Investigation of the capabilities of a POD-based flow estimation technique

Inthiswork,the apabilitiesofaPOD-basedowpredi tionandestimationte hniquesare

in-vestigatedfortheowaroundasquare ylinderatReynoldsnumbers(basedonthefreestream

velo ityand ylinder edgelength):

Re = 150

,

Re = 300

and

Re = 22000

.We assumethatthe

velo ity eld is expanded linearly in terms of empiri al basis fun tions obtained by Proper

OrthogonalDe omposition(POD)appliedtovelo ityeldsobtainedthroughdire tnumeri al

simulations.Inorder to in rease therepresentativeness of thePODmodes,twoltering

pro- eduresofthevelo itysnapshotsusedtoobtainthePODbasisare onsidered.Bothmethods

are based on a onvolution with a box lter, the rst in time and the latter in spa e.

Low-order models are obtained by a Galerkin proje tion of the Navier-Stokes equations onto the

POD basis and bya alibration pro edure whi h allows theunresolved s ales of the ow to

bemodeled.Stabilityand a ura yofthepredi tionsofthedynami models forthedierent

ltering pro eduresare analyzed. Furthermore,twoowestimation pro eduresstarting from

a limited number of measurements are employed. The rst one estimate the oe ients of

themodalexpansion bymeans ofa LeastSquare approa h(LSQ).Theother oneis basedon

theideathat the oe ients give thebest approximation of theavailable measurements and

at thesame time satisfy thenon-linear low-order modelas losely aspossible (KLSQ).Both

pro edures are tested and evaluated in the ases of POD basis extra ted from the ltered

Nelpresentelavoro,vengonoindagatele apa itàdistimaepredizionedite ni heimprontate

sul metodo POD per un usso attorno ad un ilindro a basequadrata, per valoridel

nume-ro di Reynolds (basato sulla velo ità asintoti a e sulla lunghezza del lato del ilindro) pari

a

Re = 150

,

Re = 300

e

Re = 22000

. Si assume he il ampo di velo ità sia espresso per

mezzodiuna ombinazionelinearedifunzioniempiri heottenutepermezzodiProper

Ortho-gonal De omposition(POD), appli ataa ampi divelo itàistantanei prodottida simulazioni

numeri he.Conl'obiettivo dimigliorarelarappresentativitàdeimodiPOD,si appli anodue

pro edurediltraggiodei ampidivelo itàistantaneida uisiri avanolebasiPOD.

Entram-biimetodisibasanosuuna onvoluzione onunafunzionerettangolare:inun asoneltempo

e nell'altro nello spazio. Modelli di ordine ridotto sono ottenuti attraverso una proiezione di

Galerkin delle equazioni di Navier-Stokes sulle basi POD ed un pro edimento di

alibrazio-ne, he permette di modellizzare le s ale non risolte del usso. Nei vari asi di ltraggio, si

analizzano stabilità e a uratezza delle predizionidei modelli dinami i. Inoltre, si utilizzano

due pro edure distima del usso basate sull'usodi un numero limitato dimisure. La prima

permettedistimarei oe ientidell'espansionemodaleinterminidiminimiquadrati(LSQ).

L'altrometodosibasasull'idea hei oe ientifornis anolamiglioreapprossimazioneperun

ertonumerodimisureeallostessotemposoddisnoilpiùa uratamentepossibileilmodello

diordineridotto(KLSQ).Tuttelepro eduresonoappli ateeanalizzatenel asodibasiPOD

1 Introdu tion 5

2 POD-GALERKIN model 7

2.1 ProperOrthogonal De omposition . . . 7

2.2 POD-Galerkinmodel . . . 11

2.2.1 Averagedtime velo ityeld . . . 12

2.3 Calibration pro edure . . . 12

3 Non linear observer. 15 3.1 Stati estimation: LeastSquare Approa h . . . 15

3.2 Dynami estimation:a non-linear observer . . . 16

4 Test ases: ows around a square ylinder 17 4.1 Briefdes ription ofnumeri solver . . . 17

4.2 Test- asesdes ription . . . 19

4.2.1 Domain andowset-up at Re=150 andRe=300 . . . 19

4.2.2 Domain andowset-up at Re=22000 . . . 20

5 Moving average lter 23 5.1 Introdu tion . . . 23

5.2 Moving average lter inthetimedomain . . . 23

5.3 Moving average asa onvolutionintime . . . 25

5.4 Moving average asa onvolutioninspa e . . . 26

5.5 Multiple passmovingspa e lter . . . 27

5.6 Appli ationof theltersto numeri simulations . . . 29

6 Preliminary analysis: two dimensional ase, Re=150 37 6.1 Denition of PODtimeaveraged databases . . . 37

6.2 Representation apabilityof TAVextra ted POD basis . . . 38

6.3 TAV: alibrations and low-order models . . . 38

6.4 TAV:LSQ andKLSQestimation te hniques . . . 46

7 Three dimensional ase: Re=300 56 7.1 Denition of PODtimeaveraged databases . . . 56

7.2 Representation apabilityof TAVextra ted POD basis . . . 57

7.3 TAV: alibrations and low-order models . . . 57

7.5 Spa eltered databases:representation apabilityofPOD basis. . . 71

7.6 FILT: alibrations and low-order models.. . . 73

7.7 FILT:LSQ andKLSQestimation te hniques . . . 80

7.8 Teston the apabilityofthetimeand spa e lters . . . 85

8 Three dimensional ase: Re=22000 91 8.1 POD database setting-up . . . 91

8.2 Representativityoftheextra ted POD basis . . . 98

8.3 Calibrations andlow-order models . . . 98

8.4 LSQ andKLSQestimation te hniques . . . 112

Introdu tion

In ontrolandoptimizationinuiddynami s,inwhi halargenumberofow omputationsis

required,numeri alsimulationsbasedonthedis retizationoftheNavier-Stokesequationsare

inmost asesunaordable,duetothehuge omputationalrequirements.Thus,theproblemof

des ribingthedynami sofan innitedimensionalsystemusing asmall numberofdegreesof

freedom arise.The fun tional spa ein whi hwe seek thelow-dimensional solution isderived

using Proper Orthogonal De omposition (POD) introdu ed by Lumley in 1967 [16℄. A vast

literature on erningthiswayofmodelinguidowsexists,e.g.[14℄,[2℄,[13℄andsomeresults

showthepossible interestof using PODinow ontrol.

A POD-based non-linear observer for unsteady ow estimation using a limited number of

measurements has been derived by Buoni et al. [1℄. A des ription and appli ation of this

te hnique to a onned square ylinder at

Re = 150

and

Re = 300

was presented in [1℄. It

wasshown that thenon-linear observeroutperforms preexisting stati estimation te hniques

su has lassi alLeastSquaresorLinear/Quadrati Sto hasti Estimationintwo-dimensional

periodi ows. When more omplex dynami s appear in the ow, the estimation pro edure

givesana uratepredi tion onlyforthePODmodesthatarerelatedtothevortexshedding.

For the remaining modes, the a ura y is lower. It appears that for ows hara terized by

omplex dynami s, the major limitation of all estimation te hniques based on POD is the

abilityof theretainedPOD modesto adequatelyrepresenting theoweld.

In the present work, we investigate the possibility of in reasing the a ura y of the POD

modes using two ltering pro edures of the DNS solution database. A priori, we x whi h

s alesoftheowwewant toreprodu eusinglow ost omputationallters. Inparti ular,the

high frequen y dynami sof the velo ityelds are smoothed by onvolution witha box lter

either in timeor inspa e. Multiple passesof the spa e ltering te hnique arealso employed

to re reate thesame ee tofa spatialGaussian lter.

The present manus riptisorganizedasfollows:inChapter

2

,wedes ribethePODte hnique

and the alibration pro edure used to obtain the low-order model of the dynami s of the

ow. In Chapter

3

,thestate estimation of theowbya few measurements ina leastsquare

sense and an approa h that ombines su h stati estimation with an appropriate non-linear

low-dimensionalowmodelarepresented.

Su h te hniques of predi tion or estimationof theowbased on POD modesextra ted from

ltereddatabasesareappliedtothe aseofa onnedsquare ylinderat

Re = 150

inChapter

6

and

Re = 300

inChapter

7

.The same owusing unlteredvelo itysnapshots wasstudied

Chapter

4

.

In Chapter

8

, a more omplex ow is investigated: an un onned square ylinder at

Re =

22000

. A LES simulation, des ribed in Chapter

4

, is used to obtain the ow snapshots. A

sensitivity analysisto sampling rates is performed to hoosea urately the database for the

derivation of the POD basis. Databases formed by either ltered or unltered velo ity eld

solutions are onsidered. Low-order models with dierent number of POD modes are built.

Stati anddynami estimationareemployedtore onstru ttheowforthe onsideredvelo ity

POD-GALERKIN model

Manyofthetoolsofdynami alsystemsand ontroltheoryhavegonelargelyunusedforuids,

be ausethegoverningequations aresodynami ally omplex,bothhigh-dimensionaland

non-linear. Model redu tion involves nding low-dimensional models that approximate the full

high-dimensionaldynami s.

The modelredu tion te hnique dis ussed here falls inthe ategory of proje tion methods, in

that it involves proje ting the equations of motion onto a subspa e of the original spa e. In

ourworkwe usetheso- alledPOD-Galerkin method.

The Proper Orthogonal De omposition (POD) provides a set of basis fun tions with whi h

we an identify a low-dimensional subspa eon whi h onstru ta modelbyproje tion ofthe

governing equations. POD was introdu ed in the ontext of uid me hani s (turbulen e) by

Lumely (see [16℄).

2.1 Proper Orthogonal De omposition

We suppose we have done a numeri al simulation, the results of whi h have been pla ed in

N tensors

©

U

(1)

_{, U}

(2)

_{, . . . , U}

(N )

ª

, where ea h term

U

(k)

represents a snapshot of the whole

velo ityeldat theK-th instant,thatis:

U

(k)

=



















u(x

1

, t

k

)

,

v(x

1

, t

k

)

,

w(x

1

, t

k

)

u(x

2

, t

k

)

,

v(x

2

, t

k

)

,

w(x

2

, t

k

)

. . .

,

. . .

,

. . .

u(x

M

, t

k

) , v(x

M

, t

k

) , w(x

M

, t

k

)



















Theterm Mindi ates the numberof gridpointsand it an beveryhigh, for example,inthe

three-dimensional asestudiedinthis resear h,is morethan6 millions.

We andenethespa e

L

:

L = SP AN

n

U

(1)

, U

(2)

, . . . , U

(N )

o

Now the purposeof all thePOD is to nd a subspa e of

L

that represents thebest

approx-imation of thesame

L

.At rst, we have to lo ate a unit norm ve tor

φ

whi h hasthe same

stru ture of

U

(k)

φ =



















φ

1

(x

1

)

,

φ

2

(x

1

)

,

φ

3

(x

1

)

φ

1

(x

2

)

,

φ

2

(x

2

)

,

φ

3

(x

2

)

. . .

,

. . .

,

. . .

φ

1

(x

M

) , φ

2

(x

M

) , φ

3

(x

M

)



















, k φ k= (φ, φ) = 1

Thisve tor

φ

hasto be determined inorder to make the proje tion of the spa e

L

on

φ

the

bestpossible representation.Thus,we have to maximizethis fun tional:

J =

N

X

k=1

³

U

(k)

, φ

´

2

We willusethefollowing symbols:

u(x

j

, t

k

) ≡ u

1

jk

v(x

j

, t

k

) ≡ u

2

jk

w(x

j

, t

k

) ≡ u

3

jk

φ

h

(x

j

) ≡ φ

h

j















, j = 1, . . . , M ; k = 1, . . . , N ; h = 1, 2, 3

UsingtheEinstein notation,theproblem ofthemaximization of fun tional

J

be omes:

 To ndthatve tors

φ

thatmaximize thefun tional:

J

1

= φ

h

j

u

h

jk

u

l

ik

φ

l

i

underthe onstraint:

φ

h

_j

φ

h

_j

= 1

.

A lassi al method to resolve this problem is to employ the Lagrange multipliers. We an

usethese be ause the onstraints, thatarerepresentedbyonly oneequation,arefun tionally

independent.In thisway,theinitialproblem, wherewe have to nd themaximaofthe

fun -tional

J

1

,issubstitutedbyanotherone,inwhi hwehaveto al ulatetheextremaofthisnew fun tional

J

2

without onstraints:

J

2

= φ

h

j

u

h

jk

u

l

ik

φ

l

i

− λ(φ

h

j

φ

h

j

− 1)

.

In this fun tional, the variable

λ

represent the Lagrange multiplier and there is only one

be ause the onstraints arerepresentedbyonlyone equation.

Considering the fun tional

J

2

, we have to work out the eigenvalues and the eigenve tors of a very big matrix, that is

u

h

jk

u

l

ik

, the dimensions of whi h are

[M · M]

. If the grid used for numeri al omputationoftheNavier-Stokesequationsis omposedbyavery highnumberof

points, the omputationbe omes veryheavy.

To avoid this problem, we go on in a ordan e with the method of snapshots proposed by

Sirovi h[15℄:the entralideaisto expressve tors

φ

byalinear ombinationofthesnapshots

U

(k)

asfollowing:

φ =

N

X

n=1

b

n

U

(n)

Bythesubstitution ofthis expressionof

φ

inthefun tional

J

2

we obtain:

φ

h

_j

= u

h

_jk

b

k

⇒ J

2

(λ, b

1

, . . . , b

N

) = b

k

³

u

h

_jk

u

h

_jr

´ ³

u

l

_ir

u

l

_is

´

b

s

− λ

³

b

k

u

h

jk

u

h

jr

b

r

− 1

´

.

Tondtheextremaofthisfun tional,wehavetoimposethevanishingofGâteauxderivatives

withrespe tto all of its unknowns. Letstart onsideringthe derivativeswith respe t to the

oe ients

b

k

:

∂J

2

∂b

k

≡ lim

α→0

J

2

³

λ, b

1

, . . . , b

k

+ αeb

k

, . . . , b

N

´

− J

2

(λ, b

1

, . . . , b

N

)

α

Withoutto expli itthepassages, wearriveto:

∂J

2

∂b

k

= eb

k

nh

u

h

_jk

u

h

_jr

u

l

_ir

u

l

_im

b

m

+ b

n

u

h

jn

u

h

jr

u

l

ir

u

l

ik

i

− λ

h

u

h

_jk

u

h

_jm

b

m

+ b

n

u

h

jn

u

h

jk

io

=

= eb

k

h

u

h

_jk

u

h

_jr

u

l

_ir

u

_im

l

2b

m

− 2λu

h

jk

u

h

jm

b

m

i

Asresultwe obtain:

∂J

2

∂b

k

= 0 ⇐⇒ u

h

jk

u

h

jr

b

m

= λb

m

And,vanishing thederivativeof thefun tionalwithrespe tto theLagrangemultiplier

λ

,we

have:

∂J

2

∂λ

= b

k

u

h

jk

u

h

jr

b

r

− 1

∂J

2

∂λ

= 0 ⇐⇒ b

k

u

h

jk

u

h

jr

b

r

= 1

This se ond equation orrespond to the one we obtain imposing that the ve tor

φ

has an

unitarynorm.Finally,tondtheextremaofthefun tional

J

2

,wehavetosolvethiseigenvalues problem:

Rb = λb

where:

R = u

h

_jk

u

h

_jr

_≡

TIMECORRELATION MATRIX

[R] = N · N

½

N =

numberofsnapshots

N ≪ M

b = [b

1

, b

2

, . . . , b

N

]

T

We annotethattheSirovi h'smethodallowstohaveamatrix

R

ofredu eddimension

[N ·N]

insteadof

[M · M]

.Moreover,

R

issymmetri andpositivedenite,thus,itis hara terizedby

a omplete set of orthonormal eigenve tors

{f

1

, . . . , f

N

}

and bya set of positive eigenvalues

b

s

=

√

f

s

λ

s

, s = 1, . . . , N ,

Weobtainasetoforthogonaleigenve torsof

R

thatsatises theequationobtainedvanishing

thederivative of

J

2

with respe t to

λ

. Then these eigenve torsmaximizes thefun tional

J

2

. Now we an go ba k up to theeigenfun tions

{φ

1

, . . . , φ

N

}

,previously substituted bylinear ombinations:

φ

h

_jk

= u

h

_jn

b

nk

, j = 1, . . . , M ; k = 1, . . . , N ; n = 1, . . . , N ; h = 1, 2, 3.

Theseeigenfun tionsarethePOD modes.

Finally it is possible to he k that theinner produ tbetween two POD modesis ee tively

unitary:

(φ

k

, φ

s

) = φ

h

jk

φ

h

js

= b

rk

u

jr

h

u

h

jq

b

qs

= b

rk

λ

s

b

rs

= λ

s

f

rk

√

λ

k

f

rs

√

λ

s

= δ

ks

s

λ

s

λ

k

= δ

ks

The istantaneous velo ity eld an now be expressed by a linear ombination of

N

r

POD modes,asfollowing:

u (x, t) ≈

N

r

X

n=1

a

n

(t) φ

n

(x)

Letgo ba k to the original goal of this treatment:to represent a spa e

L

of large dimension

withasubspa eofitselfofredu eddimension.Thenwehavetonegle tthelessenergeti POD

modes,that is those orrespondingto thesmaller eigenvalues of thetime orrelation matrix.

In pra ti eweestimatetheenergyamount oftherst

N

r

modesasfollowing:

N

r

X

i=1

λ

i

Then,if, for instan e, we want to apture the 99% of the entire energy of the velo ity eld,

wehave to hoose

N

r

withrespe t to thisrelation:

P

N

r

i=1

λ

i

P

N

i=1

λ

i

= 99%

From a series of attempts done at variousRe, it was observed that, to apture some energy

level,

N

r

hasto in reaseastheRe in reases.Thisagreesto thefa tthat,astheRein reases, theamount of the energypresent inthe smallers ales in reasestoo, thenmore POD modes

2.2 POD-Galerkin model

Considering theadimensional formof theNavier-Stokes equations:

½

∇ · U = 0

∂U

∂t

+ (U · ∇)U = −∇P +

1

Re

△U

(2.1)

and substitutingto thevelo ityve tor

u (x, t) = {u(x, t), v(x, t), w(x, t)}

thelinear

ombina-tionpreviously indi ated, knowing that thePOD modesare divergen e free by onstru tion,

we an write:

½

_∂

∂t

a

r

(t)φ

r

+ (a

i

(t)φ

i

· ∇) a

j

(t)φ

j

= −∇p +

Re

1

△ (a

i

(t)φ

i

)

u (x, 0) = a

r

(0)φ

r

Then,proje tingtheseequationsonthe

N

r

retainedPODmodesusingtheGalerkinproje tion:

½

( ˙a

r

(t)φ

r

, φ

r

) + ((φ

i

· ∇) φ

j

, φ

r

) a

i

(t)a

j

(t) = − (∇p, φ

r

) +

_Re

1

(△φ

i

, φ

r

) a

i

(t)

(u (x, 0) , φ

r

) = a

r

(0) (φ

r

, φ

r

)

RememberingthatthePODmodesareorthonormalve tors,theseequations anbesimplied

asfollowing :

½

˙a

r

(t) + B

ijr

a

i

(t)a

j

(t) = − (∇p, φ

r

) +

_Re

1

D

ir

a

i

(t) , r, i, j = 1, . . . , N

r

a

r

(0) = (u (x, 0) , φ

r

) , r = 1, . . . , N

r

where:

B

ijr

= (φ

i

· ∇φ

j

, φ

r

)

D

ir

= (△φ

i

, φ

r

)

Thisis alledPOD-GALERKIN MODEL.

Thenwehaveto onsiderthepressureterm

− (∇p, φ

r

)

;sin ethePODmodesare divergen e-free,we an write :

− (∇p, φ

r

) = −

Z

Ω

∇p · φ

r

dΩ = −

Z

Ω

∇ (pφ

r

) dΩ ≡ −

Z

∂Ω

pφ

r

· ndσ

Thepressure term istransformedin asurfa e integral; to omputeit, it issu ient to know

thevaluesofpressureandvelo ityontheboundaryofthedomain.Inthe aseofthisresear h,

the domain is represented by a square ylinder onned by two parallel walls; then, on the

surfa esofthe ylinder andonthewallsof the ondu tthepressuretermiszero, thevelo ity

beingzero; thus :

− (∇p, φ

r

) =

Z

S

i

pφ

r

· idΓ

i

−

Z

S

o

pφ

r

· idΓ

o

where:

S

i

=

ondu t entryse tion

S

o

=

ondu t exitse tion

i =

unitversor orientedastheaxisof the ondu t.

2.2.1 Averaged time velo ity eld

We andenean averageeld

u(x)

and to write ageneri eld

u(x, t)

as:

u(x, t) = u(x, t) + e

u(x, t)

with

e

u

theu tuating partoftheow.

Consideringthe

M

snapshots theaverage eldis :

u(x) =

1

M

M

X

k=1

u

k

(x)

Dened

a

kr

= a

r

(t

k

)

,byproje ting theaveragevelo ityeldonone ofthe

M

PODmodeswe obtain:

(u, φ

i

) =

1

M

M

X

k=1

M

X

r=1

a

kr

(φ

r

, φ

i

) =

1

M

M

X

k=1

a

ki

Andproje tingon all the

M

modes:

u =

M

X

r=1

c

r

φ

r

Thus we have:

c

r

=

1

M

M

X

k=1

a

kr

= < a

r

>

Andsothe oe ients

c

r

oftheaverageeldaretheaverageofthe oe ientsofthesnapshots. Subtra ting this average velo ityeld to ea h snapshots,we obtain anew set ofelds, those

are the u tuating part of the ow and those are no linear-independent, being zero their

sum. Thus, if we arryout thePOD at this new set,thenew time orrelation matrix

R

e

has

Im = M − 1

,and thereforeaneigenvalueof theproblemiszero.

Sowe have :

e

u

k

=

M −1

_X

r=1

ea

kr

φ

e

r

Where

ea

kr

and

φ

e

r

areobtained solvingtheeigenvalueproblemwiththenewmatrix

R

e

.From hereweare alling

e

u(x, t)

as

u(x, t)

,rememberingthatthisistheu tuatingvelo ityeldand

no the ompleteone.Hen e we an onsidertheappli ationoftheSirovi h'smethodandthe

development of the POD-Galerkin model for the set of u tuating elds, all the hypothesis

beingvalid.

2.3 Calibration pro edure

Wedes ribethis pro edurefor thepreviously obtainedPOD-Galerkinmodel,seese tion 2.2.

Tosimplify thetreatment,we model, asproposedin[14℄,thepressure term inthisway:

and thevis ous term as:

C

_ir

1

_{= −}

D

ir

Re

+ ¯

C

1

ir

where

C

¯

0

r

and

C

¯

1

ir

are addedinorder to modeltheintera tion ofthe unresolved modeswith theresolved ones. Thus, thedynami modelbe ome:

˙a

r

(t) = f

r

(a

1

, ..., a

N

r

, C

r

0

, C

kr

1

) = C

r

0

+ C

kr

1

a

k

(t) − B

ksr

a

k

(t)a

s

(t)

a

r

(0) = (u(x, 0), φ

r

)

(2.2)

In [14℄theproposedmethodto ndthe oe ients

C

0

r

and

C

1

kr

issolving aninverseproblem whi h minimize thedieren e, measured in

L

2

norm, betweenthe model predi tion and the

a tual referen e solution. The model alibrated in su h this way is apable of a urately

reprodu e the omplex ow of a onned square ylinder as shown in [2℄. Although very

a urate,the omputational ostofobtainingthis modelisnotnegligiblewhenthenumberof

modesislargeor whentheowshowslargespanoftimefrequen ies.Forthisreason,weused

an alternative method shown in[1℄ that delivers a reasonable model at the ost of a matrix

inversion.Weask thattheterms

C

0

r

and

C

1

kr

are su hthat:

Z

T

0

˙a

r

(t)dt = −C

r

0

T + C

kr

1

Z

T

0

a

k

(t)dt − B

ksr

Z

T

0

a

k

(t)a

s

(t)dt

and

Z

T

0

˙a

r

(t)a

m

(t)dt = −C

r

0

Z

T

0

a

m

(t)dt + C

kr

1

Z

T

0

a

k

(t)a

m

(t)dt − B

ksr

Z

T

0

a

k

(t)a

s

(t)a

m

(t)dt

are satised

∀r, m ∈ 1, ..., N

r

. The time interval

[0, T ]

is the same as that onsidered for building the POD modes. Hen e, all theintegrals inthe above equations are known and we

set:

B

ksr

Z

T

0

a

k

(t)a

s

(t)dt +

Z

T

0

˙a(t)

r

dt = b

0

r

Z

T

0

a

k

(t)dt = I

k

1

B

ksr

Z

T

0

a

k

(t)a

s

(t)a

m

(t)dt +

Z

T

0

˙a

r

(t)a

m

(t)dt = B

rm

1

Z

T

0

a

k

(t)a

m

(t)dt = I

km

2

Z

T

0

a

m

(t)dt = I

m

2

Consequentlya setof

N

2

r

+ N

r

linear equationsis obtained for the oe ients

C

0

r

and

C

1

kr

:

C

_kr

1

I

_k

1

_{− C}

_r

0

T = b

0

_r

C

1

kr

I

km

2

− C

r

0

I

m

1

= B

1

rm

with

r, m = 1, ..., N

r

where

a

r

(t)

are the snapshotproje tions. It an be seenthat this

te h-nique amountsto aminimization ofthe modelderivativepredi tion error inthe

H

1

norm:

J = min

C

0

r

C

kr

1

Z

T

cal

0

N

r

X

r=1

³

˙a

r

(t) − ˆ˙a

r

(t)

´

dt

Non linear observer.

3.1 Stati estimation: Least Square Approa h

Our aim is to provide an estimation of the modal oe ients

a

i

(t)

starting from

N

s

ow measurements

f

k

, k ∈ {1, . . . , N

s

}

using the te hnique developed in [1℄. Let

α

¯

i

(t)

be the proje tion ofthevelo ityeld

u(t)

overthe

i

-th PODmode and

α

i

(t)

beits estimatedvalue at time

t

.We assumethat ea h measurement

f

k

is a s alar quantitywhi h depends linearly ontheinstantaneous velo ityeld

u(t)

.For instan e,

f

k

anbe apoint-wisemeasurement of a velo ity omponent,ofa shear-stress,or it an bea spatialaverage ofa linear ombination

ofvelo ity omponents.

The available spatial information may be exploited by using a LSQ approa h, as done in

[13℄. At any given time

τ

, thanksto the linearity of

f

k

with respe t to

u

and to the modal de omposition ofthevelo ityeld, equation 3.6),

f

k

an be writtenintermsof PODmodes

f

k

(u (τ )) ≃

N

r

X

j=1

a

j

(τ )f

k

¡

φ

_j

¢

(3.1) where

f

k

¡

φ

_j

¢

is obtained from the appli ation of

f

k

to the ve tor eld asso iated to mode

φ

_j

.Then,thefollowingleast-squares problemis solved forevery

τ

min

{a

1

(τ ),...,a

_Nr

(τ )}

N

s

X

k=1



f

k

(u (τ )) −

N

r

X

j=1

a

j

(τ )f

k

¡

φ

_j

¢





2

(3.2)

This problem leads to the solution a

N

r

-dimensional linear system of equations. On e this problem issolved, thePODmodal oe ientsarewritten

a

j

(τ ) =

N

s

X

k=1

Υ

kj

f

k

(u (τ ))

(3.3)

where

Υ

isaknownre tangularmatrix ofsize

N

s

× N

r

.Theerror minimization(3.2)leads to alinearrepresentation oftheestimatedmodesasafun tion ofthemeasurements.

3.2 Dynami estimation: a non-linear observer

Let us now assume that a ertain number of measurements at onse utive times

τ

m

,

m ∈

{1, N

m

}

areavailable. The main idea of thedynami -estimation approa h is to impose that

the oe ientsofthemodalexpansionofthevelo ityeldgivethebestapproximationtothe

available measurements using LSQ (3.2) and that at the same time they satisfy as lose as

possible thenon-linear low-ordermodel:

R

r

(a(t)) = ˙a

r

(t) − A

r

− C

kr

a

k

(t) + B

ksr

a

k

(t)a

s

(t) = 0

a

r

(0) = (u(x, 0) − u(x), φ

r

)

(3.4)

This isdone byminimizingthesum of theresidualsof (3.3) and theresidualsof (3.4) for all

times

τ

m

.Morepre isely,let

α

(t) : R → R

N

r

and

α

(t) = {α

1

(t), . . . , α

N

r

(t)}

,wehave

α(t) = argmin

a

(t)

N

m

X

m=1

Ã

C

R

N

r

X

r=1

R

2

r

(a(τ

m

)) +

N

r

X

r=1

(a

r

(τ

m

) −

N

s

X

k=1

Υ

kr

f

k

(u (τ

m

)))

2

!

(3.5) where

a

(t) = {a

1

(t), . . . , a

N

r

(t)}

. The parameter

C

R

weights more themeasurements(LSQ) or thedynami modelinthedenitionoftheresidualnorm.It ouldbesystemati allytuned,

or it ould be amatrix.In thenumeri alexperimentsreportedinthefollowing hapters, this

parameter hasbeensetfollowing theindi ationreportedin[1℄.

Theminimization ofthisfun tionalisredu edtoanon-linear algebrai problem.Asin[14℄,a

pseudo-spe tralapproa hisusedandea h

α

r

(t)

isexpandedintimeusingLagrange polynomi-alsdenedon Chebyshev-Gauss-Lobatto ollo ation points. Thene essary onditionsfor the

minimumresult ina non-linearset of algebrai equations for the oe ientsof theLagrange

polynomials. The solution is obtained by a Newton method, whi h, in the present

appli a-tions,usually onvergesinafew(typi ally5to8)iterations.Thesolutionoftheproblem(3.5)

provides an estimation for the POD modal oe ients for all modes and for all instants at

whi h measurements are available. This allows the re onstru tion of the entire ow eld at

the sameinstantsthrough equation:

u(x, t) = u(x) +

N

r

X

n=1

a

n

(t)φ

n

(x)

(3.6)

Therefore,theabovemethodrepresentsanon-linearobserveroftheowstate.Inthefollowing,

Test ases: ows around a square

ylinder

In this hapter, we des ribe the numeri al solver used to obtain the POD databases. The

databasesusedat

Re = 150

and

Re = 300

arethesameanalyzedanddes ribedin[2℄whereas

Re = 22000

istheben hmarkproblemshownin[4℄.

4.1 Brief des ription of numeri solver

Inthe ase

Re = 150

and

Re = 300

,toobtainthevelo ityeldsnapshotsforthe onstru tion

of thePOD-Galerkinmodel, we usea3Dsolverof theNavier-Stokesequationsfor

ompress-ible uid based on a mixer nite-volume/nite-element dis retization in spa e appli able to

unstru tured grids [4℄. Theadopted s heme isvertex entered,thatis all thedegreesof

free-domarelo atedatthevertexes.P1Galerkinniteelementsareusedtodis retizethediusive

terms.

A dualnite-volume grid is obtained bybuilding a ell

C

i

aroundea h vertex

i

throughthe rule of medians. The onve tiveuxes aredis retized on thistessellation, that is, intermsof

uxes through the ommonboundaries sharedbyneighboring ells.

TheRoes heme [6℄is adoptedfor thenumeri alevaluationofthe onve tive uxes

F

:

Φ

R

(W

i

, W

j

, ~n) =

F (W

i

, ~n) + F (W

j

, ~n)

2

− γ

s

P

−1

_{|P R|}

W

j

− W

i

2

(4.1) where:

• Φ

R

_(W

i

, W

j

, ~n)

= numeri al approximation of the ux between the

i

-th and the

j

-th

ells;

• W

i

=solution ve tor at the

i

-th node;

• W

j

=solution ve torat the

j

-thnode;

• ~n

=outwardnormalto the ell boundary;

• P (W

i

, W

j

)

=Turkel-typepre onditioning term,introdu edtoavoida ura yproblems at lowMa hnumbers[7℄.Notethat,sin e itonlyappearsintheupwindpartofthe

nu-meri aluxes,thes hemeremains onsistentintime,and anthusbeusedforunsteady

owsimulations;

•

The

γ

s

parametermultipliestheupwindpartofthes heme,andthusitpermitsadire t ontrol of the numeri alvis osity, leadingto a full upwind s hemefor

γ

s

= 1

,and to a entered s heme when

γ

s

= 0

.

The spatial a ura y of this s heme is only rst order. The MUSCL linear re onstru tion

method (Monotone Upwind S hemes for Conservation Laws), introdu ed by Van Leer [8℄

is employed to in rease the order of a ura y of the Roe s heme. This is obtained by

ex-pressing the Roe ux as a fun tion of the re onstru ted values of

W

at the ell interfa e:

Φ

R

_(W

ij

, W

ji

, ~n

ij

)

, where

W

ij

is extrapolated from the values of

W

at nodes

i

and

j

. A re onstru tionusinga ombinationofdierentfamiliesofapproximate gradients(P1-element

wise gradients and nodal gradients evaluated on dierent tetrahedra) is adopted, whi h

al-lows a numeri al dissipation made of sixth-order spa e derivatives to be obtained [9℄. For

theintegration intime, inthis ode animpli it time mar hing algorithmis used, basedon a

se ond-ordertime-a urate ba kwarddieren e s heme.In the aseat

Re = 22000

,we

intro-du ea LES te hnique to modelthe turbulen e. We perform an e onomi al pro edure based

onvolumeagglomerationtoseparateapriorithesmallestandthelargestresolveds ales.The

unresolved s ales are modeled by the Smagorinsky eddy vis osity model only added to the

Figure4.1: Computationaldomain.

4.2 Test- ases des ription

4.2.1 Domain and ow set-up at Re=150 and Re=300

The ow around a square ylinder symmetri ally positioned between two parallel walls is

onsidered here; this onguration is sket hed in gure 4.1. The ratio between the ylinder

side

L

and thedistan e between thewalls

H

is

L/H = 1/8

.The in oming ow is a laminar

Poiseuilleowdire tedinthe

x

dire tionandthe onsideredReynoldsnumbers,basedonthe

maximumvelo ityofthein omingowandon

L

are150and300.No-slipboundary onditions

aresettedon thewallsand on thefa es of the ylinder whereas the same inow ondition is

alsoimposedfor theoutow.

Two dierent omputationaldomains wereused, for arrying outtwo-dimensionaland

three-dimensional simulations, whi h dier only for the spanwise extent of the domain. In both

ases, withreferen eto gure4.1,

L

in

/L = 12

and

L

out

/L = 20

.For two-dimensional simula-tions,thespanwiselength adopted is

L

z

/L = 0.6

,and itwassystemati ally he ked thatthe simulatedspanwisevelo itywasnegligible.Forthethree-dimensionalsimulationat

Re = 300

,

the spanwiselength of thedomain is

L

z

/L = 6

.Thisvaluewassele tedfollowing the experi-mentalresults for theun onnedsquare- ylinderow[10℄,whi hshowa maximumspanwise

lengthofthethree-dimensionalstru turesequalto

5

.2L

andtheindi ationsgivenin[11℄and

[12℄forthenumeri alstudy ofthethree-dimensionalwakeinstabilities ofasquare ylinderin

an openuniformow.

Thesimulationparametersaresummarizedintable4.1for two-dimensional ase,andintable

4.2for three-dimensional ase.For the

Re = 150

ase,thedistribution of theelementsinthe

grid is hosen to havethe maximum resolutionintheproximityof the ylinder. Thenumber

ofgridnodesisapproximately

7

.5 · 10

5

(

758015

).Thelteredandunltereddatabaseof

snap-shotsaredes ribed in hapter 6.Thesnapshotshavebeen olle tedstarting from

t = 138.6 s

whentheowis ompletelydeveloped,andthus theamplitudeofthelift oe ientvariation

is onstant. For three-dimensional ase, the total number of the grid nodes is more than 6

millions,pre isely6 505 397.Theunltered database ofsnapshotsis des ribed in hapter7.

Sin ewe areinterested hereto in ompressibleowsthesimulations havebeen arriedoutby

assumingthatthemaximumMa hnumberoftheinowproleis

M = 0.1

.Thisvalueallows

ompressibilityee tstobereasonablynegle tedanddoesnotimplyseriousproblemsforthe

numeri . The pre onditioning term of thenumeri solver (seese tion 4.1) isused to in rease

oe ient nearthe stagnation point inthe upwind fa eof the ylinder, improving themean

value of the drag oe ient. Conversely, the time u tuations of the for e oe ients were

insensitivetothepre onditioner.Con erningthenumeri alvis osity,theupwindparameter

γ

s

issetto

γ

s

= 1.0

onthenodeswithinadistan eequalto

0.1L

from the ylinder and

γ

s

= 0.1

intherest ofthe domain.This hoi e ensuresthe stabilityof all thesimulations arriedout,

and,at thesame time, allowsthe pre onditionerto be parti ularlyee tive intheproximity

ofthe ylinder.

Parameter Value Des ription

L/H

1/8

blo kageratio

L

in

/L

12

inowlength

L

out

/L

20

outowlength

L

z

/L

0.6

spanwiselength

Table4.1: Domaingeometryvaluesusing

L

, ylinder edgelength,asa referen e.

Parameter Value Des ription

L/H

1/8

blo kageratio

L

in

/L

12

inowlength

L

out

/L

20

outowlength

L

z

/L

6

spanwiselength

Table4.2: Domaingeometryvalues withreferen e to

L

, ylinderedge length.

The used grid at

Re = 300

is obtained byrepli ating the grid for two-dimensional ase ten

timesinthespanwise dire tion.

4.2.2 Domain and ow set-up at Re=22000

Theowpastan un onnedsquare ylinder at Ma hnumber

M = 0.1

andReynoldsnumber

Re = 22000

is onsidered. The omputational domain istheone shown ingures4.2 and4.3

with theparameterssummarizedintable 4.3using asreferen ethesket h4.1.

Parameter Value Des ription

L/H

2/13

blo kageratio

L

in

/L

4.5

inowlength

L

out

/L

9.5

outowlength

L

z

/L

4

spanwiselength

Table4.3: Domaingeometryvalues withreferen e to

L

, ylinderedge length.

The omputationaldomainisdis retizedby200000nodesand1100000tetrahedra.

Horizon-tal and verti al ut-planes of theunstru tured mesh are shown ingures 4.2 and 4.3. Inthe

spanwisedire tion,approximately40nodesareusednearthe ylinder,whi h orrespondstoa

spanwiseresolution

δz ≃ 0.1L

.Theaveragedistan eofthe losestpointstothe ylinderwallis

0.05L

.Theboundary onditionsatthewallareenfor edthroughRei hardt'swalllawtoavoid

Figure4.2:Computationaldomain:verti al ut-planeofthemesh,andzoomaroundthesquare

ylinder.

belowthe ylinder.TheinowandoutowboundariesaretreatedbytheStegerWarmingux

de omposition.Finally,theupwind parameter

γ

is setto

γ = 0.2

inorder to ensure thatthe

ee tofturbulen emodelingispreponderantwhen omparedtothatofnumeri aldissipation.

Figure4.3:Computationaldomain:verti al ut-planeofthemesh,andzoomaroundthesquare

Moving average lter

5.1 Introdu tion

The movingaverageisa very ommonlterinDigitalSignalPro essing,mainlybe auseitis

theeasiestdigitalltertobeused.Inspiteofitssimpli ity,themovingaveragelterisoptimal

for a ommontask:redu ingrandomnoise whileretaining asharpstepresponse. Thismakes

itthepremier lter for timedomain en oded signals. Figure 5.1shows an example of howit

works.Thesignalin(a)isapulseburiedinrandomnoise.In(b)and( ),thesmoothinga tion

ofthemovingaveragelterde reasestheamplitudeoftherandomnoise,butalsoredu esthe

sharpnessoftheedges.Ofallthepossiblelinearltersthat ouldbeused,themovingaverage

produ es thelowest noise fora givenedge sharpness.

Tounderstandwhythemovingaverageisthebestsolution,imaginewewantto designalter

with a xed edge sharpness. For example, let us assume that we x the edge sharpness by

spe ifyingthatthereareelevenpointsinthegradient orrespondingtothestep.Thisrequires

the lter kernel to have eleven points. The question is: how do we hoose the eleven values

in thelter kernel to minimize thenoise on theoutput signal? Sin e thenoise we aretrying

to redu eis random,none of theinputpointsis spe ial;ea h isjust asnoisyasits neighbor.

Therefore,itisuselesstogivepreferentialtreatmenttoanyoneoftheinputpointsbyassigning

italarger oe ientinthelterkernel.Thelowestnoiseisobtainedwhenalltheinputsamples

are treated equally, i.e., the moving average lter. In se tion 5.5, we show that other lters

are essentially as good but the point is, no lter is better than the simple moving average.

However, the moving average is the worst lter for frequen y domain en oded signals, with

little abilityto separateone bandof frequen iesfromanother.Similarto themoving average

lter are Gaussian,Bla kman, and multiple-pass moving average. Thesehave slightly better

performan e inthefrequen ydomain, at theexpenseof in reased omputationtime.

5.2 Moving average lter in the time domain

Asthenameimplies,themovingaveragelter operatesonea hpointof adomain(e.g.spa e

or time) by averaging over a sele ted interval. In this ase, the aim is to apply the lter in

time on ea h omponent

u

i

(with

i = 1, 2, 3

) of the ow velo ity eld. For a ontinue time

(a) (b)

( )

Figure5.1:Exampleofamovingaveragelter.In(a),are tangularpulseisburiedinrandom

noise. In(b)and( ), thissignalis lteredwith11 and51 pointmoving average,respe tively.

Asthenumberofpointsinthelterin reases,thenoisebe omeslower;however,theedges

be- ominglesssharp.Themovingaveragelteristheoptimalsolutionforthisproblem,providing

¯

u

i

(x, y, z, t) =

1

T

Z

t+T /2

t−T /2

u

i

(x, y, z, t) dt

(5.1)

The periodi ity of the Von Karman street leads to remark that averaging in time an be

seen as averaging in spa e in the

x

-dire tion of vortexes translation. Indeed, assuming that

the velo ity eld snapshots are, substantially, repeated every period of vortex shedding, we

an suppose that the omponent

u

i

is a signal whi h translate a ording to the fun tion

r(x, t) = x − C · t

(

C

,meanvelo ityofvortexestranslation).Consequently,we an write(for

simpli itywe negle tthey,z dependen e):

¯

u

i

(x, t) =

1

T

Z

t+T /2

t−T /2

u

i

(x, t) dt =

1

T

Z

t+T /2

t−T /2

u

i

_{(x − C · t) dt}

(5.2)

Using a variable hange,itfollows:

¯

u

i

(x, t) =

1

C · T

Z

x−C·T /2

x+C·T /2

u

i

_{(x − C · t) dx}

(5.3)

Computationally speaking,thetimemoving averagelter operatesbyaveraginga numberof

snapshots of

u

i

jk

= u

i

(x

j

, t

k

)

(

j = 1, ..., N

s

, with

N

s

numberof grid points and

k = 1, ..., N

t

velo ityeld snapshotsinDNSsolution database)to produ eea hsnapshot

t

k

intheoutput signal

u

¯

i

jk

. In order to avoid shifting in the signal ltered, the group of snapshots from the input signal an be hosen symmetri ally around the output point. In equation form, the

enteredaverageis written:

¯

u

i

_jk

=

1

M

k+

M −1

2

X

˜

k=k−

M −1

2

u

i

j˜

k

(5.4)

Symmetri al averagingrequiresthatMmustbe anodd number.

5.3 Moving average as a onvolution in time

Supposing for simpli ity to onsider a generi fun tion

f (t)

, we apply the entered moving

average andobtain:

F (˜

t) =

Z

˜

t+T /2

˜

t−T /2

f (t)dt

(5.5)

whereTistheintervalofaverage.Now,we anexpand

f (t)

inFourierseriesasinthefollowing:

f (t) =

+∞

X

n=−∞

c

n

e

jω

n

t

(5.6)

with

j

,imaginary unitand

ω

n

= n

2π

T s

Bysubstituting 5.6in5.5, we have:

F (˜

t) =

Z

˜

t+T /2

˜

t−T /2

+∞

X

n=−∞

c

n

e

jω

n

t

dt =

+∞

X

n=−∞

c

n

·

e

jω

n

t

jω

n

¸

˜

t+T /2

˜

t−T /2

=

+∞

X

n=−∞

c

n

"

e

jω

n

t

˜

_{· e}

jω

n

T /2

jω

n

−

e

jω

n

˜

t

_{· e}

−jω

n

T /2

jω

n

#

=

+∞

X

n=−∞

c

n

e

jω

n

˜

t

jω

n

³

e

jω

n

T /2

_{− e}

−jω

n

T /2

´ 2j

2j

andusing therelation:

sin(α) =

e

jα

_{− e}

−jα

2j

itfollows:

=

+∞

X

n=−∞

c

n

sin(ω

n

T /2)

ω

n

T /2

e

jω

n

˜

t

Comparing5.6with5.3, dueto ltering,ea h term oftheFourierseriesis multipliedby:

G(ω

n

) =

sin(ω

n

T /2)

ω

n

T /2

(5.7)

Byinverselytransforming

G(ω

n

)

,are tangularpulseisobtained.Thus,we anassertthatthe moving average lter is a onvolutionof an inputsignal

f (t)

witha re tangular lter having

unitarea:

¯

u

i

(x, t) =

Z

∞

−∞

G (τ ) u

i

_{(x, t − τ) dτ}

(5.8)

and

G(ω

n

)

isthekernel of thelter:

G (τ ) =

1

T

· H

µ

1

T

− |τ|

¶

(5.9)

with

H(·)

Heaviside stepfun tion.

Figure 5.2 shows the frequen y response of the moving average lter: it is mathemati ally

des ribedbytheFouriertransformofthere tangularpulse,asseenbefore.Clearly,themoving

averagelter annotseparateonebandoffrequen iesfromanother.Goodperforman einthe

timedomain results in poor performan e in the frequen y domain, and vi eversa. In short,

themovingaverageisanex eptionallygoodsmoothinglter(thea tioninthetimedomain),

buta bad low-passlter (the a tioninthefrequen y domain).

5.4 Moving average as a onvolution in spa e

Inorderto remove thehighfrequen y noise inthespatial domain,webuild up alter based

onavolumetri averageoverasphere ofradius

r

f

andvolume

V

appliedto thevelo ityeld of ea h snapshot:

¯

u

i

(x, t) =

Z

∞

−∞

G (r) u

i

_{(x − r, t) dr}

(5.10)

Figure 5.2: Frequen y response of the moving average lter. The moving average is a very

poorlow-passlter,due to itsslowroll-oandpoor stop-bandattenuation.These urvesare

generated by eq. (5.7) varying the time interval: we use 3, 11 or 31 uniformly distributed

points.

and

G(r)

:

G (r) =

1

V

H (r

f

− |r|)

(5.11)

In the grids used in the simulations, for any given tetrahedral mesh, a orresponding dual

mesh denedby ellsor ontrol volumes is derived. Su h ells are denedfor ea h vertex of

the mesh by means of the medians of every fa e joining the vertex. In ea h ontrol volume,

thevalueofthesolutionis onstantandequaltotheaverageofthesolutiononthewhole ell.

Thus, to reate a simple and fast algorithm from the ltering pro ess des ribed above, the

ideaisavolumeweightedaverageofea h omponentofthevelo ityeldovertheneighboring

ells.Usingthenotation

u

i

_(x

j

, t

k

) = u

i

jk

asthevelo ity omponent

i

onea hvertex

x

j

ofthe mesh at thetime

t

k

,we obtainthelteredeld

u

ˆ

i

jk

inthis way:

ˆ

u

i

_jk

=

P

p∈I

j

V ol(C

p

)ˆ

u

i

pk

P

p∈I

j

V ol(C

p

)

(5.12)

with

j = 1, ..., N

s

,

N

s

numberofgridvertexes,

i = 1, 2, 3

velo ity omponentsand

I

j

,ensemble

oftheneighboring ellsof

j

in ludeditself.

5.5 Multiple pass moving spa e lter

The multiple pass moving average lter onsists in applying to the signal a moving average

ltertwo ormore times.Figure5.3shows theoveralllter kernelresultingfromone,twoand

four passes. Two passesare equivalent to using a triangular lter kernel (a re tangular lter

kernel onvolved withitself).After four or morepasses, theequivalent lter kernel looks like

a Gaussian.

Indeed, the Central Limit Theorem states that the sum of a large number of independent

and identi ally distributedrandom variables will be approximately distributedasa Gaussian

urve.Regardinga boxlter asa dis rete uniformprobabilitydistribution,thelter window

Figure5.3:Chara teristi s ofmultiple-passmovingaverage lters.Figure (a)shows thelter

kernels resulting from applying a seven point moving average lter to the data on e, twi e

andfour times.Figure(b)showsthe orrespondingstepresponses,while( )and(d)showthe

µ =

n

X

i=1

i · p(i) =

n

X

i=1

i

n

=

n + 1

2

(5.13) obtained using

P

n

i=1

i =

n(n+1)

2

;varian e, whi h anbe seenasameasure ofthelter size:

σ

2

=

n

X

i=1

(i − µ) · p(i) =

n

X

i=1

µ

i −

n + 1

₂

¶

1

n

=

n

X

i=1

µ

i

2

n

+

(n + 1)

2

4n

−

i(n + 1)

n

¶

=

(n

2

_{− 1)}

12

(5.14)

obtained using theprevious and

P

n

i=1

i

2

=

n(n+1)(2n+1)

6

.

Hen e, the entral limit theorem implies that, to a hieve a Gaussian urve of varian e

σ

2

g

,

m

lters with windows of varian es

σ

2

1

, ..., σ

m

2

with

σ

2

g

= σ

2

1

+ ... + σ

2

m

must be applied.

Consequently,

m

movingaverages withsizes

n

1

, ..., n

m

yielda standarddeviation:

σ =

r

n

2

1

+ ... + n

2

m

− m

12

(5.15)

Intheequation5.12,forea hpointofthedomain,thenumberof ellsinvolvedintheaverage

isthesameinea hpass(

I

j

remainthesame),thus,we an assume

n

onstantforea hvertex and onsequently,we obtainthattheamplitude (

σ

) of thelter inea hzoneis

σ ∝

√

m

.For

example, withve passes(

m = 5

),theltering operation ison azoneof about

√

5

timesthe size of thegridaroundthe point onsidered.

Looking at thestep response (see gure(b), 5.3), multiple passesprodu ean "s" shaped,as

ompared to the straight line of the single pass.The frequen y responses in ( ) and (d) are

given byequation 5.7multipliedbyitselffor ea hpass.That is, ea h onvolutionresults ina

multipli ationinthethefrequen y domain.

Figure5.4showsthefrequen yresponseoftwootherlterssimilartothemovingaveragelter.

When a pure Gaussian is used as a lter kernel, the frequen y response is also a Gaussian.

TheGaussian isimportantbe auseitistheimpulseresponseofmanynaturalandman-made

systems (seeCentralLimit Theorem).The se ondfrequen y responseinFig.5.4 orresponds

to using a Bla kman window as a lter kernel. The exa tshape of the Bla kman window is

given ingure5.5; however, itlooks mu hlike a Gaussian.

At rst sight, these ltershave better stop-band attenuation than themoving average lter.

Se ondly, the step responses are smooth urves, rather than the abrupt straight line of the

moving average. Anyway, the moving average lter and the other lters behave similarly in

redu ingrandom noisewhile maintainingasharpstepresponse. Themaindieren e inthese

lters is their ost: moving average is the fastest digital lter available. Multiple passes of

the moving average will be orrespondingly slower, but still very qui k. In omparison, the

Gaussian and Bla kman ltersareextremelyslower, be ause theymustuse onvolution.

5.6 Appli ation of the lters to numeri simulations

We usethe timemoving average (see equation 5.4) to lter a database of DNS velo ityeld

snapshotsobtained at

Re = 150

andat

Re = 300

.For both ases,weperformastudy varying

Figure 5.4: Frequen y response of the Bla kman window and Gaussian lter kernels. Both

theseltersprovide better stop-bandattenuation thanthemoving averagelter.Thishasno

advantageinremovingrandomnoisefromtimedomainen odedsignals,butit anbeusefulin

mixed domain problems.The disadvantageof these ltersis thatthey mustuse onvolution,

aslowalgorithm.

Figure5.6: 2Drepresentation ofvorti ity snapshot omponent

ω

z

(-0.5<

ω

z

< 0.5)at Re= 300, (a)unltered, (b)TAV3,( )TAV7.

5.7represent the ee ts on thevorti ity omponent

ω

z

in a se tion of the domain using the dierentaverageintervals:TAV standsforTimeAverageonVelo ityeldsanditisfollowed

bythenumberof snapshotsused

M

.

Thesmallandafterwardsthewidestru turesintheowdiuseanddisappearastheamplitude

oftheaverageintervalin reaseuptoTAV21whi hisanaverageofthewholeperiodofvortex

shedding. In this last ase, we have a re ir ulation zone behind the square ylinder without

the Von Karman vortex street: this is pra ti ally the mean velo ity eld. Moreover, we see

an elongation of the vortexes ores in the

x

dire tion when the interval is in reased be ause

averagingintime orrespondsto averagingin

x

.From athree dimensionalpoint ofview,the

Re = 300

oweldis hara terizedbysmall vorti alloops(see

ω

x

isosurfa es) whi h onne t

thevortex tubesof the Von Karman streetalso distorted by 3Dee t (see [2℄), as shownin

gure5.8. InTAV7 lteredeld,the

ω

x

stru turesaresmallerthanintheunltered aseand theylookstraight inthe

x

dire tion insteadof bendedaroundthe vortex ores.Asremarked

before, this lastones arealsofairly elongatedintheowdire tion.

In the aseat

Re = 300

and

Re = 22000

,welter thedatabases ofsolutions, respe tively,of

DNSandLES numeri simulationsbymeansof themoving multiplepass volumetri average

te hnique.Ingures5.9and5.10,weshowtheee tsofthespa elterwithdierentnumbers

ofpasseson a

Re = 300

oweld bymeans of a2D visualizationof thevorti ity omponent

ω

z

on a

z

plane. At a rst sight, we note as the iso ontours arefairly uneven ompared to the unltered ones. This is a onsequen e of the dis retization of the lter, whi h realizes

theaveragenot exa tlyon a sphere but onits approximations onthe tetrahedralmesh from

Figure5.7: 2Drepresentation ofvorti ity snapshot omponent

ω

z

(-0.5<

ω

z

< 0.5)at Re= 300, (a)TAV9,(b)TAV11,( ) TAV21.

Figure 5.8: Isosurfa es of vorti ity snapshot omponents

ω

z

(red,

ω

z

= 0.4

and green

ω

z

=

(a)

(b)

(c)

Figure5.9: 2Drepresentation ofvorti ity snapshot omponent

ω

z

(-0.5<

ω

z

< 0.5)at Re= 300, (a)notltered, (b)1timeltered, ( )5 timesltered.

oflter passes, we tendto have onlythe vortexes ores of theVon Karmanstreet. Inthe3D

view(seegure5.11),thevorti alloopsareredu edinnumberandsizebutstillhavethesame

behavior.Inaddition,thevortextubesofVon Karmanstreetpreservesubstantiallyunbroken

theirthree-dimensional hara teristi s.

The ow eld at

Re = 22000

is totally turbulent: the Von Karman street is hara terized

byapronoun edthree-dimensionalitywhi htransformsdeeplytheshapeof thevortextubes;

furthermore, there arelots ofsmall stru turesinsteadofvorti alloopsasat

Re = 300

.

The ee t of ltering is more enhan ed in this ase be ause the mesh is less rened than

at

Re = 300

. In gures 5.12 and 5.13, using the same visualization as before, we see the

z

vorti ity omponent: it is lear as thelter spread the vorti ity and eliminate several

stru -tures with a redu ed number of passes. In support of this, we show the isosurfa es of the

vorti ity omponents

ω

z

and

ω

x

for the asenotlteredand lteredwith5 passes: thesmall stru tures are disappeared but the vortex tubes are unbroken and reveal their signi antly

Figure5.10: 2Drepresentation ofvorti itysnapshot omponent

ω

z

(-0.5<

ω

z

<0.5)at Re= 300,(a)10 timesltered, (b)20 timesltered, ( )50 timesltered.

Figure5.11: Isosurfa es of vorti itysnapshot omponents

ω

z

(red,

ω

z

= 0.4

and green

ω

z

=

Figure5.12: 2Drepresentation ofvorti itysnapshot omponent

ω

z

(-0.5<

ω

z

<0.5)at Re= 22000,(a)unltered, (b)1 timespa eltered, ( )5 timesspa e ltered.

Figure5.13: 2Drepresentation ofvorti itysnapshot omponent

ω

z

(-0.5<

ω

z

<0.5)at Re= 22000,(a)10 times spa eltered, (b)20 timesspa eltered.

Figure5.14: Isosurfa es of vorti itysnapshot omponents

ω

z

(red,

ω

z

= 0.4

and green

ω

z

=

−0.4

) and

ω

x

(yellow,

ω

x

= 0.4

andskyblue

ω

x

= −0.4

) at Re =22000, (a)unltered, (b)5 times spa eltered

Preliminary analysis: two dimensional

ase, Re=150

Inorder to test the apabilityof using ltereddatabases for POD,we employthis te hnique

for the ow around a onned square ylinder at

Re = 150

. LSQ and KLSQ estimation

withoutltering havebeeninvestigated in[1℄,thus we onsiderthis results asareferen e for

theanalysis.The DNSdatabase,whi h we lter,istheone presentedin[3℄.

6.1 Denition of POD time averaged databases

TheProperOrthogonalDe ompositionisappliedtoagroupofsnapshotvelo ityelds,named

POD db, and we test the possibility of re onstru ting the ow of another group, alled

OUTSIDEdb. Thetimemovingaverage lteris usedonbothdatabases: at ea hunltered

velo ity snapshot we substitute another one obtained by the entered average made on a

ertainnumberofsnapshotsaroundtheone onsidered.Clearly,for therst andlastvelo ity

elds,therearen'tallthesnapshotsbeforeorafterasthe enteredaveragerequires.Therefore,

we dis ard a few snapshots ompared to not ltereddatabase at thebeginning and the end

of thedatabase inorder to use thesame entered average lter for all the velo ityelds. In

Tab.6.1,weshowthedatabase details:ea honeis alledTAV(itstandsforTimeAverageon

Velo ityelds) followed bythenumberof snapshotsinvolved inea h averageoperation.

POD db OUTPODdb

POD ase

N

o

snp Timestep

N

o

snp Timestep unlt.

95

136.83 < t < 150.78

95

165.03 < t < 178.99

TAV3

93

136.98 < t < 150.64

93

165.18 < t < 178.84

TAV9

87

137.43 < t < 150.19

87

165.63 < t < 178.39

TAV15

81

137.87 < t < 149.75

81

166.07 < t < 177.95

TAV21

75

138.32 < t < 149.30

75

166.52 < t < 177.50

TAV27

69

138.76 < t < 148.85

69

166.96 < t < 177.06

TAV33 63

139.21 < t < 148.41

63

167.41 < t < 176.61

TAV39 57

139.65 < t < 147.96

57

167.85 < t < 176.17

TAV43

53

139.95 < t < 142.67

53

168.15 < t < 175.87

6.2 Representation apability of TAV extra ted POD basis

Withthepurpose of evaluating theee tof using time ltered databasesin POD,we study

the approximation errors in terms of mean u tuating kineti energy aptured, varying the

retained POD modes for the snapshots inside and outside the database (see g.6.1, we

re-member that 0 means non ltered database). The mean energy is the sum of normalized

eigenvaluesforPODdatabaseswhereasfor theOUTSIDEPODdb itisobtainedproje ting

on the modes. At this stage, no dynami s is involved but it is only a question of the POD

modesa tually spanningthe solutionmanifold([2℄).

The rst two modes apture almost entirely the energy of the bidimensional ow, nearly in

thesamequantitybothintheinsideandalsooutsideintervalofPODdatabase.Thisis learly

visiblealsofromgure6.2whi hshowtherstteneigenvaluesnormalizedbythetotalsumof

them for PODofea hTAVdatabase.The rsttwo modesarerelatedto thevortexshedding

whereas theothersare onne tedwithsmallstru turesintheowwhi h,inthelaminar ase,

aresubstantiallyperiodi andrepresenta littlepartoftheenergy.Withadeepersight atthe

rst two eigenvalues (g.6.3), in the unltered ase,

λ

1

and

λ

2

are quite lose but through movingaveragethemagnitudeoftherstonegrowsandtheotherredu euntilTAV15.Then,

with TAV21 thetenden y hanges and omesba k loser tothe ondition nonltered.

For the mean u tuating energy, the same situation o urs: as the number of points in the

time average in rease, we have a paraboli trend withthe maximum lo ated around TAV21

and TAV27 with

N

r

= 2

.Taking morePOD modes,all the asesmovetowardsthethreshold of100%.ThepeakatTAV21/27 orrespondstoatimeaverageonaperiodofahalfthevortex

shedding.

6.3 TAV: alibrations and low-order models

The low-order modelhas beenobtained for the ases TAV3, TAV9, TAV15, TAV21, TAV27

and TAV43 retaining

N

r

= 6

modes. The alibration of the model is performed inthe POD database(detailsshownintable6.1)using81 ollo ationpoints.Ea h y leofvortexshedding

is omposedofabout43snapshots:thus,thetimeintervalofea hPODdatabases oversfrom

onetotwoperiodsdependingonthe ase.Forall ases,weshowtheproje tion oftheltered

Navier-Stokessimulation overthePOD modes ompared to theintegration ofthedynami al

systeminsidethe alibrationintervalfor some representative oe ients (seegures 6.4, 6.5,

6.6, 6.7, 6.8 and 6.9). In the ase TAV3, all the oe ients are almost periodi al and well

predi tedbythelow-ordermodel.In reasingthetimeaverageinterval,weobservearedu tion

intheamplitude and a hangeto anon periodi al behaviorofthemodal oe ientsstarting

from thelast ones whi h brings to an ina urate predi tion in some ases. Thereare visible

errors for the alibrated model for TAV15 and TAV27 where the modal oe ient trend

has more irregularbehaviorin the time interval and the values in module are lower. In this

ondition,the alibrationpro edurebasedonthederivativeofthemodal oe ientsintrodu es

errors in the models.Anyway, withthe ost of wider omputational resour es, the problems