• Nie Znaleziono Wyników

State Observer data assimilation for RANS with time-averaged 3D-PIV data

N/A
N/A
Protected

Academic year: 2021

Share "State Observer data assimilation for RANS with time-averaged 3D-PIV data"

Copied!
13
0
0

Pełen tekst

(1)

Delft University of Technology

State Observer data assimilation for RANS with time-averaged 3D-PIV data

Saredi, E.; Tumuluru Ramesh, Nikhilesh; Sciacchitano, A.; Scarano, F.

DOI

10.1016/j.compfluid.2020.104827

Publication date

2021

Document Version

Final published version

Published in

Computers & Fluids

Citation (APA)

Saredi, E., Tumuluru Ramesh, N., Sciacchitano, A., & Scarano, F. (2021). State Observer data assimilation

for RANS with time-averaged 3D-PIV data. Computers & Fluids, 218, [104827].

https://doi.org/10.1016/j.compfluid.2020.104827

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

(2)

ContentslistsavailableatScienceDirect

Computers

and

Fluids

journalhomepage:www.elsevier.com/locate/compfluid

State

observer

data

assimilation

for

RANS

with

time-averaged

3D-PIV

data

Edoardo

Saredi

a,∗

,

Nikhilesh

Tumuluru

Ramesh

b

,

Andrea

Sciacchitano

a

,

Fulvio

Scarano

a a TU Delft, Faculty of Aerospace Engineering, Delft, The Netherlands

b University of Waterloo, Mechanical and Mechatronics Engineering, Waterloo, Canada

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 7 July 2020 Revised 4 December 2020 Accepted 22 December 2020 Available online 29 December 2020

Keywords:

Data assimilation Robotic Volumetric PIV RANS

state observer

a

b

s

t

r

a

c

t

State observertechniques areinvestigated forthe assimilation ofthree-dimensionalvelocity measure-mentsintocomputationalfluiddynamicssimulationsbasedonReynolds-averagedNavier–Stokes(RANS) equations.Themethodreliesonaforcingterm,orobserver,inthemomentumequation,stemmingfrom the differencebetweenthe computedvelocity andthe reference value,obtained bymeasurements or high-fidelity simulations. Two different approaches for the forcing term are considered: proportional andintegral-proportional.Thistechniqueisdemonstratedconsideringanexperimentaldatabasethat de-scribesthetime-averagethree-dimensionalflowbehindagenericcar-mirrormodel.Thevelocityfieldis obtainedbymeansofRoboticVolumetricPIVmeasurements.Theeffectsofthedifferentforcingterms and the spatialdensity ofthe measurementinput to thenumerical simulation arestudied.Thestate observerapproachforceslocallythesolutiontocomplywiththereferencevalueandtheextent ofthe regionmodifiedbytheforcinginputisdiscussed.Thevelocitydistributionandflowtopologyobtained withdataassimilationarecomparedwithattentiontotheobjectwakeandthereattachmentpointwhere thelargestdiscrepancyisobservedbetweenthedifferentapproaches.Theresultsshowthattheintegral termismoreeffectivethantheproportionaloneinreducingthemismatchbetweensimulationandthe referencedata,withincreasingbenefitswhenthedensityofforcedpoints,orforcingdensity,isincreased. © 2020PublishedbyElsevierLtd.

1. Introduction

The advances of computational fluid dynamics (CFD) coupled with theincreasingly affordable computational powerhave made CFD oneofthemain toolsforaerodynamicstudyanddesign op-timization for industrial problems [40]. The computational cost of high-fidelity simulations (i.e. LES, DNS), instead, remains rel-atively high for their use in optimization studies, especially for highReynoldsnumberflows.Asaresult,CFDanalysismakesmost often use of the simplified approach offered by the Reynolds-averagedNavier-Stokes(RANS)formulation[1,8].Consideredtobe theworkhorseinaerodynamicengineeringformanyyearstocome [40], RANS solvers model the entire turbulent spectrum, achiev-ing lower computational cost andhigherrobustnesswithrespect tomentionedhigherfidelitymodels.Thecostoftheamountof re-quiredmodellingistheaforementionedloweraccuracy[5].Several sourcesofuncertaintyarepresentedinliterature;thosestemming fromthechoiceoftheturbulencemodelanditsparameter,

nowa-∗Corresponding author.

E-mail address: e.saredi@tudelft.nl (E. Saredi).

daysbasedmainlyonexpertjudgment,arereportedtooften dom-inatetheoveralluncertaintyofthesimulation[40].

Conversely,theexperimentalapproachoffersaccurate measure-ments,directlyfromscaledmodels.Whenflow quantitiesare ob-tainedsuchasvelocityorpressure,themeasurementsusually suf-ferfromlimitedspatialrangeandspatio-temporal resolution.For thisreason,itshouldberetainedinmindthatthesoleexperiment is often insufficient for the analysis of the flow problem, which rendersthesimulationsnecessaryforacompletedescriptionofthe aerodynamicbehaviour.Experimentaldata isoftenused “offline”, to verify the validity of a given simulation, ormore specifically, theadequacyofturbulencemodelsparameters(alsoreferredtoas “calibration” [6,27]).Thelatter approach is often iterative and sev-eralsimulationsareconducteduntilmodelcalibrationisachieved. In order to overcomethe above limitations, direct integration ofexperimentaldatawithinCFDsimulationshasbeenconsidered. The resultingprocess iscalleddata assimilation(DA).The aimof thisapproachistoproduceamoreaccurateflowsimulation,which iscompliantwiththedatagatheredduringanexperiment.

Originallyintroducedinthefieldsofmeteorologyand oceanog-raphy,asreviewedbyNavon[24],DAhasseenagrowingnumber of applications in fluid mechanics [11]. Among the DA

method-https://doi.org/10.1016/j.compfluid.2020.104827

(3)

ologies presented in the literature, three main categories have emerged:variational methods ([9,35];amongothers), Kalman fil-ter(KF;[17,18])andstateobservermethods(reviewedin[11]).

Variational methods are based on the application of optimal controltheorytofindtheminimumoftheerrorfunctionbetween the simulationandareference,herein representedby the experi-mental data.Thecalculation oftheadjointandthe needtosolve an optimization problem entails a relatively large computational cost. KF andstate observertechniques are sequential algorithms, which means that the resultof the previous iteration is used at eachfollowingiteration.Thesolutionoftheassimilatedsimulation is expectedtoasymptotically convergetothe referencealongthe simulation[11].Acomparisonbetweenavariationalmethodanda KF applicationcan be found inMonset al.[23]. Ifthe equations ofstateobserverandKFarecompared,acertainsimilaritycanbe noted[36].KFcanbe consideredatype ofstate observermethod too,butdesignedforstochasticsystems,whilestateobserver tech-niques are typically applied to deterministic systems. While in a deterministic system, thevariables of interestare represented by a scalaroravector ateach point,inastochastic systemtheyare representedbyprobabilitydensityfunctions,withacorrespondent mean and standard deviation. Between the three of them, state observer methods require both a computational cost andan im-plementation effortthatare significantlylower comparedto vari-ational methods, making them attractive for aerodynamicdesign andoptimization[11].

Thestateobserveralgorithm,originallyproposedbyLuenberger [20] isnowadaysa keyconcept ofthecontrol theory,basedona feedbackterminthemodellingequation.Theintensityofthe feed-backisafunctionofthedifferencebetweenthemodelresultsand the reference(viz.experimental)datamultiplied bya gainfactor. The latterisusually tunedtomaximiseconvergencerate.The de-signofthefeedbackandits lawofapplicationisthekey pointof astateobserveralgorithm[11].HayaseandHayashi[12]applieda state observer algorithmto improvethesimulation ofa fully de-velopedturbulentsquare duct.The authorsmodifiedthepressure boundaryconditionsattheinletandtheoutletwithaproportional lawin ordertoreduce the errorbetweenthesimulationandthe ground truth, which inthat casewasa pre-calculated numerical solutionperformedonthesamemeshwithadifferentinitial con-dition.Theapplicationofthefeedbackgaveanaccelerated conver-genceofthesimulationandareductionofoneorderofmagnitude inthefinalerroracrossthewholedomain.Thisapproachalleviates the errorsgivenbythe erroneousboundaryconditionsbutisnot abletotacklestheerrorgivenbytheapproximationperformedin theturbulencemodelling.

ImagawaandHayase[13]performedaMeasurement-Integrated (MI)simulationfortheturbulentflowalongasquareductand ap-pliedafeedbackbymeansofabodyforceterminthemomentum equation. The forcing term was linearly proportional (DASOP) to thedifferencebetweenthevelocityreturnedbytheunsteady sim-ulation andthat froma referencesimulationperformedwiththe samenumericalschemes onthesamegridbutwithdifferent ini-tialconditions.Theapplicationofthefeedbackled toareduction ofthesteadystateerrorbyfourordersofmagnitudewhenallthe velocity components were forcedat all thegrid points. However, interruptingtheassimilationduringthesimulationcausedthe sys-tem to convergence again towards the non-assimilated solution. Theauthorsalsoconsideredthecaseofalimiteddensityofforcing points,forcingfromoneplaneeveryfourdowntoonlyoneplane. Theyconcludedthat,withtheirforcing term,tuningthegain fac-tor, it was possible to obtain the sameerror reduction consider-ing one planeevery four.Furthermore,disabling theforcing term at a quarter ofthe total running time, the simulation was natu-rally convergingtowards thenot assimilatedsimulation,withthe effectof theforcing termeffect thatwasvanishing.Thisshowsa

typicalbehaviour ofaproportionalcontroller, withtheefficacyof the forcing termthat reduces when the error reduces. Neeteson andRival[25]appliedthestate observeralgorithmtotheKarman vortexsheddingproblematRe=102,usingacomputational

refer-ence.TheyintroducedaProportional-Integral-Derivative(PID) con-trolin the pressure equation, which wasformulated as feedback law.Thelatterwasmoreelaboratethanthesoleproportionallaw usedin mostprevious worksandaimed atsolving the reduction of effectiveness of the forcing termat the low error stages. The applicationofthePIDcontrollaw(DASOPID)improvedtheresults comparedtothosegivenbythesoleproportionalfeedbacklaw, re-turningavortexsheddingbehaviourclosertothatofthereference dataintermsofsheddingfrequency.Thestateobserveralgorithm hasbeenused alsowithexperimental dataasareference. Yama-gataetal.[39]studiedtheunsteadybehaviouroftheKarman vor-texstreetbehindatruncatedcylinderatRe=1.2 × 103

perform-ing a MIsimulation withplanar PIV datainput. The assimilation allowedreproductionofthelarge-scaleunsteadinesswhicharenot obtainedbyordinarysimulations.

Thediscussionaboveshowsthatthestateobserverdata assimi-lationhasbeenconsideredtoenhancetheaccuracyofRANS-based CFDsimulations, mainly inthe unsteady flow regime andatlow

Renumber.However,manyapplicationsofindustrialaerodynamics areprimarilyconcernedwiththeaccuracyofthesteady-stateflow solution.Moreover,manyrelevantproblemsinvolvehighReynolds numberflowsandafullyturbulentregime forwhichmany ques-tionsabouttheapplicabilityandthepotentialofDAremain unan-swered.

This study considers steady-state simulations of the flow around a wall-mounted bluff obstacle, representing the geom-etry of a simplified car mirror presented by de Villiers [37]. The problem is studied at Re = 8 × 104. The reference

time-averaged velocity field is obtained from experiments that make useofRoboticVolumetric PIV[15].Twoformulations oftheState Observer forcing term are considered, namely Data-Assimilation-State-Observer-Proportional (DASOP) and Data-Assimilation-State-Observer-Proportional-Integral (DASOPI). Moreover, the work in-vestigates theextent oftheregions affected bythe forcing point, withtheaimofidentifyingacriterionforoptimumdata assimila-tionforcedpointsdensity.Finally,a metricisintroduced to quan-tifythe effectofDA methodsand parameterson the accuracyof theassimilatedsolutionswithrespecttothereferencevelocity. 2. StateobserverdataassimilationforRANS

Reynolds-averaged Navier-Stokes (RANS)equations are consid-ered here asthe framework for the numerical simulation of in-compressible turbulent fluid flows. The baseline simulation is in-tendedastheprocessleadingtoasolution(baselinesolution)with nouseofa-prioriinformationfromtheexperiments.Theworking principle ofdataassimilation isto introducea certain amountof referencedata,indicatedhereasuref,andforce thesimulation to complywithit.Inthiswork,theresultingprocessiscalledthe as-similatedsimulation,whichreturnstheassimilatedsolution.The ob-jectiveistodrivetheassimilatedsolutiontowardsamorecorrect evaluationof the relevant features present in theflow field (e.g. flow topology, separation, reattachment, pressure gradient distri-bution,aerodynamicloadsandforces),thusminimisingsimulation errors,suchastheonesassociatedwiththeturbulencemodelling. The underlying concepts and equations of data assimilation through astate observer (DASO)algorithm are recalledhere. Ap-plying Reynolds decomposition andtime averaging, the dynamic behaviour ofasteady, incompressible,viscous andturbulent flow canbedescribedbytheRANSequations,formedby the combina-tionoftheconservationofmassandmomentum,respectively:

(4)

(

¯u·

)

¯u= −

¯p +

ν



¯u

· R (2) imposingconsistentinitialconditions(IC)andboundaryconditions (BC) andwhereRi j= ui ujrepresentsthe Reynoldsstresstensor. Due to theincompressible flowcondition, p representsthe static pressure divided by the constant density. As shown by Imagawa andHayase [13],theassimilation ofthedata througha state ob-server algorithm is achieved by introducing a body force term f intothemomentumequation:

(

¯u·

)

¯u= −

¯p+

ν



¯u

· R+f (3)

Underthehypothesisofsteadyflowconditions,inthefollowing discussiontheoverbarindicatingthetime-averagehasbeen omit-ted,althoughallthequantitiesaremeanttobetime-averaged.The relative strength of the body force termrepresents the feedback anditisproportional tothelocaldifferencee betweenthe simu-lationandtheexperimentaltime-averagedvelocities:

e =



uref− u



(4)

whereuref= P(uref

)

,withP beingtheoperatorthatprojects ve-locityinformationfromtheexperimentalgridtothecomputational grid.FurtherinformationonP aregiveninSection4.1.

In this work, two different forms of the feedback term have been considered: data assimilation based on state observer with proportional feedback (DASOP)andon proportional-integral feed-back (DASOPI). The feedback control law used in the DASOP methodreadsas:

fDASOP= Kp

De

|

e

|

(5)

where Kp is the proportional feedback gain, the symbol ◦

rep-resents the Hadamard product [34] and

|

e

|

is the component-wise absolutevalue oftheerror

|

e

|

=[

|

ex

|

,

|

ey

|

,

|

ez

|

]T .The

forc-ingtermfortheDASOPImethodreadsas:

fDASOPI= Kp D e

|

e

|

+ Ki D N−1  n=1 en◦

|

en

|

(6)

where N is the current iteration and Ki is the integral feedback

gainanden isthelocalerrorvectorattheiterationn.Whilestate

observersaretypicallyappliedtounsteadysystemsandactinthe timedomain,inthemethodologyhereproposedthestateobserver actsintheiterationdomain,forthedeterminationofasteady-state solution.ThegainsKpandKiarescalarsastheyareassumedequal

for each component ofthe error andconstant along the simula-tionandspatiallyintheentiresimulationdomain.Thevalues cor-responding to cells whereuref is not available are set to 0,thus disabling the forcing term. The quadratic term implemented in boththeproposedfeedbacktermsresemblesanerror-squared con-troller,asproposedinShinskey[32].Itsadvantageisthatthe forc-ing termisstrengthenedwithrespecttoalinearcontrollerwhen theerrorislarge,penalizingtheregions wherethedifference be-tweenthereferenceandthesimulatedvelocityarelower.

Theresultsobtainedattheendofthesimulationrepresentsthe assimilatedsolution.Thepresentedmethodologyimpliestheusage ofasinglesimulationtoreachthefinalassimilatedsolution, con-trary to ensemble Kalman filters and variationalmethods, which requiremultiplesimulationstoreachthefinalassimilatedsolution. 3. Setupofthereferencedatasetexperiment

3.1. Windtunnelandmodel

ExperimentsareperformedattheTUDelftAerodynamics Lab-oratories in an open-jet open-circuit low-speed wind tunnel (W-tunnel).Thetunnelfeaturesa4:1areacontraction,afterwhichthe

Fig. 1.. Side view (top) and top view (bottom) of the experimental setup. The dashed green line schematically represents the total measurement volume. The shadowed green region indicates the instantaneous measurement volume for a given robot position. (For interpretation of the references to color in this figure leg- end, the reader is referred to the web version of this article.)

airflow reachesthefree-stream velocityU=12m/sin a cross-sectionof60× 60cm2.Aflatplate1.5mlongisinstalledat10cm

height above the bottom edge ofthe exit (Fig.1). The platehas asharpleadingedge andis equippedwitha zig-zagtripping de-viceat5cmpasttheleadingedgethatforcestheboundarylayer tothe turbulentregime. Themodelisa halfcylinderofdiameter

D=10cmtoppedbyaquartersphere(Fig.1) withatotalheight of15cm,asthatusedintheinvestigationsofdeVilliers[37].The Reynoldsnumberbasedonthe modeldiameterisReD = 8×104.

Thisgeometryrepresentsabenchmarkforautomotive aerodynam-ics for the study of the flow around appendices and in particu-lar the side mirror. It reproduces the essential features of junc-tureflow and bluff body aerodynamics,including horseshoe vor-tex, sharp separation at the back, and large-scale fluctuations of thewakeandofthereattachmentlocation[37].

3.2. RoboticVolumetricPIV

The three-dimensional velocity field is measured by Robotic Volumetric PIV [15]. An illustration of the system configuration duringtheexperimentsisshowninFig.1.Thesystemiscomposed ofacoaxialvolumetricvelocimeter(CVV),containing4high-speed cameras,andlaserilluminationthroughopticfiber.TheCVVis in-stalled on a robotic arm, UR5 from Universal Robots. The robot motionsequenceisprogrammedthroughtheproprietarysoftware

RoboDKandoperatedthroughtheLaVisionsoftwareDaVis10. Syn-chronizationofilluminationandimageacquisitionismadethrough aprogrammabletimingunit(LaVisionPTU9). Helium-Filled-Soap-Bubbles (HFSB) are used as flow tracers ([3,28]; among others). The bubbles are nearly neutrally buoyant with a median diame-ter of approximately 0.3 mm [7]. A rake composed of 10 verti-cal elementshosting 200bubblegenerators[4]isinstalled inthe settlingchamber ofthewindtunnel.Soap,airandheliumsupply iscontrolled througha LaVisionfluidsupplyunit (FSU).The flow is seeded at a concentration of approximately 0.3 bubbles/cm3.

Table 1 summarises the experimental parameters. The reader is referred to the works of Jux et al. [15,16] and Schneiders et al. [30] for a deeper discussion of the working principles of this

(5)

Table 1

Characteristics of the Robotic Volumetric system and the experimental parameters. Seeding Neutrally buoyant HFSB, ~300 μm diameter Illumination Quantronix Darwin-Duo Nd:YLF laser

(2 × 25 mJ @ 1 kHz)

Recording Device LaVision MiniShaker Aero system: 4 x CCD cameras

(640 × 452 @ 857 Hz) 4.8 μm pitch

Imaging f = 4 mm, f # = 8

Acquisition frequency fTR = 857 Hz

Pulse separation time Time-resolved: t = 1/ f TR = 1.17 ms

Magnification factor M ~ 0.01 at 30 cm distance Number of recordings per region N = 20,000

measurement technique. Here the experimental procedure is dis-cussed. Priorto themeasurements, acalibration ofthe robot po-sitionw.r.t.themeasurementdomain(windtunnel,plateand ob-ject)isperformed.Eachmeasurementvolumespansapproximately 30 × 20 × 40 cm3. At each robot position, 20,000 recordings

areacquiredintime-resolvedmodeatafrequencyof857Hz(Trec

~ 23 s). The time elapsed from one position to the subsequent isapproximately270s. Theoverallmeasurementencompasses15 views, wherethemeasurementsystemisdirected byroboticarm manipulation. Attheendofthemeasurements, therawdata fea-tures a coverage of the measurement domain, with the datasets obtainedfromeachviewingposition.Theprocedurefordata pro-cessingandreductionisdiscussedinthenextsection.

3.3. Dataprocessingandreduction

Therawimagesarepre-processedtoreducebackground reflec-tionsusingtheButterworthhigh-passfilter[31].Thetracermotion analysis is performedwith the Lagrangian particle tracking algo-rithmShake-The-Box[29].Trackscontainingmorethansix appear-ances oftheparticletracer areacceptedasvalid. Ateach record-ing, a sparse measurement of the tracers’ velocity is obtained in thesub-volume.

The measurements fromdifferent robot positions are merged into a singledataset usingthe robotcalibration data.The result-ingdomainisinterrogatedwithincubicvoxels(orbins)of15mm side length. Within each bin, the tracers’ velocity is ensemble-averaged, yielding the time averaged velocity vector distribution in a Cartesian grid. The averaging process inside the bin follows a spatiallyweightedalgorithmwithrespecttothecentroidofthe bin. AGaussian weighting function (witha widthof halfthe bin size)is applied,followingAgüeraetal.[2].Partialoverlapby3:4 ofneighbouringvoxelsyieldsvelocityvectorsspacedby3.75mm. The resultisrenderedina domainof55× 30 × 25 cm3 (Fig.1)

with a grid of 151 × 84 × 69 data points describing the time-averagevelocityuref foruseinthedataassimilationprocedure. 4. Numericalsimulations

Bothbaseline andassimilated simulationsare based ona RANS solver using the open-source C++ toolbox OpenFoam 1706 [14]. Eqs. (1) and (2) are discretised using the finite volume method on a collocatedgrid andsolved usingtheSIMPLE algorithm[26]. For the baseline simulation, the steady solver simpleFoam imple-mented inOpenFoamisused.Fortheassimilatedsimulations,the feedback termis included within an in-house developed version ofthesamesolver.Turbulencemodellingisbasedonthek-

ω

SST model[22].Thecomputationaldomainisasquarecuboidwith di-mensions30 × 20 × 10diameters.Theobjectisplacedalongthe centrelineofthedomain,withtheoriginofthecoordinatesystem posedattheintersectionbetweentherearsurfaceofthecar

mir-Table 2

Boundary and initial conditions of the simulations for velocity and pressure.

Velocity Pressure Inlet X/D = [-10,20]; Y/D = [-10,10]; Z/D = [0,1] Dirichlet, u = (12,0,0) m/s Neumann, ∂p ∂x = 0 Outlet X/D = [-10,20]; Y/D = [-10,1]; Z/D = [0,1] Neumann, no backflow u ∂x = 0 Dirichlet, p = 0 Pa (atmospheric pressure) Object surface no-slip condition

Bottom wall Z/D = 0 no-slip condition Lateral walls

Y/D = [-10,10]

slip condition

Top of the domain

Z/D = 10 slip condition Velocity Pressure Initial condition (entire domain) u = (12,0,0) m/s p = 0 Pa

Fig. 2. Computational mesh around the object along the symmetry plane of the simulation domain.

ror andthe ground. The orientation of the coordinate system is showninFig. 1.An inlet conditionisimposed atX/D=−10and anoutletconditionisimposedatX/D = 20.No-slipconditionis imposedatthegroundfloorandtheobjectsurfaces,whereasslip conditions are applied to the top and side faces of the domain. A summary of the boundary and initial condition is reported in Table2. The samehexahedral meshis used forboth thebaseline

andtheassimilatedsimulations, formedby~18.5 Mcellsand cre-atedwith thecommercialsoftware CFMesh+.The region closeto thesurfaceoftheobject(−7<X/D<15and−7< Y/D<7)is re-finedatseveralstages,asshowninFig.2,wherethecell dimen-sionreduces from10mm to 0.78mm,corresponding to y+ < 5. Allthesimulationsadvancefor5000iterationsandconvergenceis verifiedbyreachingarelativevariationoftheaverageddrag coef-ficientoftheobject

|

C¯d,n− ¯Cd,n−1

|

below10−3,wheretheaverage

(6)

Fig. 3. Section at Y = 0 of the forced region. Crosses represent the points forced to evaluate the response function presented in Sec. 5.3 .

isperformedintheinterval[n-500…n].Thesameinterval isused toaveragethevelocityfieldtotakeoutnumericaloscillations.

4.1. Assimilationprocedure

After conducting the baseline simulation, several assimilated simulations are run to investigatethe effectsof assimilation.The analysis presented here aims atdetermining the effectof sparse forcing varyingtheconcentrationofforcedcells.Thelatteris rep-resentedintermsofthenormalisedmeandistancebetween neigh-bouring forcing cells

γ

=

λ

/D,where

λ

is themean distance be-tweenneighbouringforcingpoints.Inordertoevaluate

λ

,the fol-lowingequationisused:

λ

= 3



4

3

π

C (7)

whereC=Nf/V, withNf equal to thenumberof forcedcells,

se-lectedrandomly,andV thetotalforcedvolume.

The small positionalmismatchbetween cellswhereforcing is applied andthe locationswhereexperimental datais availableis accommodatedbylinearinterpolationtotheclosest experimental datapoints,obtainingthevelocityfielduref.Thisoperationisthen fully justified in cases in which the expectedinterpolation error arenegligible,asinthecaseconsideredinthiswork.Furthermore, in order to damage the assimilated solution, outliers have to be removedfromuref beforetheinterpolationstep.Intheothercells, both thevalues ofuref ande areset tozero.At eachiteration of theassimilated simulation,thevelocityfield uref iscomparedwith thevelocityfieldun−1 ofthepreviousiterationandtheerrorfield

eisexplicitlyconstructedaccordingtoEqs.(5)and(6),respectively fortheDASOPandDASOPIfeedbacklaw.Thelimitedspatial reso-lutionoftheensemble-averagedPIVvelocitydataclosetothewall leadsto prescribed referencevelocityu∗ref toohighandnot com-patiblewiththeno-slipconditionassignedtothewallsofthe sim-ulation.Forthisreason, PIVdatacloserthan3cm(twoaveraging sub-volumesinthePIVanalysis)havenotbeenconsideredtobuild theforcingtermf.Inrelationtothis,Fig.3showsasectionalong thesymmetryplaneoftheregionswheretheforcingtermwas ac-tivated,withdifferentforcingcelldensity,dependingonthechoice of

γ

.

5. Resultsanddiscussion 5.1. Referenceflowfield

The measured velocity field uref isvisualized inFig. 4, where thestreamwisevelocitycomponenturef isshownatthesymmetry

plane andina wall-parallelplaneata heightZ/D=0.5. The po-tentialflowregioninfrontoftheobjectisvisible,wheretheflow deceleratesandeventuallystagnatesattheobstacleduetothe ad-versepressure gradient(Fig.4-bottom).Theboundarylayerthat hasdevelopedalongtheplateisalsovisibleinsectionY=0(Fig.4 -top).Accelerationatoptheobstacleisconsistentwiththe stream-linescurvaturethatfollowstheobjecthead(Fig.4-top).Thelocal

Fig. 4. Visualization of the normalized time-averaged streamwise velocity u_ref/U_ ∞ with superimposed streamlines, measured by Robotic Volumetric PIV: (top) at the symmetry plane and (bootom) at Z/D = 0.5.

velocity in thisregion exceeds the free-stream value by approxi-mately20%. The boundarylayer developingalong the object sur-face separatesabruptlydueto thesharp truncationofthe object atthetrailingedge.Withintheseparatedzone,areverseflow re-gionis formed.Thisfeature hasbeenobserved pasta numberof wall-mounted obstacles and reportedfrequently in the literature ([21,37,38];amongothers).Theseparatedregionterminates down-streamwithaground-reattachmentlinewithitsmostdownstream positionatXR inthesymmetryplane. Inthe presentexperiment,

thelengthofseparationisobservedtobeXR/D=2.4.

At the edge ofthe shear layer andthe backflow region,from Fig. 4 (bottom) is also possible to visualize two counter-rotating vortices fromthestreamlines pattern. Thecorrespondent foci are labelledF2andF3inFig.4andintherestofthiswork.Asreported

by de Villiers(2006)[37], thesearepart ofaU-shaped structure thatappearsinthetime-averagedvelocityfield,positioned upside-down w.r.t.the objectandtilted towards the backsurface ofthe mirror. The point in which this structure crosses the symmetry planeisdetectableinFig.4(top)bythefocinamedF1.

Theanalysisofthevorticity field,illustratedinFig.5,provides furtherinsightsonthereferenceflowtopology.Theadverse pres-suregradient atthefrontoftheobjectcausestheboundarylayer to detach from the ground andform the horseshoe vortex [33], whichdevelopsaroundtheobject,eventuallyaligning streamwise andfurtherdevelopdownstreamoftheobjectwake.The develop-mentfromthefrontoftheobjectofthehorseshoevortexisshown inFig.5bythetwoportions oftheiso-surface wherethe magni-tudeofthestreamwisevorticity

|

ω

x

|

>200Hz.Asdescribedbyde

Villiers[37],thetime-averagedseparationregionisboundedbyan arc-likefreeshearlayer, whichdevelopsdownstreamandbounds aregionofflowrecirculationthatalsofeaturesarc-likeshape.

(7)

Fig. 5. Iso-surface of vorticity magnitude |ω| = 200 Hz, coloured by streamwise vorticity ωx .

Fig. 6. Streamwise velocity contour obtained by the baseline simulation with over- lapping streamlines: (top) at the symmetry plane and (bottom) at Z/D = 0 . 5 .

5.2. Baselinesolution

The flow field topologyreturned by thebaselinesimulation,as shown inFig. 6,largely followsthe flow patternobserved inthe experiments. The boundarylayer developmentunderthe adverse pressuregradientcausedbytheobstacleisvisible,withthe forma-tionofasmallrecirculationaheadoftheobject(headofthe horse-shoevortex) atapproximatelyX/D= -0.8.The overallflow decel-eration ahead ofthe object andsubsequent acceleration towards

Fig. 7. Colour-contours of streamwise velocity from the baseline solution and over- lapping isolines of the reference velocity field (dashed contour lines for negative values). Symmetry plane (top) and at Z/D = 0 . 5 (bottom).

Fig. 8. Relative error εof the baseline simulation: (top) at the symmetry plane and (bottom) at Z/D = 0 . 5 .

thetrailingedgearealsovisiblebyregionsatavelocityexceeding thefree-streamvalue.Intheregionupstreamoftheobject,the re-sultobtainedbythebaselinesimulationcompliesratherwellwith thereference,asvisibleinFig.7.ThisisexpectedbeingRANSable topredict accuratelyflow featureinpotential flowregions. A no-table exception is instead the region close to the ground, where the simulation is not forced.The topological differences are dis-cussedintheremainder.Theflow separationatthesharptrailing

(8)

Fig. 9. Region of influence of DASOPI for points in the external flow (left); inside the shear layer (middle) and in the separated region (right). Coarse distribution of forcing points γ= 0 . 5 (distance of 5 cm).

edge is also well reproduced. Whereas the most significant dis-crepancy withrespect to the referencevelocity field is produced in theseparatedflow regionandthe reattachmentregion in par-ticular.The RANS datapredicta longerwake,asclearlyvisiblein Fig. 7, with a reattachment occurring atXR/D = 3.1, along the

symmetryplane.Thisbehaviourhasalreadybeenreportedin pre-vious works ([19], among others), where RANS simulations tend to overestimate the length of thewake of abluff body. Together withtheelongationofthewakebehindtheobject,other topologi-calfeaturesinthewakearedistorted.AsvisibleinFig.6, combin-ing theinformationobtainedbythetwosections,itispossibleto infer the presenceof thearc-shaped vortex.The latterhas, how-ever,apronouncedoffset(judgedfromthelocationthetwofociat

Z/D = 0.5)fromX/D= 1.25ofthereferencedata,toX/D=1.85. Inthesymmetryplaneinstead,thefocusF1islocatedclosertothe

ground(Z/D = 0.8inthesimulation,comparedtoZ/D = 1.05of the referencedata).The resultingarc-vortexinthe baseline

simu-lation is more elongated in the streamwise direction and witha

lowerpositionofitstopportion.Thelatterisalsovisible consider-ing theregionofreverseflow(inFig.7by thedashedlineofthe referencedataissuperimposedtothebaselinesimulation) appear-ingflattenedtowardstheground.

Further smalldifferencesare theupstream separationforming the head ofthe horseshoe vortex, a smallrecirculation region at the free endoftheobject andasmallregion ofacceleratedflow closeto theobjecttrailingedge (Fig.6). Thesefeaturesare, how-ever,notcapturedwithintheexperimentsduetothelimited spa-tialresolutionofthemeasurements.

An L-2 metrics is introduced to quantify the discrepancy be-tween the baselinesimulation andthereferencedata.The follow-ingrelativeerrornormalisedwiththefreestreamvelocityU has beenused:

ε

=



(

u− uref

)

2+

(

v

v

ref

)

2+

(

w− wref

)

2

U (7)

Thereferencedataisavailableonagridthatdoesnotcoincide with that ofthe numericalsimulations. The value of

ε

is there-foreobtainedbylinearinterpolationofthenumericalsimulations ontothegridofthereferencedata.Thespatialdistributionof

ε

at thesymmetryplaneisshowninFig.8.Therelativeerror distribu-tion can beclustered infourdifferentregions. The widestregion of error can be found forX/D>2 around the centerline, where

ε

>0.3is reached. The source of thiserror lieson the length of thewake,havingthereferenceflowalreadyrecoveredmomentum atthat stage compared tothe simulation.Anotherregion ofhigh relative erroriscentred atX/D=1.25andZ/D=0.55.Thisis re-lated to the height of the backflow region. Since the simulation predicts a backflow region shorter in height with respect to the reference, thiscauses a deficit ofmomentum and an increase of

ε

inthatflowregion.Furthermore,

ε

peaksin the close proximity

Fig. 10. Spatial-averaged relative error ( ¯ε ) with respect to the integral and pro- portional gain ( K p and K i ). Variability of the results as a function of the varia-

tion of K p ( K p = [ 0 . 001 , 0 . 01 , 0 . 1 , 1 , 10 ] ) is expressed by the vertical bars (up-

per bound: maximum error; lower bound: minimum error). Simulations performed with γ= 0 . 03 . ( ) Artifacts appear at the edge of the forced domain.

ofthe object.This isdueto the lackofresolution forthe exper-imental data previously mentioned. Because the discrepancies in thelatterregioncannotbe ascribedtothenumericalsimulations, anintegralevaluationoftherelativeerrorwillexcludetheregions closetotheobjectsurfaceandthegroundwall.

5.3. Assimilatedsimulations

Before presenting the results of the assimilated simulations, the local response to the forcing function at individual points is discussed. The effect of the forcing is expressed through the re-sponse functionH=

|

(

ubas− u

)

/

(

|

ubas− u∗ref

|

)

|

, whereubas

repre-sents the baseline solution and u the assimilated solution. The analysis is performed on cells widely separated spatially (

γ

=

λ

D=0.5 or5cm distancebetween forced cells), as illustrated in Fig. 3. This is done to consider the result only dependent on thelocalforcing, withnointerferenceamongneighbouringforced points. For each forced cell withcenter Xc=

(

Xc, Yc, Zc

)

, a 5 ×

5 × 5cm3 cube is considered, with its center on X

c, and

dis-cretized with 101 × 101 × 101 elements. For each cube, the effect of the forcing is averaged over a local coordinate system X∗=

(

X− Xc, Y− Yc, Z− Zc

)

permitting the alignment of results

amongdifferentpoints.TheparameterH isevaluatedaroundXc.

Becausetheforcingtermisintroducedinthemomentum equa-tion, the extent and shape of the region affected by the forcing

(9)

Fig. 11. Streamwise velocity contours of DASOP (left) and DASOPI (right) assimilated simulation . Contours of reference data are superimposed (dashed lines for negative values). Y/D = 0 in the top row; Z/D = 0.5 in the bottom row. Forcing density by γ= 0 . 03 .

isexpectedtofollowlocaldiffusionandconvection.Toassessthe region ofinfluence ofthe forcing,three flow regimesare consid-ered, namely:a)theouter (potential) flowatapproximately free-streamvelocity;b)theshearlayeremanatingfromtheobject trail-ingedge;c)therecirculatingflowinsidetheseparatedregion.

Fig.9showsthespatialdistributionofH forthe three consid-eredregionsobtainedapplyingDASOPI.Thelocaleffectofthe forc-ingalgorithmextendstoaregionwithlengthnotexceeding2cm (0.2D).Suchlocalisedeffectoftheforcinghasbeenalsoobserved in previous works,namely, in the studyby Imagawa andHayase [13].Thisresultsuggestsinadvancethat dataassimilationat val-uesof

γ

>0.2mayresultinlocalisedratherthanglobal modifica-tionsofthesimulationresult.

Theextentandshapeoftheregionneighbouringtheforced lo-cation appear to be affected by the localflow properties.In the outer flow region,theeffectofthe forcingis comparativelyweak andhighlyelongatedalongtheconvectiondirection.Intheregions of lower local velocity and high turbulent diffusion, such asthe shearandtherecirculationregions,thespatialresponsefunctionis closetoisotropic.InFig.9,alsothelocalvalueoftheturbulent ki-neticenergyknormalisedbythelocalkineticenergyispresented. TheextentandshapeofHfollowthatofk.Asimilaranalysis per-formedwiththeDASOPmethodyieldsnovisiblechangein veloc-ity,withvaluesofHtypicallybelow0.1%.

Intheremainderofthiswork,thelocationoftheforcingpoints ischosenrandomlywithadistributioncontrolledthroughthe den-sityparameter

γ

=[0.01…0.45];thecorrespondingmeandistance

λ

between forcing points varies between 4.5 cm (0.45 D) and 1mm(0.01D).

Beforecomparingtheresultsobtainedbythetwoproposed al-gorithms, the effect of the two parameters Kp and Ki is investi-gated. ThestudyhasbeenperformedusingtheDASOPIalgorithm forthecaseof

γ

= 0.03.Inordertoevaluatetheeffectivenessof the chosen valuesofKpandKi,theerror

ε

hasbeenaveraged in thevolumewhereexperimentaldatawereavailable;thesymbol

ε

¯ indicates the resultofthisoperation.Since theresultshave been found to be mostly insensitive to the value of Kp, Fig. 10 shows the obtained

ε

¯ varying Ki andaveraging theresults obtained for 0.001<Kp<10.The bars ateach data point representsthe vari-ation obtainedvarying Kp for a givenKi. The results show a

de-Fig. 12. Streamwise velocity contour obtained by the assimilated simulation by DA- SOPI algorithm and γ= 0 . 03 with super-imposed streamlines at Z/ D = 0 . 5 . cayof

ε

¯whenKiisincreased. Thisisexpectable,sincewhenKiis increased, theintegral partoftheforcing termbecomesstronger, witha subsequencestrongercorrectiontowardsthebiaserrorsof thesimulation. When Ki=1,while thesimulation resultsexhibit a lowererror withrespectto thereference, artificialedge effects start to appear at the boundary of the forcedregion. For values ofKi>1, thesimulations show numericalinstability anddiverge. Forthisreason,ithasbeendecidedtosettheparametersasKp=1 andKi=0.1forallthesimulationspresentedintheremainingpart ofthemanuscript.

Fig.11showstheresultsobtainedbyboththeforcingmethods DASOPandDASOPIwhen

γ

= 0.03.Asforthebaselinesimulation, inthe potential flowregion, bothDASOPandDASOPI assimilated resultsagree well withthereferencemeasurement. Larger differ-ences between the two forcing methods are found in the wake region. The applicationof the proportional forcing term(DASOP) doesleadtoa reductioninthelength oftherecirculationregion, andfurthestpointofthereattachmentlineismovedtoX/D=2.9, closer to the reference than that of the baseline simulation. The height ofthe backflowregion,however, remains lower compared to the referenceas itcan be seen inthe section atZ/D=0.5in Fig.11;furthermore,thebackflowisunderestimatedbymorethan 50%.

(10)

Fig. 13. Streamwise velocity contours of DASOP (left) and DASOPI (right) in the assimilated simulation . Contours of reference data are superimposed (dashed lines for negative values). Y/D = 0 in the top row; Z/D = 0.5 in the bottom row. Forcing density γ= 0 . 01 .

TheuseoftheDASOPIalgorithmvisiblyimprovesthefidelityof thevelocityfieldobtainedbyassimilation.Thereattachmentpoint upstreamtoX/D=2.4,practicallymatchingthereferencedata.The DASOPIresultalsorecoversthetopologyoftherecirculationregion givenin thereferencedataforwhatconcerns thepositionofthe foci(Fig.12).

The effectoffurtherrefiningthedensityofforcingpoints(the distance between forcing points is reduced to 0.01 D) is illus-trated inFig.13.Inthiscase,all gridcellsintheselectedvolume are forced.TheDASOPalgorithmwithagreaternumberofforced pointsyieldssomeimprovementintermsofsimilaritytothe refer-encedata.Whileinthepotentialflowregionnodramaticchanges arevisible,thelengthofthewakeisshortened,withthe reattach-mentpointfoundatX/D=2.28atthesymmetryplane.The short-eningofthewakeisalsocoupledwithanincreaseofthebackflow withrespecttothesimulation obtainedby DASOPwith

γ

=0.03. However, attheplane Z/D=0.5,thebackflowmagnitudepeak is stillunderestimatedwithrespecttothereference.

ForwhatconcernsthecomparisonbetweenDASOPandDASOPI, Fig. 13 confirms the trend observed with

γ

=0.03, withDASOPI overperforming. In the region where the forcing is applied, the contourlinesoftheassimilatedsolutionwithDASOPIoverlapwith theonesrepresentingthereference.Itisnoticed,however,thatthe simulationattemptstoconvergeeverywheretothereferencedata, even where the latter is affected by outliers. This behaviour can benoticedinthetop-leftcornerofthesymmetryplaneofFig.13. Furthermore,whenthisdensityofforcingpointisused,the exten-sionofthevolumewheretheforcingisactivatedbecomesdirectly visibleinthevelocityfield,withartefactsatitsedges.

Theglobalbehaviouroftheerrorrelativetothereferencedata isquantifiedforbothalgorithmswithitsspatialaverage

ε

defined in Eq. 7. The variation of

ε

¯ as a function of the forcing density

γ

isrepresented inFig.14.Inorderto capturethe mostrelevant sourceoferrorshowninFig.8,theaveragehasbeenobtained tak-ing into consideration only the volume downstream of the body showninFig.3.Theresultgivenbythebaselinehasbeenplotted withadottedlineasreference.Theassimilatedsimulationsappear tobetoocoarselyforcedaslongas

γ

≥ 0.1.Underthiscondition,

Fig. 14. Spatial-averaged relative error ¯ε as a function of the relative mean forced point distance γ. The horizontal dashed line represents the relative error of the baseline simulation. The vertical dashed-dotted line represents the vector spacing of the PIV measurements ( γPIV = 0.023).

bothDASOPandDASOPImethodsproduceanerrorcomparableto thebaselinesolution.Thedistancebetweentheforcingpointsistoo hightohaveaglobalreductionoftheerrorcomparedtothe base-lineandtheeffectoftheforcingremains local.When

γ

<0.1,the behaviourofthetwomethodsdiffers.DASOPmethodisnotableto reduce theerroruntil

γ

<0.05.If

γ

isfurtherreduced, theerror reduces,reaching

ε

¯=0.06when

γ

=0.01.Comparedtothe base-lineerror,anerrorreductionof28%isobtained.

The DASOPI algorithm becomes effective for

γ

<0.1. As also shownin Figs. 10 and 12, DASOPI is more effective than DASOP, yieldinglowererrorvaluesinawiderrangeofvaluesfor

γ

.When

γ

=0.05,the relativemean errorbecomes

ε

¯=0.055. Further re-ducing

γ

to0.01, yields

ε

¯ = 0.023,corresponding to a reduction of72%comparedtothebaseline.

Thereductionof

ε

¯atdecreasing

γ

canbeexplainedbythe re-sponsefunction shownby Fig.9.When

γ

decreases, theamount

(11)

Fig. 15. Local relative error ε ( Eq. (7 )) at the symmetry plane Y = 0 for: (first row) DASOP and DASOPI with γ= 0 . 1 , left and right respectively (second row) DASOP and DASOPI with γ= 0 . 03 , left and right respectively (third row) DASOP and DASOPI with γ= 0 . 01 , left and right respectively.

of the flow field that is affected by the assimilation increases. Thedimensionofthevolumeaffectedbytheforcing proportional-integral forcingtermshowninFig.9explainsthebehaviourof

ε

forwhatconcernsDASOPI.Themajorityofthesourceof

ε

is con-centratedinwakeoftheobject.Sincetheaveragevolumeof effec-tivenesscan be representedasasphere withdiameter~O(1cm), the method startsbeingeffective when such volumesstart over-lapping,correspondingtothecondition

γ

<0.1.

Fig. 15 compares the spatial distribution of the relative

er-ror

ε

for the baseline simulation, DASOP and DASOPI, with

γ

=

[0.1,0.03,0.01].Thespatialdistributionof

ε

illustratesthatthe er-ror is confinedin specific regions ofthe flow: incloseproximity of the object; around the reattachment region; within the recir-culating flow. For

γ

=0.1,the spatialdistribution of

ε

presented by the assimilated simulations is close that given by the baseline, with asmall reduction shownby DASOPI inthe recirculation re-gion and the far wake.When

γ

is reducedto 0.03, both DASOP andDASOPIyieldareductionof

ε

comparedtothebaseline.While DASOP still shows areas characterized by

ε

>0.2 in the recircu-lation region and the far wake, DASOPI achieves

ε

<0.05 every-where except in proximity of the object and in the shear layer close to separation. Finally, for

γ

=0.01, a further reduction of

ε

is noticed in the entire field. It can be noted that, for what concerns DASOPI, some regions of high erroroutside the forcing area arise, as introduced in the previous paragraph. The optimi-sation of the location of the forcing points to minimise the er-ror of the assimilated simulation will be investigated in future works.

Overall considerationsontheeffectofdataassimilationtothe flowtopologycanbedoneconsideringthehorseshoeandthe arc-vortex axislines,alongsidethe specificvelocity contouru =0 at

Z/D=0.5(Fig.16).

Fig. 16. Comparison of the topology obtained by PIV (reference), baseline simula- tion, DASOP with γ= 0 . 01 and DASOPI with γ= 0 . 01 . Dahs-dotted lines represent the ( X,Y ) projection of the trajectory of the horseshoe vortex. Continuous lines rep- resent the contour line u = 0 at Z/ D = 0 . 5 . Crosses represented the position of the foci F 2 and F 3 at Z/ D = 0 . 5 .

Consideringthe horseshoevortex, noneofthe simulations re-produces the path showed by the reference. With a larger sepa-ration distance [10], the horseshoe vortex showed by the simu-lations presents an offset toward negative X values with respect to the reference. The assimilation, both through DASOPand DA-SOPI, doesnot influencethe pathof thehorseshoe vortex before

X/D<1. Itmust be notedthat, beingthe horseshoe vortexclose tothe ground, itlays outsidethe regionwherethe forcing is ac-tive,asrepresentedbyFig.3.ForX/D>1,theassimilationthrough DASOPIoftheflow abovethevortexinfluences itspath,reducing its distancewithrespect tothe reference. The continuouslinein Fig.16 representsthe contourline u=0atZ/D=0.5.At this

(12)

lo-Fig. 17. Number of iterations required to reach convergence as a function of the forcing spacing γ.

cation,inside theforcedregion,the assimilationsperformedboth withDASOPandwithDASOPIreducesthedistancebetween simu-lationandreference.As shownbyFig.14,DASOPIovercomes DA-SOPinreproducingthewakeshapeofthereference.The shorten-ingofthewakebroughtbytheassimilationcausesalsoareduction inthedistancebetweenthefociF2andF3 betweenreferenceand

assimilatedsimulations,focithatarerepresentedbythecrossesin Fig.16.

The assimilationhasalso an effectonthe convergencerateof thesimulation.Asimulationhasbeenconsideredconvergedwhen the moving standarddeviation(kernel equalto 500iterations) of themodeldragcoefficientCd fallsbelowathresholdvaluechosen equal to10−3.Fig.17showsthenumberofiterationsrequiredto reach convergencewithrespecttothe relativemeanforcedpoint distance

γ

.ForbothDASOPandDASOPI,theamountofiterations for convergenceislower than inthe baseline simulation (dashed line inFig. 17).Also inthiscase, decreasing

γ

has apositive ef-fect, yielding a reduction ofthe numberof iterationsrequired.It mustbenotedhoweverthatfor

γ

<0.03,thenumberofiterations neededbyDASOPIdoesnotdecreaseanymore,whichisascribedto theappearanceoftheaforementionededgeeffects.

6. Conclusions

In this study,a data assimilation framework based on a state observeralgorithmhasbeenpresented.Twoforcingterms, propor-tional(DASOP)andintegral-proportional(DASOPI),havebeen con-sidered.Theperformancesoftheproposedalgorithmsarebasedon anexperimentaldatasetconsistingofthetime-average3Dvelocity field around asimplified carmirrorgeometry.The measurements havebeenconductedwithRoboticVolumetricPIV.

Boththeforcingtermsproposedarefunctionsofthedifference between the simulated velocity and the reference experimental data.Thelocalresponsefunctionoftheforcingtermhasbeen eval-uated, confirmingtheresultsof ImagawaandHayase [13], where the effect is limited to the region closethe forced location;the shape and extent of the affected region depending on the local convectionanddiffusion.

The data assimilation of the entire domain of interest is parametrised with respect to the spatial concentration of forced pointsorforcingdensity

γ

.BothDASOPandDASOPIproducea re-duction ofthe errorcompared to the baseline simulation. DASOP requires

γ

<0.05 to produce significant effects, DASOPI yields a comparable errorreduction atalready

γ

=0.1, asa resultof the

higherstrength,giventheintegralformulationoftheforcingterm. For

γ

<0.1,DASOPIreducesprogressivelytheerrorofthe assimi-latedsimulation.Themaximumerrorreductionisobtainedatthe maximum forcing density (

γ

=0.01

)

, wherethe error of the as-similatedsimulationisapproximately25% oftheone ofthe base-linesimulation.

The topologicalanalysis based onthe reattachment point, the arc-vortex andhorseshoe vortex axes confirms that the data as-similation by DASOPI produces a realignment of the simulation towards the experimental reference, also in those regions where the forcing is not applied locally. The latter suggests that the convective-diffusivenatureoftheforcingmechanismmay extrapo-latetheeffectsoftheassimilatedregionbeyondthedomainwhere experimentaldataisavailable.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

Edoardo Saredi: Conceptualization, Methodology, Software, Writing - original draft. NikhileshTumuluru Ramesh: Investiga-tion,Writing-review&editing.AndreaSciacchitano:Supervision, Writing-review&editing,Projectadministration.FulvioScarano: Supervision,Writing-review&editing,Projectadministration. References

[1] Argyropoulos CD , Markatos NC . Recent advances on the numerical modelling of turbulent flows. Appl Math Model 2015;39:693–732 .

[2] Agüera N , Cafiero G , Astarita T , Discetti S . Ensemble 3D PIV for high resolution turbulent statistics. Meas Sci Technol 2016;27:124011 .

[3] Bosbach J , Kühn M , Wagner C . Large scale particle image velocimetry with he- lium filled soap bubbles. Exp Fluids 2009;46:539 .

[4] Caridi GCA . Development and application of helium-filled soap bubbles PhD Thesis. TU Delft; 2018 .

[5] Davidson L , Peng SH . Hybrid LES-RANS modelling: a one-equation SGS model combined with a k–w model for predicting recirculating flows. Int J Numer Meth Fluids 20 03;43:10 03–18 .

[6] Duraisamy K , Iaccarino G , Xiao H . Turbulence modeling in the age of data. Annu Rev Fluid Mech 2018;51:1–23 .

[7] Faleiros DE , Tuinstra M , Sciacchitano A , Scarano F . Generation and control of helium-filled soap bubbles for PIV. Exp Fluids 2019;60:40 .

[8] Ferziger JH , Peric M . Computational methods for fluid dynamics. Berlin Heidel- berg: Springer-Verlag; 2002 .

[9] Foures DPG , Dovetta N , Sipp D , Schmid PJ . A data-assimilation method for Reynolds-averaged Navier–Stokes-driven mean flow reconstruction. J Fluid Mech 2014;759:404–31 .

[10] Gazi AH , Afzal MS . A review on hydrodynamics of horseshoe vortex at a ver- tical cylinder mounted on a flat bed and its implication to scour at a cylinder. Acta Geophys 2020;68:861–75 .

[11] Hayase T . Numerical simulation of real-world flows. Fluid Dyn Res 2015;47:051201 .

[12] Hayase T , Hayashi S . State estimator of flow as an integrated computational method with the feedback of online experimental measurement. J Fluids Eng 1997;119:814–22 .

[13] Imagawa K , Hayase T . Numerical experiment of measurement-integrated simulation to reproduce turbulent flows with feedback loop to dynami- cally compensate the solution using real flows Information. Comput Fluids 2010;39:1439–50 .

[14] Jasak H . OpenFOAM: open source CFD in research and industry. Int J Nav Arch Ocean 2009;1(2):89–94 .

[15] Jux C , Sciacchitano A , Schneiders JFG , Scarano F . Robotic volumetric PIV of a full-scale cyclist. Exp Fluids 2018;59:74 .

[16] Jux C , Sciacchitano A , Sciacchitano A , Scarano F . Flow pressure evaluation on generic surfaces by robotic volumetric PIV. Meas Sci Tech 2020:109645 .

[17] Kalman RE . A new approach to linear filtering and prediction problems. Trans ASME 1960:35–45 .

[18] Kato H , Yoshizawa A , Ueno G , Obayashi S . A data assimilation method- ology for reconstructing turbulent flows around aircraft. J Comput Phys 2015;283:559–81 .

[19] Lübcke H , St Schmidt , Rung T , Thiele F . Comparison of LES and RANS in bluff– body flows. J Wind Eng Ind Aerod 2001;89:1471–85 .

(13)

[20] Luenberger DG . Observing state of linear system. IEEE Trans Mil Electron Mil 1964;8:74–80 .

[21] Martinuzzi R , Tropea C . The flow around surface-mounted, prismatic obstacles placed in a fully developed channel flow (data bank contribution). ASME J Flu- ids Eng 1993;115(1):85–92 .

[22] Menter FR . Zonal two equation k- ω turbulence models for aerodynamic flow. AIAA Paper 1993;93:2906 .

[23] Mons V , Chassaing JC , Gomez T , Sagaut P . Reconstruction of unsteady flows using data assimilation schemes. J Comput Phys 2016;316:255–80 .

[24] Navon IM . Data assimilation for numerical weather prediction: a review. Data assimilation for atmospheric, oceanic and hydrologic applications. Park SK, Xu L, editors. Berlin, Heidelberg: Springer; 2009 .

[25] Neeteson NJ , Rival DE . State observer-based data assimilation: a PID control-in- spired observer in the pressure equation. Meas Sci Technol 2019;31:014003 .

[26] Patankar SV , Spalding DB . A calculation procedure for heat, mass and mo- mentum transfer in three-dimensional parabolic flows. Int J of Heat and Mass Transfer 1972;15:1787–806 .

[27] Ronch AD , Panzeri M , Drofelnik J , Ippolito R . Sensitivity and calibration of tur- bulence model in the presence of epistemic uncertainties. CEAS Aeronaut J 2019 .

[28] Scarano F , Ghaemi S , Caridi GCA , Bosbach J , Dierksheide U , Sciacchitano A . On the use of helium-filled soap bubbles for large-scale tomographic PIV in wind tunnel experiments. Exp Fluids 2015;56:42 .

[29] Schanz D , Gesemann S , Schröder A . Shake-the-box: Lagrangian particle track- ing at high particle image densities. Exp Fluids 2016;57:70 .

[30] Schneiders JFG , Jux C , Sciacchitano A , Scarano F . Co-axial volumetric velocime- try. Meas Sci Technol 2018;29:06520 .

[31] Sciacchitano A , Scarano F . Elimination of PIV light reflections via a temporal high pass filter. Meas Sci Technol 2014;25:084009 .

[32] Shinskey FG . Process control systems: application, design, and adjustment. New York: McGraw-Hill Book Company; 1988 .

[33] Simpson RL . Junction flows. Annu Rev Fluid Mech 2001;33:415–43 .

[34] Styan GPH . Hadamard products and multivariate statistical analysis. Linear Al- gebra Appl 1973;6:217–40 .

[35] Symon S , Dovetta N , McKeon BJ , Sipp D , Schmid PJ . Data assimilation of mean velocity from 2D PIV measurements of flow over an idealized airfoil. Exp Fluids 2017;58:61 .

[36] Utkin VI , Guldner J , Shi J . Sliding mode control in electro-mechanical systems. 2nd ed. Boca Raton, Fla: CRC Press; 2009. ISBN:97814200656191420065610 .

[37] de Villiers E . The Potential of Large Eddy Simulation for the modeling of wall bounded flows PhD Thesis. Imperial College of Science, Technology and Medicine; 2006 .

[38] Yakhot A , Liu H , Nikitin N . Turbulent flow around a wall-mounted cube: a di- rect numerical simulation. Int J Heat Fluid Fl 20 06;27:994–10 09 .

[39] Yamagata T , Hayase T , Higuchi H . Effect of feedback data rate in PIV measure- ment-integrated simulation. J Fluid Sci Tech 2008;3:477–87 .

[40] Xiao H , Cinnella P . Quantification of model uncertainty in RANS simulations: a review. Prog Aerosp Sci 2019;108:1–31 .

Cytaty

Powiązane dokumenty

As inspection data on the condition of lateral house connections are scarce, this study adopts a statistical procedure to support proactive strategies by analysing spatial

2, Lipsk 1839-1845; O kościół w Opolu, wydane staraniem Bronisława Szlubowskiego, Warszawa 1910; Sprawozdanie Centralnego Towarzystwa Rolniczego w Królestwie Polskim za rok 1909,

Utworzenie diecezji białostockiej (1991) i podniesienie jej do rangi arcybiskupstwa, a Białegostoku do godności stolicy metropolii (1992), było uwieńczeniem pewnego procesu

De boomtuin is in zijn tegenstrijdigheid van een tuin in de lucht een other space, de term die Foucault in de jaren ’60 introduceerde om een plek aan te duiden met de

odnosi się to głównie do kazań pogrzebowo-żałobnych z cza- sów niewoli narodowej, obliczonych także na promowanie ściśle określonych osób lub grup społecznych, które –

Przypuszczam zaś, że to jest jedyna droga dojścia do prawdy. miało być uosobieniem scepty­ cyzmu Kordyana-poety, to niewątpliwie w scenie V., staw iając przed

W raporcie z 13 grudnia 1920 roku Józef Sramek informował o niepokojących doniesieniach z polskiej części Śląska Cieszyńskiego.. Według niego na tym terenie panował