Dynamics!
(CFD)!in!combination!with!(or!instead!of)!scale!models!is!that!it!is!both!expensive!and!takes!long!time!
to!build!experimental!models.!If!one!can!find!and!validate!a!numerical!model!this!can!be!used!to!
optimize!the!structure!before!building!a!model.!Different!parameters!can!more!easily!be!changed!in!a!
numerical!model!and!in!the!end!might!save!both!time!and!money![18].!
2.4.1.&Shuibuya&scale&model&project&
Previous!to!this!research!task,!experiments!were!performed!on!a!model!of!the!Shuibuya!hydropower!
plant!with!the!purpose!of!examining!the!effects!of!flood!discharge!and!possible!energy!dissipation.!As!
explained!in!section!2.1!hydraulic!structures!downstream!of!spillways!are!susceptible!to!failure!from!
scour.!The!risk!of!scour!increases!with!rapid!jet!flows!from!the!spillways!as!sediments!are!transported!
by!the!currents,!which!damages!the!structures!of!the!basin.!In!order!to!decrease!scour!in!
downstream!hydraulic!structure,!a!bed!enforcement!consisting!of!a!concrete!lining!can!be!used.!
However,!in!many!cases!the!most!economical!is!to!not!use!bed!protection.!Instead!the!solution!is!to!
build!energy!dissipating!spillway!structures![8].!At!Shuibuya!the!spillway!structure!is!designed!to!be!
used!in!this!purpose!and!it!is!a!system!with!slotted!flip!bucket!terminals.!The!spillway!has!high!head!
and!the!design!discharge!rate!is!large;!11,940!m3/s.!
The!spillway!structure!was!built!up!in!a!model!scaled!1:100!and!three!different!spillway!designs!were!
investigated![9].!Among!others!local!scour!depth,!flow!velocity!profile!in!downstream!river,!pulsating!
pressures!and!jet!nappe!orbit!was!investigated!to!evaluate!the!spillway!structures.!The!scheme!that!
showed!the!best!overall!results!is!the!same!as!the!structure!now!built!at!Shuibuya![19].!!
In!this!study!the!spillway!structure!of!Shuibuya!is!numerically!investigated!regarding!the!jet!throw!
distance!and!this!is!subsequently!compared!to!the!results!from!the!experiments!on!the!scale!model.!
The!experimental!measurements!were!made!for!eight!different!flow!rates,!from!5,000!m3/s!to!16,300!
m3/s.!The!maximum!flow!through!the!turbines!is!800!m3/s.!Up!until!the!flow!through!the!spillways!
reaches!10,000!m3/s!the!turbines!can!be!in!full!operation.!However,!for!larger!flow!rates!the!turbines!
are!shut!down!and!the!entire!flow!goes!through!the!spillways,!in!order!to!protect!the!turbines.!The!
design!flow!for!the!spillways!is!11,940!m3/s,!which!occurs!once!every!100!years.!This!study!is!focused!
on!the!larger!flow!rates!for!when!the!turbine!is!shut!down.!The!largest!flow!rate!is!not!numerically!
simulated!since!the!experimental!measurements!could!not!be!completed.!Flows!below!11,940!m3/s!
are!not!simulated!because!the!spillway!gates!are!not!fully!open.!For!a!summary!of!the!numerical!
simulations!see!table!1!in!section!1.3.!The!corresponding!experimental!results!are!summarized!in!
table!3!below.!Note!that!the!inner!edge!for!simulation!3!(i.e.!spillway!3,!flow!rate!14,810!m3/s)!is!
missing!due!to!measurement!failure.!
# #
Table#3.#Summary#of#the#experimental#results#used#for#comparison#to#the#numerical#results.#Case#1#corresponds#to#
simulation#1,#case#2#corresponds#to#simulation#2#etcetera.#
Experimental!results!
! ! !
Case! Inner!edge![m]! Outer!edge![m]! Jet!throw!distance![m]! Flow!rate![m3/s]!
Case!1! 115.0! 245.0! 130.0! 14,810!
Case!2! 105.0! 234.0! 129.0! 11,940!
Case!3! ;! 228.5! ;! 14,810!
Case!4! 133.0! 232.5! 99.5! 11,940!
Case!5! 135.0! 201.5! 66.5! 14,810!
Case!6! 130.0! 208.0! 78.0! 11,940!
!
Case!1!corresponds!to!simulation!1,!i.e.!it!is!the!experimental!result!for!channel!1!and!a!flow!rate!of!
14,810!m3/s,!case!2!is!channel!1!and!11,940!m3/s!etcetera.!!
2.5.&Computational&Fluid&Dynamics&
Today!it!is!possible!to!do!numerical!computations!and!solve!fluid!mechanical!problems!using!
Computational!Fluid!Dynamics!(CFD).!This!is!now!a!large!part!of!the!industry!and!companies!such!as!
General!Electric!and!Boeing!are!using!CFD.!With!CFD!it!is!possible!to!design!and!redesign!a!structure!
multiple!times!before!building!the!real!life!model!or!prototype![17].!Regarding!open!channel!flows!
there!has!been!extensive!research!on!how!to!discretize!and!solve!the!non;linear!partial!differential!
equations!(PDEs)!that!constitute!a!fully!three;dimensional!open!channel!flow!equation!system.!There!
are!now!both!in;house!and!commercial!codes!available!for!simulating!the!behavior!of!fluid!flows![20].!!
The!Ansys!Fluent!code!is!one!of!the!available!codes!for!simulating!fluid!flow!behavior.!Fluent!provides!
the!possibility!to!model!flow!properties!and!events!such!as!turbulence,!and!heat!transfer!for!instance!
[21].!It!is!useful!within!the!industry,!when!modeling!for!example!the!flow!of!air!over!an!aerofoil!or!
combustion!in!a!furnace,!and!is!used!by!thousands!of!companies!all!over!the!world![22].!!
2.5.1.&NavierLStokes&equations&
The!Navier;Stokes!equations!are!nonlinear!PDEs!that!are!widely!used!when!investigating!events!
within!fluid!mechanics.!They!consist!of!a!set!of!equations!in!a!differential!form!of!Newton’s!second!
law!applied!to!motion!of!fluid![16].!Fluent!solves!the!Navier;Stokes!equations!in!each!control!volume.!
The!Navier;Stokes!equations!describe!the!conservation!of!three!quantities;!mass,!momentum!and!
energy.!In!three!dimensional!space!the!Navier;Stokes!equations!make!up!an!equation!system!
consisting!of!five!fully!coupled,!time;dependent!PDEs.!These!five!equations!contain!five!unknown!
parameters,!a!velocity!vector!consisting!of!three!velocity!components!(one!for!each!direction!in!
space)!and!two!thermodynamic!quantities,!e.g.!temperature!and!pressure![23].!
When!assuming!continuity!of!the!flow!properties!the!law!of!conservation!can!be!expressed!as!!
!!
!" + ∇ ∙ ! − !! = !!!
Where!!!is!a!flow!quantity!(mass,!momentum!or!energy),!!!is!flux,!!!!is!a!source!of!a!flow!quantity!
!!through!a!volume!and!! !is!a!source!of!flow!quantity!!!through!a!surface![23].!
Applying!the!law!of!conservation!on!mass!gives!the!law!of!mass!conservation!
!"
!"+ ∇ ∙ ρV = 0!
which!is!also!called!the!continuity!equation,!where!!![kg/m3]!is!density!and!!![m/s]!is!velocity.!The!
law!of!mass!conservation!states!that!in!a!fluid!system,!mass!cannot!disappear!from!the!system!nor!be!
created.!Note!that!only!convective!transport!is!possible!for!mass!transportation!since!no!diffusive!
transportation!is!possible![23].!
Momentum!is!defined!as!the!product!of!mass!and!velocity.!Assuming!Newtonian!fluid,!constant!
viscosity!and!incompressible!fluid!gives!the!Navier;Stokes!equation!of!motion!
!!V
!" + ! V ∙ ∇ V = −∇! + ! ∆V +1
3∇(∇ ∙ V) + !!!!
where!!![Pa∙s]!is!dynamic!viscosity!and!!!!![N]!represents!body!forces!.The!conserved!energy,!!![J],!
in!a!fluid!consists!of!internal!energy!plus!kinetic!energy,!!!!!.!The!total!energy!per!unit!of!volume!is!
indicated!by!!".!The!conservation!of!energy!can!then!be!expressed!as!
!"E
!" + ∇ ∙ !v! − !∇! − ! ∙ V = !!+ !!! where!!!![N]!is!external!forces!and!!![J]!is!total!enthalpy!defined!as!
! = ! +!
!! where!!![Pa]!is!pressure![23].!
2.5.2.&Turbulence&
Turbulence!occurs!in!most!natural!flows!and!is!characterized!by!statistical!fluctuations!of!all!flow!
quantities![17].!There!is!intense!mixing!and!unsteady!flow.!The!flow!is!described!as!unsteady!when!
the!velocity!fluctuates!with!time!at!any!point!in!e.g.!a!pipe!or!a!channel.!Generally!a!flow!with!high!
velocity!is!turbulent,!and!this!is!also!the!most!common!flow.!However,!the!Reynolds!number!is!a!
more!accurate!index!to!use!for!determining!flow!character![16].!!
In!open!channel!flow!the!Reynolds!number,!!"![;],!is!defined!as!
!!!!!!" =!!!
! =!
!×!
!!!!!!!!!!!(1)!
where!!![m/s]!is!velocity,!!![mm2/s]!is!viscosity,!!!![m]!is!hydraulic!radius,!!![m2]!is!cross;sectional!
area!and!!![m]!is!wetted!perimeter.!The!flow!is!fully!turbulent!when!it!is!larger!than!750.!For!the!two!
different!flow!rates!examined!the!Reynolds!number!is!much!larger!than!750,!see!appendix!I!for!
calculations.!
!"!!"#$ ≅ 3.77 ∙ 10!≫ 750!
!"!"#!$≅ 4.25 ∙ 10! ≫ 750!
[16]!
Thus!the!flow!in!the!channel!is!fully!turbulent.!
The!standard!approach!to!treating!turbulent!flow!is!to!represent!the!velocity!as!an!average!value!plus!
a!fluctuating!quantity![16].!In!many!situations!the!fluctuations!can!be!calculated!from!the!average!
values!of!the!different!flow!quantities.!The!fluctuations!of!a!flow!quantity!often!deviate!10!%!from!
the!average!value.!However,!in!some!flow!regions!the!fluctuations!may!vary!much!more!than!10!%!
from!the!average!values.!As!computer!power!has!increased!so!has!the!possibilities!to!simulate!large;
scale!turbulent!fluctuations!from!the!time;dependent!Navier;Stokes!equations![17].!!
2.6.&Simulating&in&Fluent&
In!order!to!solve!a!CFD;problem!using!Fluent,!the!working!process!consists!of!a!number!of!steps.!
These!steps!are!explained!in!sections!2.6.1!–!2.6.3.!First,!a!preprocessing!software!is!needed.!For!this!
particular!study!Gambit!was!used!for!preprocessing.!
2.6.1.&Preprocessing&
When!performing!a!simulation!for!a!continuous!volume!or!surface,!it!is!necessary!to!represent!the!
continuous!geometry!in!discrete!points!since!a!computer!can!only!recognize!numbers.!This!process!is!
called!discretization,!and!can!be!done!in!Gambit!software.!The!result!from!the!discretization!is!a!
mesh,!or!grid,!consisting!of!cells![23].!
2.6.1.1.%Mesh%design%
When!creating!a!mesh!the!following!has!to!be!considered:!
• Setup!time,!that!is!the!time!it!takes!to!create!the!mesh!
• Computational!power!
• Numerical!diffusion,!that!is!the!error!from!discretization!of!a!continuous!volume!!
[24]!
The!results!from!a!CFD!simulation!are!extremely!depending!on!the!properties!and!the!quality!of!the!
grid![23].!Knowledge!regarding!the!different!design!possibilities!of!a!mesh!has!to!be!obtained!in!order!
to!create!a!grid!of!high!quality.!The!first!choice!to!make!is!whether!the!grid!should!be!structured!or!
unstructured.!The!second!choice!is!choosing!an!element!type,!which!represents!the!shape!of!the!cells!
[2].!!
In!a!structured!grid,!each!mesh!point!is!made!up!of!intersecting!family!lines.!There!will!be!one!family!
of!lines!for!each!space!dimension,!resulting!in!two!families!of!lines!in!2D!and!three!families!of!lines!in!
3D.!In!2D!each!mesh!point!will!consist!of!two!family!lines!intersecting,!whereas!in!3D!the!mesh!points!
will!consist!of!three!family!lines!intersecting.!Note!that!a!mesh!point!cannot!consist!of!two!lines!from!
the!same!family!intersecting,!it!has!to!be!one!line!only!from!each!family![2].!A!structured!grid!will!
consist!of!rectangles!or!squares!in!2D!and!hexahedrons!in!3D![24].!
An!unstructured!grid!consists!of!mesh!points!distributed!arbitrarily.!The!element!types!can!be!
triangular,!quadrilateral!or!polygonal!in!2D,!whereas!the!3D!element!types!can!be!represented!by!
various!polyhedrals,!for!example!prisms!or!pyramids![2].!
!
Figure#8.#The##outlet#of#the#channel#meshed#with#a#structured#grid.#
!
Figure#9.#The#outlet#of#the#channel#meshed#with#an#unstructured#grid.#
2.6.1.2.%Setup%time%
When!choosing!between!structured!and!unstructured!grid!the!geometrical!complexity!has!to!be!
considered![2].!When!modeling!complex!geometries!it!can!be!time!consuming,!or!even!impossible,!to!
create!a!structured!grid.!For!this!reason!it!can!be!very!useful!to!create!an!unstructured!grid![24].!
2.6.1.3.%Numerical%diffusion%
Numerical!diffusion!arises!due!to!representation!of!a!continuous!value!with!discrete!numbers.!The!
numerical!diffusion!can!be!minimized!by!making!the!mesh!denser!and!by!aligning!the!mesh!with!the!
flow![24].!A!dense!mesh!provides!accuracy!to!the!model!since!a!larger!number!of!cells!in!the!mesh!
will!give!a!better!approximation!of!the!continuous!geometry.!As!the!size!of!the!cells!decrease!the!
error!of!a!numerical!simulation!will!decrease!even!faster![1].!
There!is!no!such!flow!that!will!be!aligned!with!an!unstructured!grid.!A!flow!can!be!aligned!with!a!
structured!grid!though,!and!thereby!decrease!the!numerical!diffusion.!A!complex!flow!might!not!be!
aligned!even!with!a!structured!grid,!but!a!simple!flow,!such!as!a!flow!through!a!long!duct,!will!be!
aligned!with!a!structured!grid.!Using!a!structured!grid!for!such!a!simple!flow!will!result!in!a!more!
accurate!solution!with!fewer!cells!compared!to!an!unstructured![24].!
The!effects!of!numerical!diffusion!can!be!minimized!by!using!the!second!order!discretization!scheme!
(i.e.!transformation!of!continuous!values!to!discrete!values).!This!is!a!more!precise!discretization!
method!than!the!default!first!order!solution!control!in!Fluent![24].!Since!it!is!more!computationally!
expensive!it!was!not!used!in!the!original!simulations!of!this!study,!instead!it!was!tested!in!the!
sensitivity!analysis!to!investigate!the!possible!impact!of!this!choice.!
2.6.1.4.%Computational%power%
A!large!number!of!cells!will!require!a!lot!of!computational!power,!since!Fluent!solves!the!flow!
equations!for!each!cell.!Since!a!larger!number!of!cells!result!in!better!accuracy,!it!is!more!important!
to!have!a!dense!mesh!in!areas!where!the!flow!parameters!have!large!gradients.!A!structured!mesh!
will!force!cells!to!be!placed!in!regions!where!it!is!not!necessary!to!have!a!dense!mesh.!An!
unstructured!mesh!will!on!the!other!hand!allow!clustering!of!cells!in!regions!where!high!accuracy!is!
needed.!Therefore!an!unstructured!mesh!can!sometimes!make!it!possible!to!create!a!mesh!with!
fewer!cells![24].!
!
The!geometry!of!the!Shuibuya!spillways!is!not!very!complex!and!can!therefore!easily!be!meshed!with!
a!structured!grid.!The!water!will!flow!along!the!channel!and!thus!it!can!also!follow!the!structured!
grid,!which!may!cause!the!numerical!diffusion!to!decrease.!The!chosen!grid!for!the!channel!is!
displayed!in!figure!8!and!below!in!figure!10!is!the!chosen!grid!for!the!jet!nappe!shown.!Both!figures!
show!only!a!small!zoomed!part!of!the!structure.!!
!
!
Figure#10.#Structured#grid#of#the#jet#nappe,#upper#left#corner.#The#width#in#zLdirection#is#4#meters.# !
2.6.1.5.%Grid%independence%
Although!convergence!in!each!simulation!is!very!important!it!is!not!always!sufficient!to!guarantee!
correct!results.!A!simulation!may!very!well!have!converged,!but!as!the!mesh!resolution!affects!the!
solution,!the!results!may!differ!depending!on!the!mesh!density.!Therefore!it!is!important!to!examine!
whether!or!not!the!results!are!independent!from!the!grid.!Not!checking!if!the!solution!is!grid!
independent!is!a!common!cause!of!error!in!CFD!simulations.!Once!this!is!done!for!the!problem!at!
hand!the!same!mesh!size!can!be!used!again!for!similar!problems.!!To!evaluate!both!convergence!and!
grid!independence!the!parameters!that!are!most!important!for!the!results!should!be!examined![25]!
[26].!!
When!determining!which!mesh!resolution!that!should!be!used!it!is!important!to!consider!accuracy!of!
the!solution,!computational!power!and!time!constraints.!Thus!although!a!finer!mesh!is!almost!always!
better,!the!choice!of!mesh!density!is!limited!by!the!time!constraints!for!the!project!at!hand!and!
available!computational!power.!
A!grid!independence!study!is!done!by!stepwise!comparison!of!finer!meshes.!A!model!with!a!given!
mesh!resolution!is!run!to!convergence!and!the!results!are!saved.!After!this!the!mesh!is!refined!and!
the!model!is!again!run!until!it!converges.!Then!the!results!from!the!previous!density!level!are!
compared!to!the!results!from!the!simulation!with!the!finer!mesh!density.!If!the!solution!has!changed!
it!is!not!independent!of!the!grid!and!the!mesh!must!be!further!refined.!These!steps!are!continued!
until!a!grid!independent!solution!is!reached.!A!grid!independent!solution!means!that!the!solution!
from!one!level!of!mesh!density!to!the!next!does!not!change!or!changes!within!an!acceptable!
tolerance.!Once!this!is!obtained!one!should!choose!the!mesh!with!the!least!number!of!cells!that!gives!
a!grid!independent!solution,!so!as!to!minimize!the!simulation!run!time![25]![27]![24].!!
2.6.2.&Settings&in&FLUENT&
A!number!of!settings!have!to!be!chosen!in!Fluent!in!order!to!make!an!as!accurate!description!of!a!
specific!CFD;problem!as!possible.!Theory!behind!the!settings!used!to!simulate!for!the!Shuibuya!
spillway!system!is!reviewed!in!sections!2.5.2.1!;!2.5.2.7.!
2.6.2.1.%Segregated%solver
%
There!are!three!different!solvers!available!in!Fluent;!segregated,!coupled!implicit!and!coupled!
explicit,!see!figure!11!below.!All!three!will!provide!accurate!solutions!for!a!wide!range!of!CFD;
problems,!depending!on!the!problem!one!of!the!solvers!might!perform!better.!The!default!setting!in!
Fluent!is!the!segregated!solver.!One!of!the!coupled!solvers!might!be!a!better!choice!when!solving!for!
high;speed!compressible!flows,!highly!coupled!flows!with!strong!body!forces!or!if!a!very!fine!mesh!is!
being!used![28].!
!
Figure#11.#Solver#option#window#in#Fluent#[28].#
It!is!not!possible!to!simulate!for!volume!of!fraction!(VOF)!models!with!the!coupled!solvers![28],!which!
is!why!the!segregated!solver!was!used!in!this!study.!See!section!2.6.2.3!for!more!information!
regarding!VOF!models.!
2.6.2.2.%Unsteady%solver%
When!performing!simulations!with!Fluent!software!either!a!steady!or!a!unsteady!solver!is!used,!see!
figure!11!above.!The!unsteady!solver!solves!the!Navier;Stokes!PDEs!with!respect!to!time,!whereas!
the!steady!solver!does!not!involve!time!derivatives,!but!uses!only!space!derivatives![29].!!
!
When!choosing!between!steady!and!unsteady!solver!the!properties!of!the!flow!needs!to!be!
considered.!!When!modeling!for!large!flows!in!the!spillways!of!the!Shuibuya!dam!the!boundary!
conditions!were!set!to!be!stationary!and!the!flow!was!of!turbulent!character!according!to!equation!1!
in!section!2.5.2!above.!Also!see!appendix!I!for!turbulence!calculations.!
!
When!solving!for!a!time;dependent!problem!the!unsteady!solver!should!be!enabled![28].!Turbulence!
has!a!time;dependent!behavior,!but!depending!on!what!length!scales!that!are!studied!a!turbulent!
flow!might!appear!steady.!Considered!at!very!small!scales!all!turbulent!flows!are!unsteady.!However,!
it!might!not!always!be!interesting!to!study!the!fine!scales!and!when!looking!at!larger!scales!a!
turbulent!flow!can!appear!to!be!steady.!It!is!preferred!to!work!with!a!model!that!discovers!unsteady!
behavior!at!a!desired!scale.!According!to!professor!Hirsch,!author!of!Numerical*Computation*of*
Internal*and*External*Flows,!it!is!therefore!only!recommended!to!work!with!the!steady!solver!if!it!is!
guaranteed!that!the!flow!will!remain!steady.!Even!in!the!case!of!a!flow!that!will!remain!steady!for!
sure,!no!advantages!will!be!gained!from!using!the!steady!solver!and!the!unsteady!solver!might!as!
well!be!used!for!this!kind!of!steady!flow![1].!!
!
From!the!statements!above,!it!was!decided!that!the!unsteady!solver!would!be!the!best!choice!for!the!
Shuibuya!spillways.!
2.6.2.3.%Multiphase%model%
A!multiphase!model!means!that!two!or!three!phases!occur!in!the!model.!A!free!water!flow!is!an!
example!of!a!system!that!requires!a!multiphase!model![30].!Since!the!Shuibuya!spillways!consist!of!
open!channels,!a!multiphase!model!had!to!be!used!when!performing!simulations.!!
There!are!two!different!approaches!for!solving!multiphase!flow!problems!in!Fluent;!the!Euler!;!Euler!
approach!and!the!Euler!–!Lagrange!approach.!The!Euler!–!Lagrange!approach!is!beneficial!to!use!
when!modeling!for!events!where!the!volume!fraction!of!one!of!the!phases!are!much!smaller!than!the!
volume!fraction!of!the!other!phase,!e.g.!liquid!fuel!combustion.!Within!the!Euler!–!Euler!approach!
neither!one!of!the!volume!fractions!can!be!neglected.!Phasic!volume!fraction!is!introduced,!which!
means!that!the!sum!of!the!volume!fractions!of!the!different!phases!in!a!control!volume!is!equal!to!
one![30].!!
There!are!three!multiphase!models!available!within!the!Euler!–!Euler!approach;!the!mixture!model,!
the!Eulerian!model!and!the!VOF!model.!The!VOF!model!was!chosen!since!it!is!adapted!for!two!or!
more!phases!that!are!immiscible,!which!is!the!case!for!free!surface!flows!such!as!the!Shuibuya!
spillway!channels![30].!
When!using!the!VOF!model,!the!volume!fraction!of!the!phases!involved!is!calculated!in!each!control!
volume![30].!This!function!was!used!when!interpreting!the!results!in!this!study,!which!is!described!in!
section!3.4.3.!
2.6.2.4.%Open%channel%flow%
The!setting!for!open!channel!flow!was!not!enabled!since!the!simulation!will!not!consider!a!uniform!
flow.!There!will!not!be!a!free!water!surface!downstream!the!spillway!(in!the!channel!model)!since!the!
water!flowing!out!from!the!channel!will!have!jet!flow!characteristics![5].!If!open!channel!flow!is!used!
in!Fluent!and!the!Froude!number!is!below!one,!Fluent!allows!for!the!downstream!water!level!to!
affect!the!upstream!water!level![31].!However,!since!the!channels!in!this!study!have!downward!
slopes!and!open!end!into!air!the!downstream!water!level!cannot!affect!the!upstream!water!level.!
The!Froude!number,!!"![;],!for!this!flow!is!calculated!with!equation!2!and!is!below!one.!The!open!
channel!setting!is!therefore!not!enabled!during!the!simulations!in!order!to!prevent!the!downstream!
water!level!from!affecting!the!upstream!water!level![5].!See!appendix!I!for!more!details!regarding!the!
calculation!parameters.!
!" = ! !
! ∙ !!!!!!!!!!!(2)!
Where!!!is!velocity!magnitude![m/s],!!!is!gravity![m/s2]!and!!!is!length!scale![m],!which!is!defined!as!
the!distance!from!the!bottom!to!the!free!water!surface.!
!" = ! 7.53788!!/!
9.81!
!!∙ 19.80!!
= 0.54!
[16]!
2.6.2.5%Turbulence%model%
In!a!turbulent!flow!the!velocity!fields!are!fluctuating!and!thus!the!transported!quantities,!such!as!
momentum!and!energy,!are!mixed.!These!fluctuations!can!be!of!very!high!frequency!and!small!scale.!
Thus!it!is!generally!too!computationally!expensive!to!solve!the!Navier;Stokes!and!conservation!
equations!directly!in!practical!engineering!solutions.!Therefore!a!set!of!adapted!Navier;Stokes!
equations!is!made!up!from!different!modifications!of!the!instantaneous!governing!equations.!The!
exact!equations!can!be!time;averaged,!ensemble;averaged!or!adjusted!in!other!ways,!but!these!
modifications!add!unknowns!to!the!equation;set!and!a!turbulence!model!is!needed!to!determine!
these!variables!in!terms!of!known!quantities.!There!are!a!number!of!turbulence!models!available!in!
Fluent!and!the!choice!has!to!be!made!depending!on!the!conditions!of!the!specific!case!that!is!to!be!
modeled![32].!For!the!flow!conditions!in!this!study!it!was!assumed!that!the!k;epsilon!model!was!the!
most!suitable.!In!the!turbulence!models!transport!equations!are!added!to!the!set!of!equations!that!
are!solved.!Which!additional!equations!that!are!used!depend!on!the!choice!of!turbulence!model.!For!
all!flows,!laminar!as!well!as!turbulent,!conservation!equations!for!mass!and!momentum!are!solved.!If!
the!simulated!flow!includes!heat!transfer!or!compressibility!an!equation!for!energy!conservation!is!
added![33].!The!flow!simulated!in!this!study!does!not!include!heat!transfer!or!compressibility!and!
thus!the!energy!equation!was!not!enabled.!Which!turbulence!model!to!choose!depends!on!the!
physics!of!the!flow,!the!established!practice!for!a!specific!class!of!problem,!how!much!computational!
power!and!time!there!is!available!for!the!simulation!and!the!level!of!accuracy!required.!
There!are!two!modeling!options!available!in!Fluent;!models!using!Reynolds;averaging!(or!ensemble;
averaging)!or!filtering.!Large!Eddy!Simulation!(LES)!“filters”!the!exact!Navier;Stokes!equations!to!
remove!the!eddies!that!are!smaller!than!the!size!of!the!filter.!Thus!by!modeling!less!of!the!turbulence!
(and!resolving!more)!the!error!due!to!the!turbulence!modeling!can!be!reduced.!This!is!very!
computationally!expensive!for!flows!with!high!Reynolds!numbers![32].!Since!the!flow!simulated!in!
this!study!has!a!very!high!Reynolds!number!and!the!computational!power!available!was!limited,!the!
other!option!was!used!(i.e.!Reynolds;averaging).!
The!Reynolds;averaged!Navier!Stokes!(RANS)!is!also!the!most!commonly!used!since!it!has!lower!cost!
in!computational!power!and!run!times.!The!method!used!decomposes!the!solution!variables!in!the!
instantaneous!Navier;Stokes!equations!into!the!mean!and!fluctuating!components,!e.g.!velocity!is!
divided!into!mean!velocity!and!fluctuating!velocity!components.!However,!this!leaves!the!equations!
with!additional!unknown!terms.!These!are!named!Reynolds!stresses!and!represent!the!effects!of!
turbulence.!The!set!of!equations!for!continuity,!momentum!and!energy!are!thus!not!closed,!and!by!
using!a!turbulence!model!to!resolve!the!Reynolds!Stresses!the!set!of!equations!are!closed.!Again!
there!are!two!ways!to!do!this,!either!an!isotropic!value!for!the!turbulent!viscosity!value!is!used,!this!is!
called!an!Eddy!Viscosity!Model,!or!a!Reynolds!Stress!Model!(RSM)!is!used.!The!later!uses!extra!
equations!to!solve!the!6!Reynolds!Stresses!and!dissipation!rate!to!get!an!anisotropic!solution.!Thus!it!
is!much!more!computationally!expensive!than!an!Eddy!Viscosity!Model.!Depending!on!the!flow!
situation,!it!is!not!always!worth!the!longer!time!and!higher!computional!cost.!It!is!most!commonly!
needed!for!high!swirling!flows!such!as!a!cyclone!separator.!As!this!study!does!not!concern!that!type!
of!flow!it!was!determined!that!an!Eddy!Viscosity!Model!was!to!be!used![32]![34].!!
There!are!seven!different!RANS!Eddy!Viscosity!Models!to!choose!from!in!Fluent.1!One!of!them!is!the!
k;epsilon!model.!This!is!a!two!equation!semi;empirical!model!most!suitable!for!free!shear!and!non;
wall!bounded!flow!behavior.!It!solves!model!transport!equations!for!the!turbulence!kinetic!energy!(k)!
and!its!dissipation!rate!(epsilon)!separately.!It!is!widely!used!due!its!robustness,!economy!and!
reasonable!accurate!solutions!for!a!wide!range!of!turbulent!flows.!However,!the!model!was!derived!
under!the!condition!that!the!flow!is!fully!turbulent![32]![34].!The!flow!in!the!spillways!simulated!is!
fully!turbulent!(see!2.4.2.!Turbulence).!After!discussions!with!the!supervisors!for!this!thesis!it!was!
concluded!that!this!model!was!the!best!option!for!this!study.!The!flow!rate!is!so!large!so!the!effect!of!
the!boundary!layers!was!assumed!to!be!negligible!and!thus!the!advantage!of!the!free;stream!
independence!of!the!k;epsilon!is!preferable.!Also!the!computational!power!and!time!constraints!
spoke!in!favor!of!using!a!robust!and!economical!model.!
However,!the!Shear!Stress!Transport!k;omega!model!(SST)!is!described!as!the!new!industrial!standard!
model!for!a!wide!range!of!flows.!Because!of!this!it!was!decided!to!test!it!in!the!sensitivity!analysis.!
SST!is!more!commonly!used!because!it!combines!the!free!stream!advantage!of!the!k;epsilon!model!
with!the!near;wall!advantages!of!the!k;omega!model.!The!standard!k;omega!model!includes!
modifications!for!low;Reynolds;effects!and!is!used!for!wall;bounded!flows.!Since!it!was!assumed!that!
these!effects!could!be!neglected!(in!choosing!the!k;epsilon!model)!the!sensitivity!analysis!will!
examine!the!potential!difference!to!when!this!is!included.!
For!both!turbulence!models!the!default!settings!were!used.!
2.6.2.6.%Boundary%conditions%
All!boundary!conditions!used!maintained!the!default!settings!except!for!the!velocity!inlet.!The!
boundary!conditions!are!all!presented!in!table!4!below.!The!water!flowing!towards!the!channel!inlet!
was!assumed!to!enter!entirely!perpendicular!to!the!cross;section!area!of!the!channel.!The!velocity!
could!therefore!be!calculated!from!equation!3!for!the!two!different!flow!rates,!see!appendix!I!for!the!
resulting!velocities.!
!!!!!!!!!!! = !
5!!!!!!!!!!!(3)!
Where!!!is!velocity![m/s],!!!is!flow!rate![m3/s]!and!!!is!area![m2].!
! !
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
1!Only!the!models!used!in!this!study!will!be!described.!For!more!information!on!different!turbulence!models!see!the!Fluent!
User!Guide!chapter!11.!!
Table#4.#Boundary#conditions#used#for#the#surfaces#in#the#models.#
Boundary!conditions!
Area! Type! Description!
Channel!inlet! Velocity!inlet! Allows!both!phases!to!flow!into!the!channel!with!a!given!velocity.!
Channel!walls! Wall! Does!not!allow!any!fluid!to!flow!through!the!surface.!Surface!
roughness!can!be!set.!
Channel!air!
surfaces! Pressure!inlet! Atmospheric!pressure!was!defined.!Allows!both!phases!to!flow!
through!the!surface.!
Channel!outlet! Pressure!outlet! Allows!both!phases!to!flow!through!the!surface.!No!pressure!was!set.!
Jet!nappe!inlet! Velocity!inlet! Allows!both!phases!to!flow!into!the!air!volume!with!a!given!velocity.!
The!input!parameters!was!set!from!the!profile!of!the!channel!outlet.!
Jet!nappe!
bottom! Wall! Does!not!allow!any!fluid!to!flow!through!the!surface.!
Jet!nappe!sides!
and!top! Pressure!inlet! Atmospheric!pressure!was!defined.!Allows!both!phases!to!flow!
through!the!surface.!
Jet!nappe!distal!
(in!x;direction)! Pressure!outlet! Allows!both!phases!to!flow!through!the!surface.!No!pressure!was!set.!
!
The!boundary!condition!wall!was!used!for!the!channel!walls.!During!the!original!simulations!the!
default!setting!for!surface!roughness!(0.5)!was!used!because!the!impact!of!the!surface!roughness!
was!assumed!to!be!negligible!due!to!the!high!flow!rate.!This!was!altered!during!sensitivity!analysis!II,!
see!section!3.4.1!for!further!information.!!
A!start!guess!is!required!for!all!equations!to!be!solved!during!the!simulation.!For!all!channel!
simulations!performed!during!this!study!the!default!guess!was!used!for!all!parameters!except!
velocity!in!x;direction.!That!is!velocities!in!y!and!z;direction,!turbulence!kinetic!energy!(k)!and!
turbulent!dissipation!rate!(ε).!For!the!jet!nappe!simulations!the!profile!values!of!the!channel!outlet,!
i.e.!the!flip!bucket!terminal,!was!set!as!initial!values!for!those!parameters!and!also!for!the!volume!
fraction.!
2.6.2.7.%Initializing%the%solver%
When!using!an!unsteady!solver!one!has!to!set!both!time!step!size!and!number!of!iterations!per!time!
step.!In!order!for!the!solution!to!converge!the!time!step!size!must!be!sufficiently!small.!Furthermore!
the!number!of!iterations!per!time!step!must!be!enough!for!the!solution!to!converge!within!each!time!
step![28].!
2.6.3.&PostLprocessing&
In!order!to!draw!any!conclusions!from!performed!simulations,!the!results!have!to!be!post;processed.!
The!first!thing!to!do!is!to!make!sure!that!the!simulations!have!converged.!If!the!simulations!are!
converged,!the!results!can!be!analyzed.!
2.6.3.1.%Convergence%
Numerical!simulation!is!an!iterative!process!and!the!flow!equations!will!be!solved!reputedly!in!order!
to!improve!the!solution.!When!the!solution!does!no!longer!change!with!the!iterations!or!if!the!
relative!error!has!reached!an!acceptable!tolerance!it!has!converged![34].!
There!are!several!different!ways!to!check!the!convergence!of!a!simulation.!The!following!criteria!
• The!residual!has!stagnated!at!a!pre;defined!value!
• Monitor!points,!showing!relevant!parameters,!has!stagnated!
• The!imbalance!of!the!solution!is!less!than!1!%!
[25]!
The!residual!shows!the!residual!sum!for!the!equations!used!in!a!simulation,!which!means!that!the!
number!of!residual!sums!varies!depending!on!the!simulation!and!how!many!equations!that!need!to!
be!solved!for!that!particular!simulation.!A!residual!is!the!imbalance!,!or!relative!error,!between!
current!iteration!and!previous!iterations!summed!over!all!computational!cells.!There!is!a!scaled!and!
an!unscaled!residual!to!choose!from!in!Fluent,!but!it!is!recommended!by!Fluent!to!use!the!scaled!one!
when!judging!convergence.!The!scaled!residual!is!the!default!setting!in!Fluent.!Fluent!recommends!a!
convergence!criterion!of!10;3!for!the!residuals,!when!this!is!achieved!the!relative!error!is!less!than!10;
3![28].!However,!examining!residuals!might!not!be!suitable!to!all!kinds!of!simulation!problems!due!to!
the!residual!definition.!This!is!why!more!convergence!criterion!should!be!examined.!For!example,!
due!to!the!definition!of!the!continuity!residual,!a!very!good!start!guess!may!result!in!a!high!residual!
value![35].!!
2.6.3.2.%Validation%
In!order!to!examine!the!accuracy!of!the!simulation!results!they!can!be!validated,!i.e.!compared!to,!
other!results.!Turbulence!is!one!of!the!most!difficult!behaviors!to!model,!which!is!why!it!is!very!
important!to!verify!numerical!methods!with!experimental!measurements![16].!
2.6.3.3.%Analytical%validation%
The!numerical!results!were!further!evaluated!with!equation!4!below.!This!is!an!empirically!based!
equation!that!is!used!for!computing!flip!bucket!throw!distances!and!was!produced!by!the!U.S.!Army!
Corps!of!Engineers![36].!
!!!!!!!!!!!!!!!!
ℎ!= !"#2!!+ 2!"#!! !"#!!!+ !
ℎ!!!!!!!!!!!(4)!
The!horizontal!position!of!the!jet!trajectory,!!![m],!is!calculated!using!this!equation.!The!other!
parameters,!velocity!head!ℎ!![m],!angular!difference!between!horizontal!plane!and!initial!velocity!
vector!!!![rad]!and!the!vertical!distance!from!downstream!water!level!to!the!initial!position!of!the!jet!
trajectory!!![m],!were!obtained!from!the!first!part!of!the!spillway!simulation.!It!is!important!when!
computing!the!trajectory!for!a!free!jet!to!use!either!the!initial!velocity!of!the!jet!itself,!or!as!in!this!
case!the!initial!velocity!head,!equation!5.!
ℎ!=!!!
2!!!!!!!!!!!(5)!
Where!!!!is!initial!velocity!of!the!jet![m/s]!and!!!is!gravity![m/s2].!
[36]!
2.6.3.4.%Sensitivity%analysis%
A!sensitivity!analysis!can!be!performed!In!order!to!investigate!how!much!a!specific!parameter!affects!
a!model.!Changing!the!parameter!of!interest,!and!keeping!all!other!parameters!constant,!makes!it!
possible!to!evaluate!the!impact!on!the!result!from!the!altered!parameter.!The!evaluation!is!
performed!by!comparing!the!resulting!outputs!from!the!different!input!settings.!!
3.&Method&
The!spillway!geometry!was!set!up!in!Gambit!by!using!coordinate!information!from!maps!of!the!
Shuibuya!spillway!system.!As!explained!in!section!1.2,!the!spillway!was!divided!into!two!volumes;!a!
channel!and!a!jet!nappe.!The!two!parts!of!the!simulation!model!are!shown!in!figure!12!below.!Both!
volumes!were!meshed!with!a!structured!grid,!the!required!density!of!the!grids!was!evaluated!in!a!
grid!independence!study,!see!section!3.3!below.!
!
& &
Figure#12.#The#volume#to#the#left#represents#a#channel,#and#the#volume#to#the#right#represents#a#jet#nappe.#
3.1.&Settings&in&Fluent&
An!overview!of!the!chosen!settings!when!simulating!the!channel!can!be!seen!in!figure!13!below.!
!
Figure#13.#Flow#chart#of#the#settings#made#in#Fluent#when#simulating#for#the#channel#flow.#
An!overview!of!the!chosen!settings!when!simulating!the!jet!nappe!can!be!seen!in!figure!14!below.!
!
Figure#14.#Flow#chart#of#the#settings#in#Fluent#when#simulating#for#the#jet#nappe.#
3.1.1.&Boundary&conditions&
Equation!3!in!section!2.6.2.6.!was!used!to!calculate!the!velocity!of!the!water!entering!the!channels!
during!the!two!different!flow!rates,!see!appendix!I!for!the!results!from!these!calculations.!Velocity!
inlet!was!used.!To!transfer!results!from!the!channel!simulation!a!profile!of!the!channel!outlet!was!
written,!containing!all!relevant!parameters!(velocity,!k,!epsilon,!volume!fraction!of!water).!This!was!
used!as!input!to!the!jet!nappe!inlet.!All!other!boundary!conditions!was!run!with!the!default!settings!
and!are!displayed!in!table!4!in!section!2.6.2.6.!&
3.1.2.&Solver&
In!order!to!find!solver!settings!suitable!for!a!specific!problem!it!is!possible!to!try!some!different!set!
ups.!It!was!found!that!a!time!step!of!0.05!s!with!40!iterations!per!time!step!was!suitable!for!this!
problem.!
3.2.&Grid&independence&study&
A!grid!independence!study!was!performed!by!simulating!spillway!1,!with!a!flow!rate!of!14,810!m3/s,!
with!several!different!mesh!sizes.!The!study!was!performed!by!evaluating!the!velocity!for!the!channel!
part!of!the!simulation,!and!the!volume!fraction!of!water!for!the!jet!nappe.!!
Nine!points!evenly!distributed!in!the!channel!outlet!were!chosen!for!evaluation!of!mesh!density.!The!
value!of!the!velocity!magnitude!in!a!point!was!measured!for!each!mesh!resolution.!A!comparison!
between!one!level!of!mesh!resolution!and!the!previous!mesh!density,!i.e.!a!more!coarse!mesh!
resolution,!was!performed!in!order!to!determine!which!mesh!size!that!would!be!sufficient!for!this!
study.!Five!different!mesh!densities!was!investigated!for!the!channel!and!three!different!mesh!
FLUENT!senngs!
Define!
Models!
Solver!
Time:!Unsteady! Mulpphase!model!
Model:!Volume!of!fluid!
VOF!parameters!
VOF!scheme:!Implicit!
(open!channel!flow!
disabled)!
Viscous!model!
Model:!k;epsilon!
k;epsilon!model!
Standard!
Materials!&!phases!
Create!primary!phase:!
Create!secondary!air!
phase:!water!
Operapng!condipons!
Operapng!pressure!
Atmospheric!pressure:!
101325!Pa!
Reference!pressure!
locapon!
At!the!top!of!the!end!
of!the!box,!
coordinates:!(636,!
297.5,!6)!
Specified!operapng!
density!
Density!of!air:!1.225!
kg/m3!
Gravity!
Enable!gravity!
Gravitaponal!
accelerapon:!;9.81m/
s2!
Boundary!condipons!
Box!inlet!
Type:!velocity!inlet!
Phase!;!water!
Volume!fracpon:!from!
profile!of!channel!
outlet!
Phase!;!mixture!
Velocity!specificapon!
method:!components!
x;velocity:!from!profile!
of!channel!outlet!
y;velocity:!from!profile!
of!channel!outlet!
z;velocity:!from!profile!
of!channel!outlet!
Turb.!kinepc!energy!
(k):!from!profile!of!
channel!outlet!
Turb.!dissipapon!rate!
(epsilon):!from!profile!
of!channel!outlet!
Solve!
Solupon!inipalizapon!
Compute!inipal!values!
from:!box!inlet!
Iterate!
Time!step!size:!0.05!
Number!of!pme!steps:!
5000!
Max!iterapons!per!
pme!step:!40!
