A Galerkin-free model reduction approach for the Navier–Stokes equations

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Eprints ID : 15804

To link to this article : DOI:10.1016/j.jcp.2015.12.051

URL :

http://dx.doi.org/10.1016/j.jcp.2015.12.051

To cite this version : Shinde, Vilas and Longatte, Elisabeth and Baj,

Franck and Hoarau, Yannick and Braza, Marianna A Galerkin-free

model reduction approach for the Navier–Stokes equations. (2015)

Journal of Computational Physics, vol. 309. pp. 148-163. ISSN

0021-9991

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A

Galerkin-free

model

reduction

approach

for

the

Navier–Stokes

equations

Vilas Shinde

a

,

∗

,

Elisabeth Longatte

a

,

Franck Baj

a

,

Yannick Hoarau

c

,

Marianna Braza

b

a_{IMSIA,EDF–CNRS–CEA–ENSTAParisTechUMR9219,ClamartCedex,France} b_{IMFT,Av.duprof.CamilleSoula,31400Toulouse,France}

c_{ICUBE,Strasbourg,France}

a

b

s

t

r

a

c

t

Keywords:

Navier–Stokesequations Properorthogonaldecomposition Reduced-orderflowmodeling Flowcontrol

GalerkinprojectionoftheNavier–Stokes equations onProperOrthogonalDecomposition (POD)basisispredominantlyusedformodelreductioninfluiddynamics.Therobustness forchangingoperatingconditions,numericalstabilityinlong-termtransientbehavior and thepressure-termconsiderationaregenerallythemainconcernsoftheGalerkin Reduced-Order Models(ROM). Inthisarticle, wepresent anovel procedure toconstructan off-referencesolutionstatebyusinganinterpolatedPODreducedbasis.Alinearinterpolation ofthe POD reducedbasisisperformedbyusingtwo referencesolution states.The POD basisfunctions areoptimal incapturing theaveraged flowenergy.The energydominant POD modes and corresponding base flow are interpolated according to the change in operatingparameter.The solutionstate isreadilybuilt withoutperformingthe Galerkin projectionoftheNavier–StokesequationsonthereducedPODspacemodesaswellasthe followingtime-integrationoftheresultedOrdinaryDifferentialEquations(ODE)toobtain the POD time coefficients. The proposed interpolation basedapproach is thus immune from the numerical issues associated with a standard POD-Galerkin ROM. In addition, a posteriori error estimate and a stability analysis of the obtained ROM solution are formulated.Adetailed case studyoftheflow pastacylinder atlowReynolds numbers is considered for the demonstration of proposed method. The ROM results show good agreementwiththehighfidelitynumericalflowsimulation.

1. Introduction

ComputationalFluidDynamics(CFD)simulationsareindispensableelementoftheengineeringresearchtoday.Although

thereisaconsiderableadvancementinthecomputingpowerinlast coupleofdecades,theexactflowsimulations athigh

Reynolds numbersare unaffordable interms ofthe timeand computingcost.The effortsbecome enormousforresearch

applications(e.g.optimization),wherethesimulationsneedtobeperformedrepeatedly.Consequently,reduced-order

mod-els (ROM) are developedextensively inrecent years.They offersubstantial reduction in the degreesoffreedom and yet

retaining theessentialfeatures ofthe flowby meansof thereducedbasis.The reducedsystemmaylead toa better

un-derstandingoftheunderlyingmechanismandtherebyimprovements inthe empiricalflow(turbulence)models.Theflow

*

Correspondingauthor.

control,optimizationandstabilityanalysisinhydrodynamics,aero-acousticsaresomeofthepotentialapplicationsofmodel reduction(seee.g.[17]).

Thefirstimportantstepofthemodelreductioninfluiddynamicsistoformanappropriatereducedbasisoutofa

com-pletesetofbasisfunctions.Thechoiceofparticularbasisfunctionsmaybeproblemspecific.Thederivationofthereduced basis canbe ‘a priori’or‘a posteriori’.One canrefer to [9,16]forsome ofthe early workson‘a priori’formationof the basisfunctions.Recently,[7]used‘apriori’derivationofthebasisfunctions,inthecontextofProperGeneralDecomposition (PGD). Besides,thespectral discretizationmethods areoftenpreferredoverthe spatialdiscretization methodsinorderto

gaintheaccuracyforsamecomputingtimeandspacerequirements.In‘aposteriori’formation,thebasis functionsare

de-rivedusingtheexistingsolutiondatasetsandmethodssuchasProperOrthogonalDecomposition(POD)(fore.g.methodof

DynamicModeDecomposition(DMD)in[21]and[22]).ThePOD(alsoPrincipleComponentAnalysis)isapopularchoiceof

theempiricalbasisfunctionsforNavier–Stokesequations.Especially,inunderstandingtheonsetofbifurcationsor

instabili-tiesandthespatial–temporaldynamicsoftheflowstructures.Theerrorintime-averagedenergyremainsminimalcompared

toeveryothermethodforthesamenumberofmodes.Theconvergenceinextractingthespacestructures(topos)andthe

associatedtimemodes(chronos)isoptimumintermsoftheflowenergy[1].Anelaborated discussionwithmathematical

derivationsontheoptimalityofthePODmethodisprovidedin[8].

ThePOD-GalerkinROMarebuildusingacoordinatetransformationperformedbymeansofaGalerkinprojectionofthe

systemofNavier–StokesequationsonthereducedPODbasisfunctions.Generally,theflowvelocity(v)isdecomposedinto

thespatial(

φ

i)andtemporal(ai)basisfunctionsasshowninEquation(1),

v

(

x

,

t

)

≈

v[0,1,2,...n]

= ¯

v

(

x

)

+

n



i=1

φ

i

(

x

)

ai

(

t

)

(1)

Wherev

¯

(

x

)

isthetime-averagedbaseflow,n isthenumberofPODmodes.Thisequationholdsgoodundertheassumption

that theflowisstatisticallystationaryintime.InincompressibleflowswithDirichlettype boundaryconditions,thebasis

functionssatisfyboth theboundaryconditionsandthedivergence-free constrainofthecontinuityequation. TheGalerkin

projection of the momentum equations on the basis functions results in the non-linear quadratic Ordinary Differential

Equations(ODE)oftheform:

dai dt

=

Ci

+

n



j Li jaj

+

n



j,k Qi jkajak (2)

Where C, L and Q are the GalerkinROM coefficients. The indices i

,

j

,

k

=

1

,

· · · ,

n. Equation(2) is a reduced model for

the Navier–StokesEquations (NSE)withn spatialmodes. The time-integrationof Equation(2)withan appropriate initial

boundary condition gives the temporal coefficients (basis functions), andthe flow solution can be easily built by using

Equation(1).TheGalerkinprojectionideallyshouldpreservethestabilitydynamicsoftheNSE,butgenerallyitisachieved

byextrinsicstability enablers.Rempfer[19]showedhowtheGalerkinROMareinherentlypronetonumericalinstabilities.

Theenergyassociatedwiththetruncatedbasisfunctionskeepspilingon,whichresultsinadivergenceoftheGalerkin-ROM.

The concept ofartificial viscous dissipation to stabilize the Galerkin ROM was introduced in [2]. Later, [23] proposed a

spectral viscositydiffusionconvolution operatorbasedonabifurcation analysis.Inaddition,thestability ofGalerkinROM

greatly dependson parameterssuch as theflow compressibility,pressure-term considerationandtime varying boundary

conditions.Theflowcompressibilityeffectcanbeconsideredbymeansofanenergybasedinnerproductwhileformulating

aROM[20].ThePOD-penalty methodwasproposed by[24] totreat thetimedependenceoftheboundaryconditionson

thePOD-GalerkinROM.TheGalerkinprojectionofthepressure-gradienttermofNSEonthereducedbasisfunctionscanbe

neglected incaseofthe internalflows, butforopen flows thepressure termdoesnotdisappear [18]andit needs tobe

modeled.ThepressuretermisaccountedinaformulationofthepressureextendedGalerkinROMby[5].Inaddition,[15]

demonstratedthatneglectingtheinteractionsbetweenthetime-averagedbaseflowandthefluctuatingflowmayleadtoan

unstableGalerkinROM.Theauthorsalsointroducedtheconceptof‘shiftmode’correctiontechnique.Further,fromtheflow

controlapplicationspointofview[14]proposed acontinuous interpolationbasedmethod.Inthemethod,aninterpolation

betweenthestabilityeigenmodes andthePODmodesisperformedto dealwiththechanging flowconditions.Adetailed

discussiononthenumericalinstabilitiesandperspectivesofthereducedordermodelsinfluiddynamicsisprovidedby[11].

The choice of an appropriate reduced basis, the Galerkin projection of the NSE on the reducedbasis and the

time-integration of theobtained ODE are the main elements of the POD-Galerkin ROM. The POD basis functions are optimal

intermsof flowenergy,while astheGalerkinprojection ofNSEonthe reducedbasismaynot producea stableROMas

discussedabove.Inthisarticle,wepropose anovelapproach,whereitisnot requiredtoperformtheGalerkinprojection

ofNSEonthereducedbasisandalsothetime-integrationtoobtainthePODtimecoefficients.Thetime-averagedbaseflow

andthe PODspacebasis functions(topos) aredirectly interpolatedforthe changeinoperating condition. ThePOD

tem-poralbasisfunctions(chronos)arealsointerpolatedinphasespace.Theperiodicity(the periodoflimit-cycles)ofthePOD

temporalmodesisaccountedfortheenergyconservation.Furthermore,themethodisextendedforacontinuoustransition

betweentwooperatingconditions.AlsoalinearextrapolationofthePODreducedbasisisperformedtowidentherangeof

oftheproposedROM.InSection3,weprovideademonstrationofthemethodusingacasestudyoftheflowpastacylinder atlowReynoldsnumbers.Atlast,theworkissummarized inSection4.

2. Mathematical formulation

The compressibleNavier–Stokes equations(including thecontinuity andenergyequations) are consideredhere asthe

High FidelityModel (HFM).Theflow isstatisticallystationaryintime such thatEquation (1)isapplicableto thesolution (state) variables.The solutionstate vector s

=

s

(

x

,

t

)

isspanned on the spacex

∈ 

,



is the spacialflow domain. t is

thetime in

[

0

,

T∞].Let H beaHilbertspaceandastatevariablesi

(

x

,

t

)

∈

H with i

=

1

,

2

,

· · · ,

r

(

s

)

.r

(

s

)

isthenumberof

state variables.Thestandard innerproductofthestate variablessi

(

x

,

t1

)

, si

(

x

,

t2

)

andthesolutionstate vectors

(

x

,

t

)

are

respectively,

(

si

(

x

,

t1

),

si

(

x

,

t2

))



=



 si

(

x

,

t1

)

·

si

(

x

,

t2

)

dx

(

s

(

x

,

t1

),

s

(

x

,

t2

))



=

⎛

⎜

⎝

(

si

(

x

,

t1

),

si

(

x

,

t2

))



..

.



sr(s)

(

x

,

t1

),

sr(s)

(

x

,

t2

)





⎞

⎟

⎠

(3)

Theinducednormandtimeaveraging(fortimeperiodT_∞)ofastatevariableandthesolutionstatevectorarerespectively

definedas,

||

si

||



=



(

si

,

si

)

 and s

¯

i

=

1 T_∞



T∞ sidt

= 

si



T∞

||

s

||



=

⎛

⎜

⎝

√

(

si

,

si

)



..

.



(

sr(s)

,

sr(s)

)



⎞

⎟

⎠

and s

¯

=

1 T_∞



T∞ s dt

= 

s



T_∞ (4)

2.1. MethodofsnapshotsPOD

The POD or Karhunen–Loeve expansion was first introduced in fluid dynamics by [12] for the analysis of coherent

structures in the flow turbulence. Following the development of POD,[25] introduced the method of snapshots for the

experimental andnumericaldatasets.Itallowsfurtherreductionofdegreesoffreedom,comparedtothedirectmethodof

POD.

The solutionstate vector s includes allvariables varyingin thetime andspace. Let

η

be anoperating parameter(e.g.

Reynoldsnumber).ThestatevectoroftheHighFidelityModel(HFM)solutioncanbedefinedas,

s

(

x

,

t

;

η

)

=

⎛

⎜

⎜

⎜

⎝

ρ

(

x

,

t

;

η

)

v

(

x

,

t

;

η

)

p

(

x

,

t

;

η

)

..

.

⎞

⎟

⎟

⎟

⎠

(5)

Where

ρ

,

v and

p arethefluiddensity,velocityvectorandstaticpressurerespectively.Thestatevectorcanbeseparatedin

thetime-averagedbaseflowandtheunsteadypartasshowninEquation(6).

s

(

x

,

t

;

η

)

= ¯

s

(

x

;

η

)

+

s

(

x

,

t

;

η

)

(6)

= ¯

s

(

x

;

η

)

+

∞



i=1

φ

i

(

x

;

η

)

ai

(

t

;

η

)

(7)

In Equation(7), theunsteady part(s

(

x

,

t

;

η

)

) is decomposedintothe PODbasis functionsusing theGalerkin expansion. The time invariant orthonormal

φ

i

(

x

;

η

)

andthespaceinvariant orthogonalai

(

t

;

η

)

arethe PODbasis functions(modes).

Thestatevectorcanbeobtainedindiscrete(Nt)snapshotsbyperformingaCFDsimulation.Thesnapshotscanbecollected

oncetheflowbecomesstatisticallystationaryandusing(typically)aconstanttimestep(



tsn).LetNt,Npodbethenumber

of snapshotsandnumberofPODmodesrespectively, alsoNpod

≤

Nt−1.Thestate vector canbe approximatedbydiscrete

snapshotsas, s

(

x

,

t

;

η

)

≈

s

(

x

,

t1

;

η

), . . . ,

s

(

x

,

tNt

;

η

)

(8)

≈ ¯

s

(

x

;

η

)

+

N



pod i=1

φ

_i

(

x

;

η

)

ai

(

t

;

η

)

t1

≤

t

≤

tNt (9)

Wheret1 andtNt arethetimecoordinatesofthefirstandlast snapshots.Also,let Tsn

= [

t1

,

..,

tNt

]

be thetime domainof discretesnapshotscollection.Thetimestep(



tsn)ofsnapshotsrecordingandthenumberofsnapshots(Nt)dependonthe

desiredresolutioninthetemporalharmonicsofthePODmodes[18].

LetR

(η)

bethetwopointtime-correlationfunction,givenby,

R

(

η

)

=

R

(

ti

,

tj

,

η

)

=

1 Nt



s

(

x

,

ti

;

η

),

s

(

x

,

tj

;

η

)



 i

,

j

=

1

,

2

, . . .

Nt (10)

ThecorrelationfunctionR

(η)

issolvedfortheeigenvalueproblem,asinEquation(12).

R

(

η

)ψ

i

(

t

;

η

)

= λ

i

ψ

i

(

t

;

η

)

(11)

where

λ

iaretheeigenvalues.Theorthogonaleigenfunctions

ψ

i

(

t

;

η

)

arethennormalizedas,



ψ

_i

(

t

;

η

), ψ

_j

(

t

;

η

)



Tsn

= δ

i j (12)

Where,

δ

i j istheKroneckerdeltainvectorform.ThePODmodesarearrangedindescendingorderoftheirenergycontent

(the eigenvalues associated with the modes), i.e.

λ

1

> λ

2

> . . . > λ

Npod

>

0. The orthonormal ‘topos’ are obtainedusing Equation(13),suchthat



φ

i

(

x

;

η), φ

i

(

x

;

η)





= δ

i j.

φ

_i

(

x

;

η

)

=

√

1 Nt

λ

i



s

(

x

,

t

;

η

), ψ

i

(

t

;

η

)



Tsn (13)

ThecorrespondingPODtimecoefficientsaregivenby,

ai

(

t

;

η

)

=



φ

_i

(

x

;

η

),

s

(

x

,

t

;

η

)



_

=



Nt

λ

i

ψ

i

(

t

;

η

)

(14)

Generally,thenumberofreducedPODmodes(Nr) ismuchsmallercompared tothetotal PODmodes(Nr

<<

Npod).The

relativeenergycaptured(Ec)bythemostenergetic(firstfew)PODmodesissubstantial.Itcanbegivenas,

%Ec

=

Nr i=1

λ

i

Npod i=1

λ

i

×

100 (15)

2.2. PeriodicityofPODtemporalmodes

Thetotalenergy1 E

(η)

pod oftheunsteadypartofthediscretestatevectorcanbegivenby,

E

(

η

)

pod

=

1 2







s

(

x

,

t

,

η

)

2



Tsn dx

=

1 2 Npod



i=1

λ

i

=

1 2 Npod



i=1



ai

(

t

;

η

)

2



Tsn (16)

Thespacedomain(



)islimitedbyaboundary(

∂

).Similarly,let Tminbetheminimumtimewindowforwhichthetotal

energyinEquation(16)remainsthesame,suchthat,

E

(

η

)

pod

=

1 2







s

(

x

,

t

,

η

)

2



Tmin dx

=

1 2 Npod



i=1

λ

i

=

1 2 Npod



i=1



ai

(

t

;

η

)

2



Tmin (17)

Instatisticallystationaryflows,thePODtemporalbasisfunctionsobservethestablelimitcyclesinphasespace(seefor e.g.[23,13,1]).Let Tη bethetimeperiodofthelimit-cycleoffirstPODtime coefficienta1

(

t

;

η

)

.Thehigher(wellresolved

bysnapshots)PODtimemodesforthestatevectorareperiodicwiththetime Tη.ThecharacteristicPODtimecoefficients

canbedefinedas,

˜

ai

(

t

;

η

)

=

ai

(

t

;

η

)

for t

∈ [

ta

,

ta

+

Tη

]

(18) Whereta

∈



0

, (

Tsn

−

Tη

)



isanarbitrarytime.Further,thetotalenergyinEquation(17)becomes,

E

(

η

)

pod

=

1 2 Npod



i=1



˜

ai

(

t

;

η

)

2



Tη

=

1 2 Npod



i=1



ai

(

t

;

η

)

2



Tmin

=

1 2 Npod



i=1

λ

i (19)

Italsoimpliesthattheminimumtimewindow(Tmin)isthetimeperiodofthefirstPODtemporalmode(Tη).

Under the statistically stationary flow assumption and using the periodic characteristic POD temporal modes

(Equa-tion (18)), onecanreconstructtheflowwithreducednumber(Nr)ofPODbasiseven outsidethesnapshotstime domain

(Tsn)as, s

(

x

,

t

;

η

)

≈ ¯

s

(

x

;

η

)

+

Nr



i=1

φ

_i

(

x

;

η

)

a

˜

i

(

t

;

η

)

t

≥

0 (20) 2.3. Linearinterpolation

A linear interpolation is used to interpolate the right hand side terms of Equation (20) for the change in operating

parameter

η

.Theinterpolation ofthecharacteristicPODtemporalmodes(a

˜

i) ensures theappropriate flowenergy(E

(η)

)

levelsintheinterpolatedstate.

Lets

(

x

,

t

;

η

j

)

with j

=

1,2 bethetworeferencestates.Inordertobuildasolutionstatevectoratanoperatingparameter

η

∈ [

η

1

,

η

2

]

, the time-averagedbase flow

¯

s

(

x

;

η

)

, the POD spacialmodes (

φ

i

(

x

;

η

)

) andthe associated time coefficients

˜

ai

(

t

;

η

)

are obtained by the linear interpolation of the reference states. The interpolation is formulated using a vector

(β

;

η

)

inEquation(21).Itstandsforthesolutionstateaverage(

¯

s

(

x

;

η

)

)andthePODmodes(

φ

i

(

x

;

η

)

anda

˜

i

(

t

;

η

)

).

(β

;

η

)

= (β;

η

1

)

+



((β

;

η

2

)

− (β;

η

1

))

(

η

2

−

η

1

)



(

η

−

η

1

)

(21)

Here

β

iseitherx,fors,

¯

φ

i ort

∈ [

0

,

Tη

]

fora

˜

i.Apriori,theconditioninEquation(22)issatisfiedsothattheinterpolated

quantities(RHSofEquation(20))followthesignsofanyofthetwo(

η

1and

η

2)referencecases.

((β

;

η

1

), (β

;

η

2

))

β

≥

0 (22)

Thetime-averagesofthestatevectors(s

¯

(

x

;

η

j

)

for j

=

1

,

2)generallydonotaltertheirsignforthechangeinoperating

parameter(

η

j).Asymmetryintheflowgeometrycanleadtoaphasedifferenceof

π

betweenthecorrespondingPODspace

modes (

φ

i

(

x

;

η

j

)

) fordifferentoperating conditions (

η

j). The constrain inEquation (22)ensures that they donot cancel

out, whileperformingthe interpolation.Inaddition,thereferencestates

η

j needtobe closeenough,inordertoperform

the linear interpolation (Equation (21)). The characteristic POD time coefficients (a

˜

i

(

t

;

η

)

) are brought inminimal phase

differencebyusingEquation(22).TheinterpolatedbasesolutionandthePODmodesfollowanyoneofthereferencestates for thephase. Thecharacteristictime period (Tη)isalsolinearly interpolatedforthechange inoperatingparameter (

η

).

TheinterpolationROMsolution,withthereducednumber(Nr)ofPODinterpolatedbasisandforthechangeofparameter

(

η

)in

[

η

1

,

η

2

]

,canbewrittenas,

s

(

x

,

t

;

η

)

≈ ¯

s

(

x

;

η

)

+

Nr



i=1

φ

i

(

x

;

η

)

a

˜

i

(

t

;

η

)

t

≥

0 &

η

∈ [

η

1

,

η

2

]

(23)

AsmoothtransitionofaROMsolutionfromoneflowstatetoanotherisusefulintheflowcontrolapplications.A

contin-uous modeinterpolatingtechnique developedin[14] usesaparameter

κ

foracontinuous transitionbetweenthestability

matrices at a steady state to an unsteady (with periodic limit cycle) state. Similarly, a smooth transition between two

interpolatedoff-referencestates(

η

n_,

_η

n+1_{)canbeachievedby,}

(β

;

η

n+1

)

=

κ

(β

;

η

n+1

)

+ (

1

−

κ

)(β

;

η

n

)

(24)

T_ηn+1

=

κ

T_ηn+1

+ (

1

−

κ

)

T_ηn (25)

Heren isanintegerindicatorforaflowstate.Thetransitionparameter

κ

variesfrom0 to1.Asimplelinearfunctionwith anappropriatetimedelayparameter(cτ )canbeusedtoobtainarealtimetransition.Equation(26)showssuchafunction.

κ

=

cτ



t

−

tn₀+1



/

T_ηn+1 (26)

Where, tn₀+1 represents the time ofcontrol parameter change.The time delay constant (cτ ) can be used to control the

transitiontime.

In addition to the linearinterpolation, a linear extrapolationof thereference states(

η

1 and

η

2) can also be used to

widentherangeofcontrollingparameter,withacautionofthepresenceofmajorflowtransitionsinthevicinity.

2.4. Aposteriorierrorestimate

2.4.1. SnapshotsPODandtruncationerrors

The HighFidelityModel(HFM)solution canbean accurateCFDsolutiontothe fullNSEsortheexperimental datasets

for the flow under consideration. The HFM solution state vector can be expressed in terms of POD basis functionsby

s

(

x

,

t

;

η

)

hf

≈ ¯

s

(

x

;

η

)

pod

+

Npod



i=1

φ

_i

(

x

;

η

)

poda

˜

i

(

t

;

η

)

pod (27)

Thesubscript‘hf ’standsforahighfidelitysolution,whileasthesubscript‘pod’standsforquantitiesestimatedusingPOD. A posterioritheerrorinPODdiscretizationcanbegivenby,



ps

(

x

,

t

;

η

)

=

s

(

x

,

t

;

η

)

hf

−

s

(

x

,

t

;

η

)

pod (28)

Where thesubscript ‘ps’stands for aPOD based errorinthe solution state vector s. ThePOD errordepends mainlyon

thetimestepofsnapshotscollection(



Tsn),numberofsnapshots(Nt)andthetime-windowofsnapshotscollection(Tsn).

A rigorousparametricanalysisanderrorestimatestudyofthePODmethodwasperformedbyKunischandVolkwein[10].

Inordertonormalize theerrors,letusrepresenttheelementwisedivisionofvectorsu and v asu



v,fornoelementof vector v iszero(vi

=

0).Further,thetotalvariancecanbedefinedforthehighfidelitystatevectors

(

x

,

t

;

η

)

as,

σ

2

₍

_η

₎

₌







s

(

x

,

t

;

η

)

2_hf



T_∞dx (29)

Aposteriori,normalizederrorinPODdiscretizationcanbegivenby,

p

(

t

;

η

)

=



















ps

(

x

,

t

;

η

)

2dx



σ

2

(

η

)













r(s) (30)

Inaddition,theerrorintroducedbythetruncationofthehigher(

>

Nr)PODmodescanbeobtainedas,



ts

(

x

,

t

;

η

)

=

Npod



i=Nr+1

φ

_i

(

x

;

η

)

poda

˜

i

(

t

;

η

)

pod (31)

Thenormalizedtruncationerrorbecomes,

t

(

t

;

η

)

=



















ts

(

x

,

t

;

η

)

2dx



σ

2

(

η

)













r(s) (32) 2.4.2. Interpolationerror

TheinterpolationerrorsassociatedwitheachtermoftheROMsolution(Equation(23))withrespecttothePODsolution

canbedefined,



s_¯

(

x

;

η

)

= ¯

s

(

x

;

η

)

pod

− ¯

s

(

x

;

η

)



φi

(

x

;

η

)

= φ

i

(

x

;

η

)

pod

− φ

i

(

x

;

η

)



a_˜i

(

t

;

η

)

= ˜

ai

(

t

;

η

)

pod

− ˜

ai

(

t

;

η

)

(33)

Let



is

(

x

,

t

;

η

)

bethetotalinterpolationerrorinsolutionstate vector(s)withrespecttothePODsolution.Itcanbe given

as,



is

(

x

,

t

;

η

)

=

s

(

x

,

t

;

η

)

pod

−

s

(

x

,

t

;

η

)

(34)



is

(

x

,

t

;

η

)

=



¯

s

(

x

;

η

)

pod

+

Nr



i=1

φ

i

(

x

;

η

)

poda

˜

i

(

t

;

η

)

pod



−



¯

s

(

x

;

η

)

+

Nr



i=1

φ

i

(

x

;

η

)

a

˜

i

(

t

;

η

)



(35)

UsingtheindividualerrordefinitionsfromEquation(33)andthetotalinterpolationerrorinEquation(35)weobtain,



is

(

x

,

t

;

η

)

=



s¯

(

x

;

η

)

+

Nr



i=1

φ

_i

(

x

;

η

)



_a˜i

(

t

;

η

)

+



φi

(

x

;

η

)

a

˜

i

(

t

;

η

)

+



φi

(

x

;

η

)



a˜i

(

t

;

η

)

(36)

Apriori,themaximumerrorboundinthelinearinterpolationcanbe givenby Equation(37),foreach interpolationerror

|



¯s

(

x

;

η

)

| ≤

1 8

(

η

)

2 _sup η∈[η1,η2]

|

α

¯s

(

x

;

η

)

|

where

α

s¯

(

x

;

η

)

=

∂

2

∂

η

2

(

s

¯

(

x

;

η

)

pod

)





φi

(

x

;

η

)

 ≤

1 8

(

η

)

2 _sup η∈[η1,η2]



α

φi

(

x

;

η

)



where

α

φi

(

x

;

η

)

=

∂

2

∂

η

2

(φ

i

(

x

;

η

)

pod

)





a_˜i

(

t

;

η

)

 ≤

1 8

(

η

)

2 _sup η∈[η1,η2]



α

a_˜i

(

t

;

η

)



where

α

a˜i

(

t

;

η

)

=

∂

2

∂

η

2

(

a

˜

i

(

t

;

η

)

pod

)

(37)

The erroris

O(

η

2

₎

_{. Here}

_η

_{= (}

_η

2

−

η

1

)

.The value of

η

canbe chosen basedonthe totalinterpolationerrorbound

|



is

(

x

,

t

;

η)

|

.Thetotalinterpolationerrorinthesolutionstatevectors

(

x

,

t

;

η

)

isinboundsas,

|



is

(

x

,

t

;

η

)

| ≤

1 8

(

η

)

2 _sup η∈[η1,η2]







α

s¯

(

x

;

η

)

+

Nr



i=1

φ

i

(

x

;

η

)

α

α˜i

(

t

;

η

)

+

α

φi

(

x

;

η

)

a

˜

i

(

t

;

η

)

+

1 8

(

η

)

2

_α

φi

(

x

;

η

)

α

α˜i

(

t

;

η

)







(38)

On the other hand, a posterioriinterpolation errorcan be directlygiven by Equation (34). The normalized interpolation

errorwillbe,

i

(

t

;

η

)

=



















is

(

x

,

t

;

η

)

2dx



σ

2

(

η

)













r(s) (39)

2.4.3. Energybasederror

Generally, theerrorinGalerkinROMis quantifiedbasedon thequadraticflow energyterms.ThePODbasis functions

(toposandchronos)aretheoptimalbasis foraROMinfluiddynamics,henceitprovidesan upperbound fortheerrorin

GelerkinROM[3,6].ThenormalizederrorinROMbasedonthekineticenergycanbeexpressedas,

e

(

t

;

η

)

=











E

(

t

;

η

)

pod

−

E

(

t

;

η

)





σ

2

(

η

)









r(s)

=













⎛

⎝

Npod



i=1

˜

ai

(

t

;

η

)

2pod

−

Nr



i=1

˜

ai

(

t

;

η

)

2

⎞

⎠ 

σ

2

₍

_η

₎













r(s) (40)

Where E

(

t

;

η

)

istheenergyofROMsolution.InthepresentedformulationofROM,theenergybasederror(

e

(

t

;

η

)

)does

notaccountfortheerrorininterpolationofthetime-averagedbaseflow(s

¯

(

x

;

η

)

)aswellasthePODspacemodes(

φ

_i

(

x

;

η

)

). ThereforethetotalerrorrelevanttotheinterpolationROMcanbedefinedas,

it

(

t

;

η

)

=

i

(

t

;

η

)

+

t

(

t

;

η

)

(41)

2.5. StabilityoftheinterpolationROM

Almost all theGalerkinROMare unstable andneedstabilizationtechniquessuch asaddition oftheartificial viscosity

terms, increasing the order of ROM. This way, either the high fidelity Navier–Stokes equation are altered or the

com-putational efforts are increased [3]. On the contrary, the interpolation based approach of ROM uses the flow statistical

stationarityassumption fortheenergybalanceinstead ofbalancingtheenergyoftruncatedPODmodesby meansofthe

empirical turbulencemodels.The timeaverageofthetotalerror it

(

t

;

η

)

inthe interpolationROM(Equation(41))can be

givenby,

(

η

)

= 

it

(

t

;

η

)



T∞

= 

it

(

t

;

η

)



Tη (42) itimplies,

∂

(

η

)

∂

Tη

=

0 (43)

The errors (



ps

(

x

,

t

;

η

)

,



ts

(

x

,

t

;

η

)

and



is

(

x

,

t

;

η

)

) in theinterpolation ROMare in boundsunderthe stationaryflow

as-sumptionforalltime.Thetotalnormalizederror

(η)

remainsafunctionoftheparameters



Tsn,Nt,Npod,Nr,

η

andthe

2.5.1. Floquetstabilityanalysis

Let No bethe numberofPODtimemodeswiththetime period Tη.The periodicbaseflow fortheFloquetinstability

canbegivenas,

so

(

x

,

t

;

η

)

= ¯

s

(

x

;

η

)

+

No



i=1

φ

i

(

x

;

η

)

ai

(

t

;

η

)

(44)

Lets _o

(

x

,

t

;

η

)

bethesmallperturbationinthebaseflow.ItiscanberepresentedintermsofthePODbasisas,

s _o

(

x

,

t

;

η

)

≈

Nr



i=No+1

χ

i

=

Nr



i=No+1

φ

_i

(

x

;

η

)

ai

(

t

;

η

)

(45)

Theperturbations _o

(

x

,

t

;

η

)

inthebaseflowisperiodicwiththeperiodTη.ThereforewecanconsiderEquation(45)forthe Floquetanalysis. The Tη periodicfunctions

χ

i can berepresentedintheform,

χ

˜

iexp

(ς

it

)

.Where

χ

˜

i arealso Tη periodic

andknownasFloquet modes. Theexponents

ς

i are calledthe Floquetexponents.Generally, theFloquet multipliers

ξ

i

≡

exp

(ς

iTη

)

areusedinthestabilityanalysis. Theperturbation(s o

(

x

,

t

;

η

)

)grows exponentiallyfor

|ξ

i

|

>

1 andtheperiodic

baseflow isunstable.Onthe other handtheperturbationdecays exponentiallyfor

|ξ

i

|

<

1 andtheperiodic baseflowis

stable[4].

TheFloquetmodes(

χ

˜

i)atatimeinstanceaftern timeperiods(Tη)canbewrittenas,

˜

χ

_in

= φ

i

(

x

;

η

)

a

˜

i

(

t

;

η

)

n (46)

Where,a

˜

i

(

t

;

η

)

n

= ˜

ai

(

t0

+

nTη

;

η

)

arethePODtimemodesatn timeperiods(Tη)afteraninitialtimet0.Thecharacteristic

PODtimemodes, asdefinedinEquation(18),areperiodicwithtime Tη.Thereforea

˜

i

(

t0

+

nTη

;

η

)

= ˜

ai

(

t0

+ (

n

+

1

)

Tη

;

η

)

,

whichleadsto

χ

˜

_in

= ˜

χ

_in+1.Furthermore,thenumberofPODmodes(Nr)usedtobuildtheROMsolutionfollowstablelimit

cycleswithtimeperiod Tη.Thus thevalue ofFloquetmultipliers

|ξ

i

|

=

1 andthecorrespondingFloquetexponents

ς

i

=

0.

Theperturbations _o

(

x

,

t

;

η

)

neithergrowsnordecayswiththetimeataparticularoperatingcondition(

η

).

3. Flow past a cylinder at low Reynolds number – a case study

Theflow pasta cylinderatlowReynolds number(Re

=

125

∼

150) in2-dimension(2D) isconsidered forthe

demon-strationoftheproposedReduced-OrderModel(ROM).Fig. 1showstheflowdomainandtheinstantaneousflowfields(u,v

andp)atReynoldsnumberRe

=

125 (Re

=

ρ

u_∞D

/μ

).ThecylinderofdiameterD

=

1 isatthecenterofthecomputational domain.The inflowstreamwise(along

+

x axis)velocity (u_∞) aswellasthe temperature(

θ

_∞) farupstream are setto1. Thedensityofthefluid (caloricallyperfectgas)is

ρ

=

1.TheMachnumberupstreamis M_∞

=

0

.

18,whileasthespecific heatratioof1

.

4 (forair)istaken.Thegasconstant R andtheinflowpressurep_∞ are22

.

05.Thedynamicviscosity(

μ

)is constant,itisestimatedusingtheReynoldsnumber(Re_∞)as,

μ

= (

ρ

v_∞D

)/(

Re_∞

)

.Theinflowtransversevelocityisv_∞

=

0. Theinternalenergy(e)andtheenthalpy(h)aregivenbyCv

θ

andCp

θ

respectively,whereCv,Cp arethespecificheatsat

constantvolumeandconstantpressurerespectively.Thetotalenergy(E)andtheinternalenergy(e)arerelatedby

e

=

E

−

1

2



u2

+

v2



)

3.1. Governingflowequationsandnumericalmethods

A compressibleNavier–Stokes flowsolver (Navier–Stokes Multi-Block – NSMB) isused witha preconditioning forthe

incompressibleflowatlowMachnumber.TheNSMBsolverisdevelopedincollaborationbetweenseveralEuropean

organi-zationswhichmainlyincludesAirbus,KTH,EPFL,IMFT,ICUBE,CERFACS,UniversityofKarlsruheandETH-EcolePolytechnique

de Zurich. The code has been developedsince early 90s. It is coordinated by CFS Engineering inLausanne, Switzerland.

NSMB is a structured code including a variety of high-order numerical schemes andturbulence modeling such asLES,

URANS,RANS-LEShybridturbulencemodeling,especiallyDDES(DelayedDetachedEddySimulations).

ThecompressibleunsteadyNavier–Stokesequationsin2Dcanbewrittenas,

∂

∂

t

(

w

)

+

∂

∂

x

(

f

−

fν

)

+

∂

∂

y

(

g

−

gν

)

=

0 (47) Where, w

=

⎛

⎜

⎜

⎝

ρ

ρ

u

ρ

v

ρ

E

⎞

⎟

⎟

⎠ ,

f

=

⎛

⎜

⎜

⎝

ρ

u

ρ

u2

+

p

ρ

uv u

(

ρ

E

+

p

)

⎞

⎟

⎟

⎠ ,

g

=

⎛

⎜

⎜

⎝

ρ

v

ρ

vu

ρ

v2

+

p v

(

ρ

E

+

p

)

⎞

⎟

⎟

⎠

Fig. 1. Computational domain and instantaneous flow fields at Re=125. f_ν

=

⎛

⎜

⎜

⎝

0

τ

xx

τ

xy [

τ

,

v]_x

−

qx

⎞

⎟

⎟

⎠ ,

g_ν

=

⎛

⎜

⎜

⎝

0

τ

yx

τ

y y [

τ

,

v]_y

−

qy

⎞

⎟

⎟

⎠

Here w isthestate vector. f , g aretheconvective fluxes,whileas _{f ν , gν are}theviscous fluxes.The componentsof

shearstresstensor

τ

intheviscousfluxesaregivenbyEquation(48).

τ

xx

=

2 3

μ



2

∂

u

∂

x

−

∂

v

∂

y



,

τ

y y

=

2 3

μ



−

∂

u

∂

x

+

2

∂

v

∂

y



τ

xy

=

τ

yx

=

μ



∂

u

∂

y

+

∂

v

∂

x



(48)

TheheatfluxiscalculatedusingFourier’slawas, qx

= −

k

∂θ

∂

x

,

qy

= −

k

∂θ

∂

y with k

=

μ

Cp

/

Pr (49)

Wherek isthethermalconductivity.ThePrandtlnumber(Pr)istaken0

.

72 (forair).

The secondorderfullyimplicitLU-SGS(Lower–UpperSymmetricGauss–Seidel)backwardA-stable schemewitha

dual-timesteppingisusedforthetimemarching.Thespacediscretizationisdoneusingforthordercentralfinitevolumescheme

in askew-symmetricform. Thepreconditioningmethod proposedin[26] toimpose theincompressibility isused,forthe

Fig. 2. POD analysis of the flow at Re=140 (η).

3.2. Resultsanddiscussion

Thestatevectors inthecasestudy(2-D,incompressibleflow)canbeconsideredas,

s

(

x

,

t

;

η

)

=

⎛

⎝

uv

(

(

xx

,

,

tt

;

;

η

η

)

)

p

(

x

,

t

;

η

)

⎞

⎠

(50)

Wherex is thespacedomainwithx and y dimensions.t representsthetime.The operatingparameter

η

istheReynolds

numberRe.ThetworeferencecasesareconsideredatReynoldsnumbers

η

1

=

Re1

=

125 and

η

2

=

Re2

=

150.Thenumber

ofsnapshots takenforeach referencecaseis Nt

=

900,thisconstitutes

≈

14 vortexshedding periods.The time step for

snapshotscollectionis



tsn

=

0

.

05.Thecorrelationmatrixwas builtforeachreferencecaseandsolvedfortheeigenvalue

problemasdetailedinSection2.1.Theoff-referencecaseisconsideredat

η

=

Re

=

140.Thelinearinterpolationofthestate

vectortime-averagesandPODmodes(bothtopos andchronos)usingthereferencestatesisperformedasperSection 2.3.

The results are build usingfirst 10 POD modes (Nr

=

10) out of 500POD modes (Npod

=

500) and compared withthe

Navier–StokesHighFidelityModel(HFM)simulationresultsatthesameReynoldsnumber.

The results of POD analysis at Re

=

140 are shown is Fig. 2, in terms of the eigenvalues and the time evolution of

thediscretizationerrorinvolvedinthemethodofsnapshotsPOD.Fig. 2(a)showsthe%energyassociatedwitheachPOD

modeofthestate variables.Italsoindicatesthatthe

≈

99

.

99% oftotalenergyiscontainedinfirst10modesofeachstate variables.Thereforethenumberofreducedbasis Nr

=

10 ischosen fortheinterpolation(ROM). Thediscretizationerrorin

themethodofsnapshotsPOD(

p

(

t

;

η

)

),asdefinedinEquation(30)isplottedinFig. 2(b).Theroot-mean-squared (rms)of

theerroris

≈

0

.

25% ofthevarianceofthestatevariable.

3.2.1. InterpolationofthePODreducedbasis

In thiscasestudy,the PODspace modes(

φ

_i

(

x

;

η

)

) are eithersymmetric orantisymmetric aboutthe x axis. The

pre-conditioningin Equation(22)isneededforthe antisymmetric modes, onlywhenthey observe aflip ofsigninchanging

operatingcondition(

η

).Fig. 3showsthelinearinterpolationperformedforthefifthspacemodeofthestreamwisevelocity (

φ

₅u).Figs. 3(a)and3(b)arethefifthPODspacemodesofthereferencecasesatRe1

=

125 andRe2

=

150 respectively.The

resultofinterpolationatRe

=

140 for

φ

₅u

(

x

;

Re

)

isshowninFig. 3(d).Fig. 3(c)showstheactualPODmode(

φ

₅u)atRe

=

140,

computedusingthemethodofsnapshotsPODforcomparisonwiththeinterpolatedmode.

Similarly,theremainingtoposfromthereducedbasiswereinterpolatedatReynoldsnumberRe

=

140.Fig. 4shows

com-parisonofthefirstfourinterpolated(ROM)modes(Figs. 4(b),4(d),4(f),4(h))versusthesnapshotsPODmodes(Figs. 4(a), 4(c), 4(e),4(g)respectively). Onecan noticethat thePOD modesact inpairs. Thefirst pairofPODmodesofstreamwise velocityu (modenumber1&2)isantisymmetric,whilethesecondoneissymmetricaboutthex axis.Ingeneralhere,the

oddpairsofPODmodesofu areantisymmetricandtheevenpairsaresymmetric.Theantisymmetryofthemodesabout

x axisisdealtbytheconstraininEquation(22)beforeinterpolatingthemodes.ThePODisabiorthogonaldecomposition

oftheflowinspaceandtime,thereisone-to-onecorrespondencebetweentoposandchronos[1].Thechangeinsymmetry

ofatoporeflectsinthecorrespondingchrono.Althoughthischangeofsign(of

φ

ianda

˜

i forthesameoperatingcondition)

doesnotalterthevalueofflowreconstructionbyEquation(23).Thephaseinformationisanywaylostbecauseofthesecond orderstatisticsusedinthePODbasisfunctions[22].Inadditiontothephase information,thechangeofoperating condi-tion(Re)leadstothechangeinorientationofthePODbasisfunctions.Theinterpolationprocedureensuresanappropriate orientationofthePODreducedbasisforanintermediateoperatingconditionsbetweenthereferencestates.

InGalerkinROMsthetime coefficientsoftenneedcorrectionsintheiramplitudes.The commonsource oferrorisdue

tothetruncationofhigherPODmodesandtheformulationoftheROMwithoutpressure-termrepresentation.Forinstance,

Fig. 3. Interpolation ofφu

5(x,·).

characteristic POD time coefficients (a

˜

i

(

t

;

Re

)

) are immune from the truncationand pressure-termerrors, since they are

extractedfromthetimecoefficientsofthePOD(ai

(

t

;

Re

)

)itselfasperEquation(18)forthereferencecases(

η

1and

η

2).The

characteristictimecoefficients,similartothefellowspacialmodesactinpairs.Theinterpolationresultsforthecharacteristic timecoefficients(chronos)areshowninFig. 5.Itshowsthecomparisonofinterpolationresultsinphasespaceforthefirst fivecharacteristictimecoefficients.Thecurvesineachplot(Figs. 5(a),5(b),5(c)and5(d))expandinsize,withtheincrease ofReynoldsnumber.Thelimit-cyclesrepresentedinredcolorareforthereferencestateRe1

=

125,whiletheonesinpink

colorareforthereferencestate Re2

=

150.Thelimit-cyclesatRe

=

140,inbluecolorareinterpolatedusingthereference

statesRe1 andRe2.ItcanbecomparedwiththecharacteristicPODtimemodesobtainedusingsnapshotsPODatRe

=

140

ingreencolor.

In addition,thecharacteristictimes(Tη)ofthereferencestatesRe1

=

125 andRe2

=

150 are TRe1

=

5

.

647 and TRe2

=

5

.

400 respectively.ThelinearlyinterpolatedcharacteristictimeatRe

=

140 is TRe

=

5

.

499 againstthevalue5

.

489 obtained

inPODanalysis.

TheeigenvaluesoftheinterpolationROMsolutionatRe

=

140 wereestimatedusingrelation,

λ

i

=



˜

ai

(

t

;

Re

)

2



TRe (51)

Fig. 6(a)showstheenergy(in %)associatedwiththereducedinterpolated(ROM)modesatRe

=

140,itiscomparedwith

theenergy(in%)ofthecorrespondingsnapshotsPODmodes(cumulativeplotinFig. 6(b)).Thetime-averagedflowenergy

estimationusingtheinterpolatedPODtimemodes(Equation(51))evincestheorthogonalityoftheinterpolatedmodes[3]. An additional orthogonalitycheck is performedaposteriori onthe interpolatedreduced basis.The angle (

θ

γ,β) between

interpolatedmodes(

γ

,

β

∈

L2

()

)iscalculatedbymeansoftheirinnerproductas,

θ

γ,β

=

arccos



(

γ

, β)

_

||

γ

||



||β||





(52)

Theangles(indegree)betweentheinterpolatedreducedbasisofstreamwisevelocity(u)aretabulatedinTable 1.Itclearly demonstratesthattheinterpolationofthePODmodesretainstheorthogonalityofboththetopos(

φ

_i)andchronos(a

˜

i).

Theerrorsquantification,asformulatedinSection 2.4isplottedinFig. 7.Thetruncationerror(

t

(

t

;

Re

)

)isnothingbut

thecontributionofhigherorderPODbasisfunctions(Npod

−

Nr)tothefluctuationsinstatevariables.Themaximum

trun-cationerroris

≈

0

.

25% ofthevariance(

σ

2_{)foreachstatevariable(Fig. 7(a)).Theinterpolationerror(}

i

(

t

;

Re

)

)isrelatively

high, themaximumofitisabout2% ofthevariance,for

η

= 

Re

=

25.ThetotalerrorrelevanttotheinterpolationROM

(

it

(

t

;

Re

)

)isalso

∼

10 timesthetruncationerror.Fig. 7(b)showstheerrors(

i, t & it)inphasespace. Thelimitcycles

illustratetheboundednessoferrorsamplitudewiththetime evolution.Ontheotherhand,maximumoftheenergybased

error e

(

t

;

Re

)

(asdefinedinEquation(40))is

≈

22% ofthevariance(Fig. 7(c)).Further,thephasediagramsinFig. 7(b)and

Fig. 7(d)showthattheerrorsfollowthestablelimitcycles,demonstratingthestabilityofinterpolationROMmethod.

3.2.2. Highfidelitysolutioncomparisons

Fig. 8(a)showstheaverageofstreamwisevelocityu

¯

(

x

;

Re

)

obtainedusingthehighfidelitycomputationalfluiddynamics

Fig. 4. Comparisonofφu

1(x,Re)toφ

u

4(x,Re)modesobtainedbythesnapshotsPODagainstthemodesobtainedusinglinearinterpolation(ROM)atRe=140.

Table 1

Orthogonality(anglebetweenthemodesindegree)oftheinterpolatedreducedbasis.

φ1u φ u 2 φ u 3 φ u 4 φ u 5 φ u 6 φ u 7 φ u 8 φ u 9 φ u 10 φu 1 00.0 89.9 89.9 90.4 90.1 90.0 90.0 90.0 90.0 90.1 φu 2 89.9 00.0 90.4 90.1 89.8 90.0 90.0 90.0 90.0 90.0 φu 3 89.9 90.4 00.0 90.4 89.8 90.5 90.1 89.9 90.0 90.0 φu 4 90.4 90.2 90.4 00.0 90.5 90.3 90.1 89.8 90.0 90.0 φu 5 90.1 89.8 89.8 90.5 00.0 89.8 90.3 89.7 90.5 89.9 ˜ au 1 ˜au2 a˜3u a˜u4 a˜u5 a˜u6 a˜u7 a˜u8 a˜u9 a˜u10 ˜ au 1 00.0 90.1 89.0 89.3 90.2 89.7 89.5 90.4 90.4 90.3 ˜ au 2 90.1 00.0 88.7 90.3 91.6 91.3 90.1 90.5 90.3 88.3 ˜ au 3 89.0 88.7 00.0 90.5 91.7 88.3 90.2 88.0 90.3 89.7 ˜ au 4 89.3 90.3 90.5 00.0 88.1 88.0 90.8 89.6 89.6 91.0 ˜ au 5 90.2 91.6 91.7 88.1 00.0 89.9 87.8 94.0 91.4 90.3

Fig. 5. Comparisonofthetimecoefficientsa˜u

i(T·;·)ofthefirstfivechronos.Thebluecurveineachplotisaninterpolatedmode(ROM)atRe=140 against

thesnapshotPODmodeatRe=140 ingreen.TheothercolorcorrespondencewithReynoldsnumbersis:Red→Re1=125 andPink→Re2=150.(For interpretationofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig. 6. Energy comparison of the interpolated (ROM) modes with the snapshots POD modes.

number(Re

=

140)usingthereferencestatesatRe

=

125 andRe

=

150 isshowninFig. 8(b).Generally,thetime-averaged

base flow shows little variation over the long range of Reynolds numbers. In addition, the dimensionless quantities of

practical importance such as Drag,Lift coefficients vary withthe logarithmic change inReynolds number. Therefore the

second derivatives

α

∗ in Equation (38),contributing to the error boundsfor the interpolation error can be expected to

be small, providing the possibilityto have larger

η

.Fig. 9(a) showsthe phase plot ofthe Drag versus Lift coefficients

estimated using pressure force, forboth the high fidelity (HFM) and interpolation ROM solutions at Re

=

140. Fig. 9(b)

showsthe comparisonoftime-averagedpressure coefficient profileon thesurface ofcylinder atRe

=

140.TheDrag, Lift

Fig. 7. Time evolution and phase diagrams of the errors.

Fig. 8. Time-averaged base flow comparison at Re=140 (u¯(x,Re)).

Cd

=

2



Lp plxdl

ˆ

;

Cl

=

2



Lp plydl

ˆ

and Cp

=

2

(

p

−

p∞

)

(53)

WhereLpistheperimeterofcylinder, pl isthepressureonthesmallsegment(dl)oftheperimeter.

ˆ

x,y are

ˆ

theprojections

oftheunitvectornormaltoalengthsegmentdl alongtheinflow(x)andflownormal( y)directionsrespectively.

Thetime signal ofstreamwisevelocity inFig. 10(a)isprobed atx

=

5, y

=

0.The time evolutionoftheDragandLift coefficientsforunit cylinderlength(estimatedusingpressureforce only)iscomparedinFig. 10(b).Itshowsafairlygood

agreementwiththehighfidelity CFDsimulationresults.TheROMtime signalsare

∼

27TRe longandtheypersistforany