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Introduction toComputational FluidDynamics (CFD) Tao Xing and Fred Stern

IIHR—Hydroscience & Engineering

C !a"#ell Stanley Hydraulics $a%oratory

 Te 'niersity o Io#a

*+,-./ Intermediate !ecanics o Fluids

ttp,00cssengineeringuio#aedu01me2-./0

Sept 34 5//3

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6utline

- 7at4 #y and #ere o CFD85 !odeling

9 :umerical metods

; Types o CFD codes

* CFD Educational Interace

. CFD

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7at is CFD8= CFD is te simulation o >uids engineering systems

using modeling (matematical pysical pro%lemormulation) and numerical metods (discreti?ationmetods4 solers4 numerical parameters4 and gridgenerations4 etc)

= Historically only @nalytical Fluid Dynamics (@FD) andE"perimental Fluid Dynamics (EFD)

= CFD made possi%le %y te adent o digital computerand adancing #it improements o computerresources

  (*// >ops4 -A;3

5/ tera>ops4 5//9)

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4

7y use CFD8

= @nalysis and Design- SimulationB%ased design instead o %uild & test!ore cost eectie and more rapid tan EFDCFD proides igBdelity data%ase or diagnosing >o#

eld

5 Simulation o pysical >uid penomena tat are

diGcult or e"perimentsFull scale simulations (eg4 sips and airplanes)Enironmental eects (#ind4 #eater4 etc)Ha?ards (eg4 e"plosions4 radiation4 pollution)

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7ere is CFD used8

= 7ere is CFDused8

= Aerospace

= Automotive

= Biomedical = Cemical

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7ere is CFD used8

Polymerization reactor vessel - prediction

of flow separation and residence time

effects.

Streamlines for workstation

ventilation

= 7ere is CFD used8= @erospacee

= @utomotie

= Kiomedical

=  ChemicalProcessing

=  HVAC

=  Hydraulics

= !arine

= 6il & Jas

=

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7ere is CFD used8

= 7ere is CFD used8= @erospace

= @utomotie

= Kiomedical

= Cemical

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!odeling= !odeling is te matematical pysics

pro%lem ormulation in terms o a continuousinitial %oundary alue pro%lem (IK

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o e ng geome ry andomain)

= Simple geometries can %e easily created %y e#

geometric parameters (eg circular pipe)= Comple" geometries must %e created %y te partial

dierential eLuations or importing te data%ase ote geometry(eg airoil) into commercial sot#are

= Domain, si?e and sape 

=  Typical approaces= Jeometry appro"imation

= C@D0C@E integration, use o industry standards suc as

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!odeling (coordinates)

x

y

z

x

y

z

x

y

z

(r,θ,z)z

r θ

(r,θ,φ)

r θ

φ(x,y,z)

Cartesian Cylindrical Spherical

General C!r"ilinear C##rdinates General #rth#$#nal

C##rdinates

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 eLuations)

= :aierBStoOes eLuations (9D in Cartesian coordinates)

∂

∂

+∂

∂

+∂

∂

+∂

∂

−=∂

∂

+∂

∂

+∂

∂

+∂

∂2

2

2

2

2

2%

 z 

u

 y

u

 x

u

 x

 p

 z 

u

w y

u

v x

u

ut 

u µ  ρ  ρ  ρ  ρ 

∂

∂+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂+

∂

∂2

2

2

2

2

2%

 z 

v

 y

v

 x

v

 y

 p

 z 

vw

 y

vv

 x

vu

t 

v µ  ρ  ρ  ρ  ρ 

( ) ( ) ( )0=∂

∂+∂

∂+∂

∂+∂

∂

 z 

w

 y

v

 x

u

t 

 ρ  ρ  ρ  ρ 

 RT  p   ρ =

 L

v   p p

 Dt 

 DR

 Dt 

 R D R

 ρ 

−=+

  2

2

2

)(2

3

C#n"ecti#n &iez#'etric press!re $radient isc#!s ter's#cal accelerati#n

C#ntin!ity e*!ati#n

+*!ati#n # state

-aylei$h +*!ati#n

∂

∂+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂+

∂

∂2

2

2

2

2

2%

 z 

w

 y

w

 x

w

 z 

 p

 z 

w

w y

w

v x

w

ut 

w

 µ  ρ  ρ  ρ  ρ 

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!odeling (>o# conditions)= Kased on te pysics o te >uids penomena4

CFD can %e distinguised into dierentcategories using dierent criteria

= iscous s iniscid (Re)=  E"ternal >o# or internal >o# (#all %ounded or not)

= Tur%ulent s laminar (Re)=  Incompressi%le s compressi%le (!a)

=  SingleB s multiBpase (Ca)

=  Termal0density eects (o# (Fr) and surace tension (7e)=  Cemical reactions and com%ustion (

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!odeling (initial conditions)= Initial conditions (ICS4 steady0unsteady

>o#s)

= ICs sould not aect nal results and onlyaect conergence pat4 ie num%er oiterations (steady) or time steps

(unsteady) need to reac conergedsolutions

= !ore reasona%le guess can speed up teconergence

= For complicated unsteady >o# pro%lems4CFD codes are usually run in te steadymode or a e# iterations or getting a%etter initial conditions

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o e ng oun aryconditions)

=Koundary conditions, :oBslip or slipBree

on #alls4 periodic4 inlet (elocity inlet4 mass >o#rate4 constant pressure4 etc)4 outlet (constantpressure4 elocity conectie4 numerical %eac4 ?eroBgradient)4 and nonBre>ecting (or compressi%le >o#s4suc as acoustics)4 etc

 .#/slip alls !0,"0

"0, dpdr0,d!dr0

nlet ,!c,"0 !tlet, pc

Periodic "oundary condition in

span!ise direction o# an air#oil#

r 

xxisy''etric

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 models)

= CFD codes typically designed or soling

certain >uid  penomenon %y applying dierent models

= iscous s iniscid (Re)= Tur%ulent s laminar (Re4 Tur%ulent models)

=  Incompressi%le s compressi%le (!a4 eLuation ostate)

= SingleB s multiBpase (Ca4 caitation model4 t#oB>uid

  model)

= Termal0density eects and energy eLuation  (o# (Fr4 leelBset & surace tracOingmodel) and

  surace tension (7e4 %u%%le dynamic model)= Cemical reactions and com%ustion (Cemical

! d li (T % l d

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!odeling (Tur%ulence and ree suracemodels)

= Turbulent models,= D:S, most accurately sole :S eLuations4 %ut too e"pensie

  or tur%ulent >o#s

= R@:S, predict mean >o# structures4 eGcient inside K$ %ute"cessie

  diusion in te separated region

 $ES, accurate in separation region and unaorda%le orresoling K$

= DES, R@:S inside K$4 $ES in separated regions= Free-surface models,= SuraceBtracOing metod, mes moing to capture reesurace4

  limited to small and medium #ae slopes

= Single0t#o pase leelBset metod, mes "ed and leelBset

  unction used to capture te gas0liLuid interace4 capa%le o 

=  Tur%ulent >o#s at ig Re usually inole %ot large and small

scale  ortical structures and ery tin tur%ulent %oundary layer (K$) nearte #all

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E"amples o modeling (Tur%ulence andree surace models)

$%, -e105, s#/s!race # criteri#n (04) #rt!r:!lent l# ar#!nd .C12 ith an$le # attac; 60

de$rees

'A, -e105, c#nt#!r # "#rticity #r t!r:!lentl# ar#!nd .C12 ith an$le # attac; 60 de$rees

'A,

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:umerical metods

=  Te continuous Initial Koundary alue

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Discreti?ation metods

= Finite dierence metods (straigtor#ard to apply4usually or regular grid) and nite olumes and niteelement metods (usually or irregular meses)

= Eac type o metods a%oe yields te samesolution i te grid is ne enoug Ho#eer4 somemetods are more suita%le to some cases tanoters

= Finite dierence metods or spatial deriaties #itdierent order o accuracies can %e deried using

 Taylor e"pansions4 suc as 5nd order up#ind sceme4central dierences scemes4 etc

= Higer order numerical metods usually predictiger order o accuracy or CFD4 %ut more liOelyunsta%le due to less numerical dissipation

=  Temporal deriaties can %e integrated eiter %y tee"plicit metod (Euler4 RungeButta4 etc) or implicit metod (eg KeamB7arming metod)

scre ?a on me o s

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scre ?a on me o s(ContQd)

= E"plicit metods can %e easily applied %ut yieldconditionally sta%le Finite Dierent ELuations (FDEs)4#ic are restricted %y te time step Implicitmetods are unconditionally sta%le4 %ut need eortson eGciency

= 'sually4 igerBorder temporal discreti?ation is used#en te spatial discreti?ation is also o iger order

= Sta%ility, @ discreti?ation metod is said to %e sta%lei it does not magniy te errors tat appear in tecourse o numerical solution process

=

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 (e"ample)

0=∂∂

+∂∂

 y

v

 x

u

2

2

 y

u

e

 p

 x y

uv x

uu ∂

∂+  

   

∂∂

−=∂∂

+∂∂

 µ 

= 5D incompressi%le laminar >o# %oundary layer

'0'1

/1

y

x

'>>'>>?1

(,'/1)

(,')

(,'?1)

(/1,')

1l 

l l mm m

uuu u u

 x x

−∂ = − ∂ ∆

1

l l l mm m

vuv u u

 y y  +

∂ = − ∂ ∆

1

l l l mm m

vu u

 y  − = − ∆

@A Si$n( )B0l mv

l 

mvA Si$n( )D0

2

1 12 22l l l m m m

uu u u

 y y

 µ  µ    + −

∂ = − + ∂ ∆

2nd #rder central dierence

ie, the#retical #rder # acc!racy

 &;est 2

1st #rder !pind sche'e, ie, the#retical #rder # acc!racy &;est 1

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 (e"ample)

1 12 2 2

1

2

1

l l l l l l l  m m mm m m m

 FDu v v yv u FD u BD u

 x y y y y y BD

 y

 µ µ µ + −

−   ∆+ − + + + −   ∆ ∆ ∆ ∆ ∆ ∆   ∆

1( )

l l l mm m

uu p e

 x x

−   ∂= −∆ ∂

2 3 1

4( )11 1 2 3 1 4     l l l l l  m m m m m B u B u B u B u p e x−

− + ∂+ + = − ∂1

4 1

12 3 1

1 2 3

1 2 3

1 2 1

4

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

l 

l 

l 

l    l mm l 

mm

mm

 p B u

 B B x eu

 B B B

 B B B

 B B   u   p B u

 x e

−

−

∂   − ÷∂       ••

    × =• • • •   ••     ••       ∂    − ÷∂  

olve it using

homas algorithm

o "e sta"le* Matri+ has to "e

$iagonally dominant,

Solers and numerical

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Solers and numericalparameters

=  Solvers include, tridiagonal4 pentadiagonal solers4

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umer ca me o s grgeneration)

= Jrids can eiter %e structured(e"aedral) or unstructured

(tetraedral) Depends upon type odiscreti?ation sceme and application

= Sceme Finite dierences, structured Finite olume or nite element,

structured or unstructured

= @pplication Tin %oundary layers %est resoled

#it iglyBstretced structuredgrids

'nstructured grids useul orcomple" geometries

'nstructured grids permitautomatic adaptie renement%ased on te pressure gradient4 orregions interested (F$'E:T)

str!ct!red

!nstr!ct!red

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:umerical metods (gridtransormation)

 y

 x# #

&hysical d#'ain C#'p!tati#nal d#'ain

 x x

 f f f f f 

 x x x

ξ η 

ξ η ξ η ξ η  

∂ ∂ ∂ ∂ ∂ ∂ ∂

= + = +∂ ∂ ∂ ∂ ∂ ∂ ∂

 y y

 f f f f f 

 y y y

ξ η ξ η 

ξ η ξ η  

∂ ∂ ∂ ∂ ∂ ∂ ∂= + = +

∂ ∂ ∂ ∂ ∂ ∂ ∂

Erans#r'ati#n :eteen physical (x,y,z)and c#'p!tati#nal (ξ,η,ζ) d#'ains,i'p#rtant #r :#dy/itted $rids Ehe partial

deri"ati"es at these t# d#'ains ha"e the

relati#nship (2A as an exa'ple)

η 

ξ 

Erans#r'

Hi ti d t

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Hig perormance computing and postBprocessing

= CFD computations (eg 9D unsteady >o#s) areusually ery e"pensie #ic reLuires parallel ig

perormance supercomputers (eg IK! .A/) #itte use o multiB%locO tecniLue= @s reLuired %y te multiB%locO tecniLue4 CFD codes

need to %e deeloped using te !assage

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 Types o CFD codes= Commercial CFD code, F$'E:T4 StarB

CD4 CFDRC4 CFX0@E@4 etc= Researc CFD code, CFDSHI

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CFD Educational Interace

-a"./ Pipe 0lo! -a" 1/ Air#oil 0lo! -a"2/ $i##user -a"3/ Ahmed car

1 Aeiniti#n # FC@A &r#cess2 #!ndary c#nditi#ns3 terati"e err#r 4 Grid err#r 5 Ae"el#pin$ len$th # la'inar

and t!r:!lent pipe l#s6 eriicati#n !sin$ @A7 alidati#n !sin$ +@A

1 #!ndary c#nditi#ns2 +ect # #rder # acc!racy

#n "eriicati#n res!lts3 +ect # $rid $enerati#n

t#p#l#$y, FC and F>eshes

4 +ect # an$le #   attac;t!r:!lent '#dels #n

l# ield5 eriicati#n and alidati#n

!sin$ +@A

1 >eshin$ and iterati"ec#n"er$ence

2 #!ndary layerseparati#n

3 xial "el#city pr#ile4 Strea'lines5 +ect # t!r:!lence

'#dels6 +ect # expansi#n  an$le and c#'paris#n  ith +S, +@A, and

  -.S

1 >eshin$ and iterati"e  c#n"er$ence2 #!ndary layer separati#n3 xial "el#city pr#ile4 Strea'lines5 +ect # slant an$le and  c#'paris#n ith +S,

+@A, and -.S

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CFD process= uid interactions or%u%%ly >o#s4 study o #ae induced massiely separated>o#s or reeBsurace4 etc

= Depend on te specic purpose and >o# conditions o tepro%lem4 dierent CFD codes can %e cosen or dierentapplications (aerospace4 marines4 com%ustion4 multiB

pase >o#s4 etc)= 6nce purposes and CFD codes cosen4 CFD process is

te steps to set up te IK< pro%lem and run te code,

  - Jeometry  5

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CFD #del

#!ndary

C#nditi#ns

nitial

C#nditi#ns

C#n"er$ent

i'it

C#nt#!rs

&recisi#ns

(sin$le

d#!:le)

 .!'erical

Sche'e

ect#rs

Strea'lineseriicati#n

Geometry

Select

Ge#'etry

Ge#'etry

&ara'eters

Physics Mesh olve Post4

Processing

C#'pressi:le

.@@

@l#

 pr#perties

Hnstr!ct!red

(a!t#'atic

'an!al)

Steady

Hnsteady

@#rces -ep#rt(litdra$, shear

stress, etc)

IJ &l#t

A#'ain

Shape and

Size

=eat Eranser

.@@

Str!ct!red

(a!t#'atic

'an!al)

terati#ns

Steps

alidati#n

'eports

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Jeometry= Selection o an appropriate coordinate

= Determine te domain si?e and sape

= @ny simplications needed8

= 7at Oinds o sapes needed to %e used to%est resole te geometry8 (lines4 circular4oals4 etc)

= For commercial code4 geometry is usuallycreated using commercial sot#are (eiterseparated rom te commercial code itsel4 liOe

Jam%it4 or com%ined togeter4 liOe Flo#$a%)

= For researc code4 commercial sot#are (egJridgen) is used

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uid properties  - Flow conditions, iniscid4 iscous4 laminar4

or  tur%ulent4 etc  5 Fluid properties, density4 iscosity4 and

termal conductiity4 etc

9 Flo# conditions and properties usuallypresented in dimensional orm in industrialcommercial CFD sot#are4 #ereas in nonBdimensional aria%les or researc codes

= Selection o models, dierent models usually

"ed %y codes4 options or user to coose= Initial and Koundary Conditions, not "ed%y codes4 user needs speciy tem or dierentapplications

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!es= !eses sould %e #ell designed to resole

important >o# eatures #ic are dependentupon >o# condition parameters (eg4 Re)4 suc aste grid renement inside te #all %oundary layer

= !es can %e generated %y eiter commercial

codes (Jridgen4 Jam%it4 etc) or researc code(using alge%raic s

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Sole

= Setup appropriate numerical parameters= Coose appropriate Solers

= Solution procedure (eg incompressi%le >o#s)

  Sole te momentum4 pressure o# eld Luantities4 sucas elocity4 tur%ulence intensity4 pressureand integral Luantities (lit4 drag orces)

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Reports= Reports saed te time istory o te

residuals o te elocity4 pressure andtemperature4 etc

= Report te integral Luantities4 suc as totalpressure drop4 riction actor (pipe >o#)4

lit and drag coeGcients (airoil >o#)4 etc= X plots could present te centerlineelocity0pressure distri%ution4 rictionactor distri%ution (pipe >o#)4 pressurecoeGcient distri%ution (airoil >o#)

= @FD or EFD data can %e imported and puton top o te X plots or alidation

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< i ('@ i i )

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ixed #scillat#ryc#n"er$ent

terati#n hist#ry #r series 60 (a) S#l!ti#n chan$e (:) 'a$niied "ie # t#tal

resistance #"er last t# peri#ds # #scillati#n (scillat#ry iterati"e c#n"er$ence)

(:)(a)

)(2

1

 LU  I 

  S S U    −=

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= @symptotic Range, For suGciently small ∆"O4te solutions are in te asymptotic rangesuc tat igerBorder terms are negligi%leand te assumption tat and are

independent o ∆"O is alid= 7en @symptotic Range reaced4 #ill %eclose to te teoretical alue 4 and tecorrection actor

#ill %e close to -=  To aciee te asymptotic range or practical

geometry and conditions is usually notpossi%le and mY9 is undesira%le rom a

resources point o ie#

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E"ample o CFD o#s (Re-;9) around ClarOy airoil

#it angle o attacO . degree is simulated= C sape domain is applied=  Te radius o te domain Rc and do#nstream

lengt $o sould %e specied in suc a #ay tatte domain si?e #ill not aect te simulationresults

E"ample o CFD

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E"ample o CFD

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E l CFD < (S l )

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E"ample o CFD

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 (Reports)

processing)

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processing)

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