Yannick Hoarau

Philippe Choquet

Christian Goetz

A Fouras

S Dubsky

Marianna Braza

S Saintlos-Brillac

Franck Plouraboué

D Lo Jacono

F Roru

D F Ghoquet

D Gf Qoetz

D Ef Pours

D F Husky

D Wf Frz

Louroué

Hf Vo Tono

Flow and Particles Deposition in Rabit and Rat Airways Under Realistic Inflow Rate

Keywords: particle deposition, rat lungs, rabbit lungs, human lungs, numerical simulations

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1 Introduction he erosol prtiles present in our environment re inresingly identi(ed s risk ftors for helthD ut erosols n lso e used for dignosti or therpeuti purposes to tret lung disesesF sn prtiulrD smll prtiles tht reh the surfe of gs exhnge in the lveolr region of lungs re onsidered most hrmful to helthF snhlersD for exmpleD re ommonly used to dminister drugs SF sn ses of lung diseseD dministrtion of medition y erosol hs the dvntge of rehing the region diretly 'eted nd requires lower doses @side e'ets re minimizedAF he morphologil desriptions of the ronhil tues of the mn ppered in the literture sine IWTQD when eiel PH introdued physil symmetri modelF rors(eld nd gumming IH hve susequently proposed n ntomil representE tion of symmetri lung tues tking the eiel model s sisF rors(eldD hrt nd ylson II hve produed two types of symmetri mthemtil models sed on dt otined from mesurement of resin st of norml humn ronhiol treeF he models permit lultion of physiologi vriles to e mde while tking symmetry into ountF rmmersley nd ylson V presented model of symmeE tri nd symmetri ifurtions in the humn lung dequte for genertions T to IPF sn PHHRD regedus et lF W provided detiled mthemtil desription of morphologilly relisti irwy of (rst (ve genertionsF he humn lung is sequene of ifurtions in two prtsF st hs n verge of PQ genertions of ifurtionsD or out IU million of ifurtionsF he lst S generE tions re overed with ellulr strutures through whih the exhnge tkes ple with the lood gs PIF he reserhers eventully wnt to understnd the )ow struture nd prtile trnsE portGdeposition ross the humn ronhil systemF his gol remin hllenge to hieve euse of the omplexity of the geometry of humn lungsF roweverD few reserhers hve mnged to simulte the )ow in models with mny generE tionsF xowk et lF IW presented the )ow (eld nd simultion results of prtile trnsport to the lung irwys otined from gEsnF gomprisons of lultion results for relisti model nd the model of eiel showed pro(le of deposition of miroEprtiles tht hd di'erenes over moderte rnge of eynolds numers of IPHEPHHH for oth stedy )ow nd unstedyF elthough gEsn provides more relisti geometryD it n not provide respirtory eyond nine genertionsD whih limits its pplition to the study of ll lung irwysF wiki et lF IV digitized nd then lulted n tul on(gurtion to the seventh genertionF irtruggen et lF T desried threeEdimensionl struture sed on the ntomy of the irwys of the lung strting from the trhe until the IUth ifurtions to simulte sttioE nry )ow with trnsport of miroEprtilesF he mximum deposition of prtiles hs een identi(ed on the ifurtion with the highest 9o'Epln9 ngle ompred to other tues of the sme genertionD thus demonstrting the role of nonE)tness on the deposition of prtiles mirometer sizeF e mesh strutured multiElok ws hosen nd re(ned until the mximum di'erene etween the veloity pro(les ws elow P7F he deposition of prtiles tkes pleD usullyD under the tion of three mehE nisms X " impt y inerti X this mode of deposition depends on the dimeter nd the veloity of the prtilesF hose whose size is greter thn IH µm impt in the orophrynx while the smller worm their wy deep into the ronhil treeF " sedimenttion resulting from the e'et of grvityD whih reveled the prtiles to the ronhil wllF st 'ets minly smll prtiles @I to S µmA nd involves only the ronhil tues nd lveoliF st is mximum for low ir)ow nd n e promoted y simple respirtory puse t the end of inspirtionF " di'usion or frownin motion is due to the hoti ustle of smll prtiles @smller thn 0.5µm why undergo ollisions with gs moleulesF his mode of deposition is insigni(nt s VH 7 of the prtiles remin in suspension nd then re removed t the expirtion IQF 2 Methods yur study fouses on the numeril modelling of the )ow nd prtiles deposition in irwys using the ommeril pkge gphEegiF he ojetive of this study is to propose relile nd omplete numeril model of trnsport of prtiles in the pulmonry )ow y tking into ount the full rething yleF por the moment the movement of the irwys imposed y the rething re not tken into ountF his study is divided into four prts X " pirst the gphEegi pkge is vlidted regrding )ow QD PS nd prtiles deposition in simple model IRD P " imple geometries derived from the eiel model re studied " en ntomilly relisti model of rt lung is simulted " e more omplex geometry of rit lung is simulted he prtiles trjetories nd deposition re otined y solving vgrngin trnsE port eqution where only the drg fore is retinedF he deposition e0ieny is omputed s the rtio of the mount of prtiles stiked t the wll over the totl numer of relesed prtilesF he rt geometry is otined fter n dult rt killed y s dministrtion of rE iturtesD ws infused with iodine ontrst gent @IH7 vGvD isipqueQPHD qi relthreD ghlfont t qilesD uAF smging of the thorx ws performed ex vivo using n Ey g snner @eplore ision IPHD qi relthreD ukeshD eAF he protool used involved QTH views over QTH¦D with I frme vergesD t IHHkD SHmeF efter reonstrution using peldkmp lgorithm of kEprojetionD the imE ging volume ws mde of ui voxels of RWm x RWm x RWmF isulistion ws perE formed using wiroiew @qi relthreD ukeshD eAF efter inversion of grey levelsD surfe rendering @mrhing ues lgorithmA ws pplied fter thresholdF he surfe ws then exported in v formt nd prepred in qwr efore eing meshed in gphEqiywF he rit geometry ws otined using synhrotronEsed dynmi omputed toE mogrphyF hetils of the experimentl tehnique nd segmenttion protool re proE vided elsewhere RF frie)yD neworn rit pups re ventilted nd imged using propgtionEsed phseEontrst imgingF he lungs re n idel smple for phseE ontrst imgingD s they exhiit lrge phse oundries t irEtissue interfesF his tehnology n provide rpid imging of the lungs without ontrst gent UF fy synE hronising the imge quisition with the ventiltion nd rotting the smpleD fourE dimensionl @RhaQhCtimeA imges of the ronhil tree n e hievedF eginD one v geometry ws prepred in qwr efore eing meshed in gphEqiywF uitok et l IS hs proposed threeEdimensionl @QhA model of the humn irwy tree using deterministi lgorithm tht n generte rnhing dut system in n orgn through the free softwre Lung4Cer @http XGGwwwUFigloeFneFjpG lungRerGindexingFhtmlAF e hve generted model with more thn U ifurtions resulting in more thn QHH outletsF rt of this work ws performed during rF slmi hh thesis IPF

∇. - → v = 0 @IA momentum ∂ - → v ∂t + ( - → v .∇) - → v = - 1 ρ ∇p + ∇.[ν(∇ - → v + (∇ - → v ) T )] @PA
where - → v is the veloity vetorD p the pressureD ρ the )uid density nd ν the )uid kinemti visosityD () T represents the trnspose of @ A QD PPF

p ∼ | - → v -- → v p |
1 nd lrge density rtioD ρ p ρD the drg fore is dominntD sed on orderEofEmgnitude rgument @gliftD qreD 8 eerD IWUV Y growD ommerfeldD 8 sujiD IWWV Y uleinstreuerD PHHQAF hese prtiles re reltively lrgeD so frownin motion n e negletedF he prtiE ulte mteril onsidered is fr denser thn )uidD using terms tht depend on the density rtioD suh s pressure foreD uoyny foreD virtul mss e'et nd fsset fore to e very smllF he lift fores in the present tokes )ow limit re negligile euse of lk of mesurle prtile spin @wgnus liftA nd the lminrD lowElevel )uid sher (elds @'mn liftAF he motion of solid prtiles is suspension in )uid is governed y the prtile trjetory eqution @glift et lFD IWUV Y uleinstreuerD PHHTA X

m p d 2-→ x p dt 2 = C DP πd 2 p 4 ρ ( - → v -- → v p ) | - → v -- → v p | 2 , @QA
where m p is the mss of singleD spheril prtileD - → x p is the displementD - → v is the )uid veloityD - → v p is the prtile veloityD nd C DP is the drg oe0ient @glift et lFD IWUVAF he (nl eqution my e stted s X

St d - → v p dt = ( - → v -- → v p )φ D U D 1 ,
@RA where is the men inlet veloityD D 1 the prent dut dimeter nd St the tokes numer de(ned s X

St = ρ p d 2 p U 18µD 1 , @SA
where ρ p is the prtile density nd µ the )uid dynmi visosityF φ D is orretion to tokes9 drg fore for (nite prtile eynolds numers sed on the slip veloity @ - → v --

→ v p A X φ D = C Dp Re p 24 ,
@TA where

C Dp = C D C slip , @UA
where

C D =        24 Rep
for 0 < Re p ≤ 1 24 Rep 0.646 for 1 < Re p ≤ 400 @VA nd the lol prtile eynolds numer is X

Re p = ρd p | - → v -- → v p | µ @WA
he orreltion for the slip ftorD C slip D is lso given y glift et lF ID PQD P X

C slip = 1 + 2λ d p [1.142 + 0.058exp(-0.999 d p 2λ )]
@IHA where λ is the men free pth in )uidF he representtive tokes numer rnge IU omputed is HFPEIFPF he deposition e0ienyD hi @7AD ws de(ned s X DE = particle number deposited in a given region total particle number entering that region @IIA 4 Results

sn this pperD we will only present the vlidtion of the softwre on Q ifurtions eiel model @(gF IA s well s the prtile deposition in the rt nd the rit lungs for stedy )ow rteF 4.1 Model validation he urrent simultion ode @gphEegiA hs een vlidted vi omprisons with experimentl dt sets provided y ho nd vieer PTD @see pigF PAF tedy )ow ws onsideredF por detils of this prtiulr se studyD gomer et lF Q nd hng nd uleinstreuer ITF por stedy )owD fullyEdevelopedD iFeFD proliD veloity pro(le n e oserved in oth numeril nd experimentl studies t lotion P @pigF P@AAF et lotion IH @see pigF P@AAD the veloity pro(le is skewed to the inner wll when the ir)ow turns from the prent to the dughter tueF hen the ir)ow ontinuously turnsD more )uid is pushed to the inner wll t lotion IR @see pigF P@eAA due to the e'ets of entrifugl fore nd seondry )owF he ir)ow ptterns in the lung irwys re determined y the )uid propertyD the inlet ondition nd geometri hrteristisF he geometri in)uene on the lol ir)ow pttern is signi(nt @see pigsF P@AD @dA nd @fAAF he xil veloity is zero ner the wll nd mximum t the enter of the irwy @pigF P@AAF pigF Q shows the omprison of our deposition e0ieny in doule ifurtion inE plne model for stedy )ows with the dt of uim 8 pisher @IWWWA IRF hue to the di0ulty in stritly ontrolling the )ow rte nd erosol dimeterD the eynolds numers signi(ntly vries during the experimentsF purthermoreD some di'erenes etween omputtionl preditions nd experimentl mesurements my e used y sutle disrepnies in model geometries nd deposition mesurementsF husD when negleting the experimentl eynolds numer e'et nd fousing on deposition e0ienyD the simultion results gree well with the experimentl dt of uim 8 pisherF he good greement indites tht the gphEegi ode is su0iently urte to simulte tul ir)ow nd prtile deposition in the humn lungF 4.2 The Rat conguration he rt geometry form medil imging is exported in stereolithogrphy formt STL to e smoothedD repired @holes geometryA nd meshed with gphEqiywD the mesh genertor of the gphEegi pkgeF e generted severl meshes E QHHk to RFQw ells E to hek mesh onvergene resultsF he reth of rt of SHHg eing of Rml.s -1 D this gives us in our setup inlet veloity of Im.s -1 so numer eynolds Re D inlet = 152 whih suggests tht the )ow is stedy nd lminrF pigure R shows the pressure drop nd the shpe of the stremlines in the lungs of rts X the )ow is reltively simple nd does not pper to e irulting in the light of the low urvture ifurtions F he di0ulty here doesn9t lerly lie on the simultion itself ut rther on the genertion of vile geometry nd mesh X in this seD the initil geometry inluded nonEphysil onstritions on multiple outputs whih mde the simultion ompletely unrelisti even lengthening out through pipesF hese onstritions hve een smoothed the outlets extended with pipesF pigure SA shows the deposition of prtiles oservedF e note tht the deposition ppers only t juntions nd ends whih is very understndle euse the geoE metry of the lungs of rts is firly liner unlike humn lungs nd we hd lredy oserved tht the deposition mehnism ws n inertil mehnismF he results for the depositions were otined with four intermedite meshes nd the onvergene of the results is otined @pigure SA on the (nest gridF egrding the deposition of prtilesD the sme onlusion s with the t geometry n e dressed @pigure UAF his is minly due to the ft tht with our unstrutured grid genertion softwreD it9s very di0ult to inrese the grid density ner the wlls without drmtilly inresing the totl numer of ellF his is the reson why we strted to develop new softwre t sgfi sed on smmersed foundry wethodsD unstrutured grtesin ells nd eutomti qrid e(nementF 5 Conclusions efter vlidtion phseD we hve simulted the )ow nd the prtile deposition in mny eiel9s generi models nd in ntomilly relisti geometries of rt nd rit oming from medil imgingF he prtile deposition is minly inertil nd inreses with the tokes numer nd the eynolds numerF yne of the mjor di0ulty of this study is the genertion of the pproprite mesh for the relisti geometriesF

3 Governing equation 3 . 1

 31 Airow equations por lminr inompressile )owD the )uid trnsport equtions n e written s X ontinuity

3. 2

 2 Particle transport and deposition equations sn this workD simultions re performed for prtiles of dimeter d p = 5µmF sn light of the smll prtile eynolds numersD Re

Figure 1

 1 Figure 1 Schematic of symmetric : a) triple bifurcation geometry [24] ; b) single bifurcation of station positions [26],

Figure 2

 2 Figure 2 Axial velocity distribution in steady ow : (Re D1 = 1000) ; a) station (2 )-(2), c) station (10 ) -(10) and e) station (14 ) -(14) [26] (see Fig.1 b)) and ow patterns : b) axial velocity pattern at Slice 2, d) at slice 10 and f) at slice 14.

Figure 3

 3 Figure 3 Comparison of the numerical and experimental particle deposition eciency (DE) in the double bifurcation in-plane model under steady ow (Experimental data were obtained from Kim & Fisher, 1999 [14]) : a) DE at the rst bifurcation ; and DE at the second bifurcation.

Figure 4

 4 Figure 4 Visualisation of the pressure drop and streamlines in rat lungs

Figure 5 a

 5 Figure 5 a) Deposition eciency and b) in rat lungs

Figure 6 a

 6 Figure 6 a) Rabbit geometry from segmentation and b) Streamlines and pressure drop on a 4.4M cells grid

Figure 7 a

 7 Figure 7 a) Deposition and b) DE in Rabbit lungs


M Elhimer
email: melhimer@imft.fr

G Harran

T Deloze

D Szubert

Y Hoarau

M Braza

Naturally separated turbulent flow around two cylinders in tandem at high Reynolds number by timeresolved PIV

Keywords: Cylinders in tandem, time resolved PIV, bistable flow, turbulence

In this study the focus is made on the properties of the bistable flow between two cylinder in tandem for a pitch ratio L/ D=3,7 at a Reynolds number of 132800. Time resolved particle image velocimetry (TRPIV) is used to measure the spatial distribution and time fluctuations of the velocity field in the wake of the upstream cylinder. From velocity spectra and statistics fields, the existence of two distinct seemingly stable flow patterns is shown, having different spatial structures and predominant frequencies.

Introduction

Multiple cylinder structures can be found in the designs for heat exchangers, cooling systems for nuclear power plants, offshore structures and cables, in both air and water flow. In many of these engineering applications, the periodic shedding of Kármán vortices is responsible for problems with flow-induced vibration and noise. Example of such cases is the triggering of fluid elastic instability in heat exchanger tubes arrays subjected to a cross-flow (Granger & Paidoussis 1996) among other. The need to predict the critical velocity of this instability in the heat exchangers of nuclear reactors has motivated several experimental studies using PIV and visualization techniques (Paul 2007[START_REF] Iwaki | PIV Measurement of the[END_REF]) that show the complex interactions between the shear layer vortices and Kármán vortex streets in the flow field between the cylinders.

The tandem configuration is among the simplest form of multiple cylinders structure. The two cylinders are in close proximity, arranged at arbitrary centre-to-centre spacing and zero angle of incidence to the oncoming flow. This arrangement can be found in several locations on a landing gear, such as multiple wheels, axes, and hydraulic lines. It has been chosen by Jenkins et al. (2005) as a canonical problem to identify and assess the effect of the interaction of unsteady flow among components of the landing gear on the airframe noise generation on large aircraft during approach.

The tandem arrangement has shown a large variety of flow patterns depending on the pitch (e.g. the distance between the cylinders) to diameter ratio, the Reynolds number and the upstream turbulence intensity. These patterns have been investigated extensively by several authors (Igararashi 1981, Xu et Zhou 2004, Sumner 2000[START_REF] Alam | An improvement to the quasi-steady model with application to cross-flow-induced vibration of tube arrays[END_REF], Lin et al. 2002) who show the existence of a critical pitch depending on the Reynolds number, concerning the transition from one pattern to another (Sumner 2010).

For the smallest values of the pitch to diameter ratio in the interval (1,2) , the tandem act as a single extended body because the downstream cylinder is located in the vortex formation zone of the upstream one, so this flow regime is called "extended body regime" in literature. The separated shear layers from the upstream cylinder wrap around the downstream cylinder, without any reattachment onto its surface, before rolling up alternately into Kármán vortices behind the downstream cylinder. The flow between the two cylinders is almost stagnant but may behave as a cavity flow (Sumner 2010).

For medium pitch ratio values in the interval (2,5), the shear layers from the upstream cylinder reattach onto the downstream cylinder, with eddies formation and their shedding in the gap region between the two cylinders. A wide variety of fluid behaviours can be observed in this regime that was first classified by Igarashi (1981). In addition Lin et al (2002) show that the gap eddies can vary substantially and intermittently in terms of their strength, asymmetry, and general behavior.

Finally, at higher pitch ratios L/ D>5 the downstream cylinder is sufficiently far away and therefore, a Von Kármán vortex shedding can occur past the upstream cylinder as well as past the downstream one. The downstream cylinder is located in this case outside the vortex formation region of the upstream cylinder and experiences the periodic impingement of shed vortices from the upstream cylinder. In this "co-shedding" flow regime, both cylinders undergo vortex shedding at the same frequency.

The transition to and from the co-shedding regime is known to be bistable, with intermittent appearance of the two flow patterns (Igarashi 1981, Xu et Zhou 2004). This bistable regime appears in several experiments with different aspects and blockage ratios (Sumner 2010). In the present study the focus is made on this bistable regime, the aim of our experiment is to fully characterize each of the two flow patterns. Time resolved PIV is used allowing resolving simultaneously the spatial structure and the spectral content of the velocity field in the wake of the upstream cylinder for each of the two configurations. A comparison between the properties of the two patterns is done. The present study shows that the bistable regime is observed for tandem cylinder with pitch to diameter L/ D=3,7 × D at Reynolds numbers ℜ=107 000 and ℜ=132 800 but not in the lower Reynolds number range, ( ℜ=31 000 and 47 000). This suggests the existence of a critical Reynolds number beyond which the bistable regime appears. In this article the focus is made on the highest Reynolds number ℜ=132 800 .

Experimental set-up

The experiments were undertaken in the wind tunnel S4 of IMFT, which delivers an upstream velocity with low turbulent fluctuations (the turbulence intensity has been measured by means hot wire anemometry and found to be of order 0,15 ). The test section is of 58 cm width and 72 cm height. Two transparent Plexiglas cylinders of diameter 8 cm have been used, yielding an aspect ratio of 7,25 and a blockage ratio of 0,11 . This compares well with Lin et al. (2002) experiments were the aspect ratio is 8,3 and the blockage ratio was 0,1 . The aspect ratio and the blockage ratio were in Igarashi (1981) 4 and 0,06 respectively and in Xu et Zhou ( 2004) 40 and 0,025 respectively.

In the present study, the pitch to diameter ratio of the tandem is L/ D=3,7 so the gap between the cylinders is 21,6 cm wide. The cylinders are positioned in cross-flow (figures 1 and 2) at the mid-height of the test section and spanned over its width. The upstream velocity in the wind tunnel was measured using a Pitot tube with pressure plugs at the inlet and the outlet of the contraction section, and was controlled by changing the fan angular velocity. The upstream velocity is U =25,5 m /s yielding a Reynolds number of ℜ=132 800 .

The velocity field measurements have been performed using a high speed PIV set-up that includes a double frame high-speed camera (Photron RS3000), a twocavity Laser (Quantronix Darwin) and a synchronizer. The planes of measurement are shown in Fig. 1 concerning the whole data base measured. In the present study, the results of plane 2 are discussed.

The camera CMOS-sensors has a resolution of 1024 × 1024 pixels and an internal memory size of 8 GB enabling an acquisition duration of 3 seconds at a frequency of 2000 Hz , the resulting PIV measurements frequency is 1000 Hz . The visualization plane was located in the wake of the upstream cylinder at its mid-width with the size of 16 cm ×16 cm . In order to reach the statistical convergence up to the second order, ten PIV acquisitions were carried out for each of the two flow patterns: knowing that the frequency of the Von Kármán shedding is about 50 Hz in the present study, each acquisition contains 150 shedding cycles, thus about 1500 shedding cycles have been covered in our measurements for each pattern.

The energy of the laser beam delivered by each cavity was set at 25mJ . Using a combination of spherical and cylindrical lenses, the Laser beam was expanded into a light sheet of 2 mm thickness. The flow was seeded with droplets of a mixture of water and alcohol using a fog generator. The droplet size distribution was measured using a Malvern granulometer and found 3,4 µm on average with a small standard deviation.

The image processing and the cross-correlation analysis were done by using DAVIS 8 La Vision software. The images intensity was inhomogeneous due to the lighting conditions and the sensor properties.

The high frequency of the PIV measurements allows extracting velocity temporal signal at any location of the visualization domain. For each location the time series from the ten acquisitions were concatenated in order to form a sufficiently long time series. The periodogram of these time series was computed using spectral averaging on multiple windows (Marple 1987) with nonoverlapping Hamming windows of size 512 samples, the resulting frequency resolution is 0,03 Hz .

Statistical and spectral analysis of the flow in the gap region between the two cylinders

As mentioned above, for the pitch ratio L/ D=3,7 D and the Reynolds number ℜ=132 800 , a random bifurcation has been observed concerning the gap flow between two flow patterns. The duration of each flow pattern is variable and it is generally of the order of several minutes, in comparison the Von-Kármán shedding period being of the order of 20 milliseconds.

The first flow pattern, named I, consists on a Von Kármán vortex shedding in the wake of the upstream cylinder (Figure 1) but with a different shedding frequency: the dimensionless Strouhal number is St =0,178 instead of St =0,21 for a single cylinder. The second flow pattern, named II, appears generally after the first one. It consists of two stable detached shear layers that reattach on the downstream cylinder and create a recirculating flow in front of this cylinder (Figure 2). This pattern displays a Strouhal number of St =0,135 close to the downstream cylinder.

The appearance of the flow pattern II always comes with 40 Pa increase of the pressure difference between the inlet and the outlet of the contraction section. Thus the upstream velocity increases by about 1m/ s while the fan's angular velocity remains unchanged. This shows the decreases of the pressure drop induced by the presence of the tandem in the test section when the flow patterns II appears, and is probably due to the decrease of shear stress in comparison with the first pattern. In our experiments, the velocity was reduced after the appearance of the second pattern to bring it back to its initial value U =25,5 m /s . The acquisition was started only two minutes later. This change of the upstream velocity does not cause the switching towards the first pattern.

FLOW PATTERN I

This pattern shows a topology and statistical properties similar to single cylinder in cross-flow. Indeed, The longitudinal average velocity field U m is symmetrical with respect to the axis y / D=0 , and one can clearly identify the non-rotational zone, the shear zone and the dead water zone (Figure 3, Figure 5). In the irrrotational zone, the upstream flow is accelerated to 1,2 ×U near the cylinder and beyond the position y / D=± 0,63 in agrement with the results of Braza ( 2006) for an isolated cylinder at a close Reynolds number ℜ=140 000 . In addition, the cylinder recirculating length, defined as the distance between the axis of the cylinder and the point were U m =0, is l c =1,27 × D which is in good agreement with the value l c =1,28 × D obtained by [START_REF] Alam | An improvement to the quasi-steady model with application to cross-flow-induced vibration of tube arrays[END_REF]. Finally, the Reynolds stresses fields (Figure 7, Figure 9, Figure 11, Figure 13, Figure 15, Figure 17,) shows a topology and non-dimensional values close to the case of an isolated cylinder.

The vertical velocity spectra have been measured at different positions on the line y / D=0,5 (Figure 19). They show a peak of power spectral density at the non-dimensional frequency f . D/U =0,178 thus defining the Strouhal number of the flow. This value is in good agreement with the results of Xu et Zhou (2004) and Igarashi (1981) for close pitch to diameter ratio but at smaller Reynolds numbers. As mentioned by Xu et Zhou (2004) the Strouhal number depends on both on the base pressure coefficient C pb and the shear layer

thickness δ St =St 0 ( 1-C pb ) 1/ 2 /(1+2δ / D)
where St 0 ≈ 0.176 is a universal Strouhal number. The shear layer thickness depends on the Reynolds number and the base pressure coefficient is modified by the presence of the downstream cylinder. This explains the dependence of the Strouhal number on both the pitch and the flow Reynolds number.

For the same pitch to diameter ratio but at higher Reynolds number ℜ=166 000 Jenkins et al ( 2005) found a higher value of the Strouhal number St =0.26 . However their experiments are not comparable to ours because these authors tripped the first cylinder in order to force a turbulent boundary layer upstream separation.

FLOW PATTERN II

The flow pattern II has a distinct topology comparing to the case of isolated cylinder. The flow structure consists of two stable shear layers issued from the upstream cylinder. This observation is confirmed by examining the longitudinal average velocity field (Figure 4).

Two irrotational zones are identified at both sides of the upstream cylinder, were the mean longitudinal velocity reaches a higher value than in the first pattern:

1,3×U at y / D=± 0,8 on the line x / D=0,71 . The shear zones located between y / D=± 0,5 and y / D=± 0,8 delimit a "dead water zone" that spans over a larger area than in the first flow pattern. Further away from the cylinder, the average longitudinal velocity becomes negative; this suggests the existence of a downstream recirculation flow that is however outside the visualization zone in these experiments. This recirculation is most probably generated by the reattachment of the shear layers on the downstream cylinder.

The normal Reynolds stresses have a symmetrical distribution with respect to the axis y / D=0 (Figure 8, Figure 12) while the Reynolds shear stress has an anti-symmetrical distribution with respect to this axis. The Reynolds stresses values are significantly smaller compared to the first flow pattern due to the attenuation of the organized large-scale flow close to the upstream cylinder.

The Reynolds stresses concentrate at the thickness of the shear zones that increase away from the cylinder due to the turbulent diffusion of the stresses. Indeed, the shear zone thickness increases from 0,25 D at x / D=-0,71 to 0,5 D at x / D=-1,5 D (Figure 10, Figure 14). The Reynolds stresses are negligible close to the cylinder in the "dead water zone" and increase as the recirculation flow get closer.

The vertical velocity spectra are shown along the line y / D=0,5 in this flow pattern (Figure 20). The spectrum is continuous near the cylinder because the large scale alternating motion is attenuated in this region of the flow in case of pattern II. However small vortex bursts are identified in the instantaneous PIV velocity fields in the shear zone. These vortices cannot be detected by a classical spectral analysis due to their high frequency appearance in respect of the sampling frequency of the TRPIV experiment ( 1000 Hz ).

For positions closer to the downstream cylinder a power spectral density (PSD) peak appears at the non-dimensional frequency f . D/U =0,135 that is different from the Strouhal number value of 0,178 in the first flow pattern. This value is however in good agreement with the findings of Xu et Zhou (2004) for close pitch to diameter ratio and at smaller Reynolds number. This peak is most probably linked to the recirculation flow in the front of the downstream cylinder and clearly appears beyond the position x / D=-1,7 . This result indicates that the Von Kármán mode appears farther downstream and is characterized by a lower frequency, because of the vicinity with the second cylinder. The fact that this Strouhal number remains constant for these downstream positions associated with the presence of two inflexion points in the mean velocity profiles indicates that this mode is a Von Kármán mode.

Conclusions

In this paper, an experimental investigation of the flow between two cylinders in tandem with a pitch L/ D=3,7 and a Reynolds number of ℜ=132 800 has been carried out. The separated shear layers from the upstream cylinder switch intermittently between roll-up with vortex shedding and reattachment onto the downstream cylinder with recirculation. The first flow pattern is similar to a Von Kármán shedding from an isolated cylinder but with a smaller Strouhal number. The recirculation is mainly outside our visualization domain but it is most probably responsible for the different value of the Strouhal number measured in the second flow pattern. Additional time-resolved PIV measurements have been undertaken, in order to fully characterize the recirculating flow and reattachment dynamics in the vicinity of the downstream cylinder, in comparison with simulation results for static and freely vibrating second cylinder. x / D=-1,5 (blue) x / D=-2,2 (green).

-0.5 0 0.5 1 -0.5 0 0.5 x / D=-1,5 (blue) x / D=-2,2 (green). x / D=-1,5 (blue) x / D=-2,2 (green). x / D=-1,5 (blue) x / D=-2,2 (green). x / D=-1,5 (blue) x / D=-2,2 (green).

Figure 1

 1 Figure 1 Measurement planes

Figure 2

 2 Figure 2 Tandem cylinder scheme and upstream velocity U=25,5 m/s U=25,5 m/s 3,7 D

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5

 12345 Figure 1 flow pattern I : Velocity field sequence at a frequency of 500 Hz . The number of vectors have been reduced and the velocity fields smoothed for clarity

Figure 6

 6 Figure6Flow pattern II : profiles of U m /U along the lines x / D=-0,71 (black) x / D=-1 (red)x / D=-1,5 (blue) x / D=-2,2 (green).

Figure 7 Figure 8 Figure 9 Figure 10

 78910 Figure7Flow pattern I : spatial field of ú2 /U in the wake of the upstream cylinder.

Figure 11 Figure 12 Figure 13 Figure 14

 11121314 Figure11Flow pattern I : spatial field of v2 /U in the wake of the upstream cylinder.

Figure 15

 15 Figure15Flow pattern I : spatial field of úv /U in the wake of the upstream cylinder.

Figure 16 Figure 17

 1617 Figure16Flow pattern II: spatial field of úv /U in the wake of the upstream cylinder.

Figure 18

 18 Figure18Flow pattern II : profiles of úv /U along the lines x / D=-0,71 (black) x / D=-1 (red)x / D=-1,5 (blue) x / D=-2,2 (green).

Figure 19

 19 Figure19Flow pattern I : vertical velocity periodogram on the horizontal line y / D=0,5 at positions from x / D=-0,76 to x / D=-2,17 by a step of -0,2344

Figure 20

 20 Figure 20 Flow pattern II : vertical velocity periodogram on the horizontal line y / D=0,5 at positions from x / D=-0,76 to x / D=-2,17 by a step of -0,2344

  Vertical Cross-flow Structure over Tube Bundles ». Experiments in Fluids 37 Jenkins, Luther, Mehdi Khorrami, Meelan Choudhari, et Cathy McGinley. 2005. « Characterization of Unsteady Flow Structures Around Tandem Cylinders for Component Interaction Studies in Airframe Noise ». American Institute of Aeronautics and Astronautics, Lin, J.-C., Y. Yang, et D. Rockwell. 2002. « Flow past two cylinders in tandem: instantaneous and averaged flow structure ». Journal of Fluids and Structures Marple, S. Lawrence. 1987. « Digital Spectral Analysis with Applications ». Englewood Cliffs, NJ, Prentice-Hall, Inc. Paul, S.S., M.F. Tachie, et S.J. Ormiston. 2007. « Experimental study of turbulent crossflow in a staggered tube bundle using particle image velocimetry ». International Journal of Heat and Fluid Flow Sumner, D., S. J. Price, et M. P. Paidoussis. 2000. « Flow-pattern identification for two staggered circular cylinders in cross-flow ». Journal of Fluid Mechanics Sumner, D. 2010. « Two circular cylinders in cross-flow: A review ». Journal of Fluids and Structures Xu, G., et Y. Zhou. 2004. « Strouhal Numbers in the Wake of Two Inline Cylinders ».

	Experiments in Fluids

Acknowledgements

This work has been carried out in the research group EMT2 (Ecoulements Monophasiques, Transitionnels et Turbulents) of the Institut de Mécanique des Fluides de Toulouse. It is based on the collaborative efforts of Sébastien Cazin, Moise Marshal, Hervé Ayroles and Christophe Korbuly for the experiments set up and the measurements techniques. Authors thank Alain Sevrain for useful discussions. This work has been carried out in the context of the ANR-BARESAFE -"Simulation of Safety Bareer reliability", coordinated by EDF, Electricité de France.


domain: spi phys

text:
Baptiste Sinquin

Caroline Kulcsár

Léonard Prengère

Henri-François Raynaud

Eric Gendron

James Osborn

Alastair Basden

Jean-Marc Conan

Nazim Bharmal

Lisa Bardou

Lazar Staykov

Tim Morris

Tristan Buey

Fanny Chemla

Matthieu Cohen

Data-based modelling of low-order modes for AO control: what do on-sky experiments tell us?

Keywords: adaptive optics, Linear-Quadratic-Gaussian control, Kalman filtering, subspace identification, turbulence modeling, on-sky validation

It is well-known that data-driven models for the tip-tilt modes significantly improve the performance of Adaptive Optics (AO) systems as it allows to compensate for vibration-induced disturbances. Whether identifying from data the temporal dynamics of more modes makes an impact on the performance has been studied on-sky with the CANARY demonstrator at the William Herschel Telescope in July 2019. In this brief paper, we report on these experiments using both Strehl ratios computed from the science camera in H band or by replaying the AO telemetry data in numerical simulations. We show that Linear Quadratic Gaussian (LQG) controllers that embed a data-driven model for the low-orders (that are not limited to the tip-tilt) improve the performance of the AO for two different dynamical behaviours of the atmospheric turbulence.

INTRODUCTION

The disturbance that reaches an AO system for a ground-based telescope is not only induced by the atmospheric turbulence but also by dome turbulence and vibrations, that all contribute to poor Strehl ratios without a dedicated controller. The integrator has soon become a standard for its simplicity and for its good rejection of the lowest temporal frequencies. Linear Quadratic Controllers (LQG) produce rejection transfer functions that allow to better shape the inverse of the turbulence Power Spectral Density (PSD). This enables to compensate for a wide class of perturbations with the aim to reduce the temporal error and achieve better performance. [START_REF] Roux | Optimal control law for classical and multiconjugate adaptive optics[END_REF] For example, vibrations are in general amplified by the standard integrator whereas LQG controllers that include a data-driven model on the tip-tilt efficiently correct these additional disturbances 2 . [START_REF] Kulcsár | Vibration mitigation in adaptive optics control[END_REF] The disturbance model that is used in such controllers should represent as accurately as possible the actual perturbation viewed by the sensor. Auto-regressive models of order two that depend on the wind speed and the seeing have been proposed and tested on-sky. [START_REF] Sivo | First on-sky SCAO validation of full LQG control with vibration mitigation on the CANARY pathfinder[END_REF] They represent a boiling behaviour of the turbulent atmosphere. There remains nonetheless unmodelled disturbances that are not captured by the model structure, for example dome turbulence (whose statistics are likely to be different) and frozen-flow. Whether using AO telemetry to model the temporal dynamics of more modes than only the tip and tilt would further improve the AO performance is investigated here.

In this paper we propose to report on the on-sky experiments from July 2019 on the CANARY demonstrator 5 at the William Herschel Telescope (WHT) on La Palma, Canary Islands, Spain.

The paper is organized as follows. Section 2 details the LQG controller and Section 3 shows first on-sky results. Conclusions are then drawn.

THE LINEAR-QUADRATIC-GAUSSIAN CONTROLLER

The LQG controller minimizes the residual phase variance over an infinite time horizon, [START_REF] Kulcsár | Minimum variance prediction and control for adaptive optics[END_REF] min lim

K→∞ 1 K K k=1 φ res k 2 2 .
(

) 1
The disturbance viewed by the sensor is the difference between the disturbance φ and the one induced by the deformable mirror φ cor . The control commands are applied to the quasi-static mirror with a delay of one frame due to the zero-order hold, such that at time instant k, the correction wavefront is expressed as

φ cor = N u k-1 .
The LQG cost function then reads

J(u) = min u lim K→∞ 1 K K k=1 φ k+1 -N u k 2 2 . (2)
We have slightly modified this criterion as in [START_REF] Sivo | First on-sky SCAO validation of full LQG control with vibration mitigation on the CANARY pathfinder[END_REF] for our on-sky experiments using rather

J(u) = min u lim K→∞ 1 K K k=1 Dφ k+1 -M int u k 2 2 , (3)
where D is the sensor matrix and M int is the interaction matrix. This allows to account for the transformations that occur between the sensor and actuator grids and to rely on the interaction matrix and not on the influence matrix. The control commands are then obtained as

u k = M com D φ k+1|k , (4)
where M com is the command matrix and φ k+1|k is the minimum variance estimate of the wavefront at time instant k + 1 using all the data up to time k, which is provided by the Kalman filter. The key in LQG controller design is the dynamical model used for describing the disturbance. Let the wavefront be expressed in a Zernike basis with 495 modes. We start from the auto-regressive model that successfully demonstrated its relevance on-sky:

4 φ k+1 = A 1 φ k + A 2 φ k-1 + v k (5)
where v k is a zero-mean white Gaussian noise with covariance matrix Σ v . This model uses priors such as the wind speed and the Fried parameter. The temporal dynamics of the first n LO Zernike modes φ LO k are modelled with another state-space:

x LO k+1 = A LO x LO k + v k φ LO k = C LO x LO k + η k . (6)
The matrices of this model are directly estimated using the system identification algorithm N4SID 7 on a temporal sequence of reconstructed wavefronts. This methods uses Pseudo-Open-Loop (POL) slopes collected before closing the loop. The low-order modes therefore consist of the sum of the two models, one based on priors (5) and the other one that describes low orders (6). The remaining modes (not modelled by ( 6)) are only modelled through the priors, as illustrated in Figure 1. The open-loop model of the wavefront sensor yields the slopes y k as follows:

y k = 0 D D LO C   φ k φ k-1 x LO k   + w k (7)
where w k is a zero-mean white Gaussian noise with diagonal covariance matrix. The Kalman gain L ∞ is computed by solving the discrete algebraic Riccati equation so that the Kalman filter writes

   x k+1|k = A -L ∞ C x k|k-1 + L ∞ y POL k φ k+1|k = C φ x k+1|k (8)
where

A =   A 1 A 2 0 I 0 0 0 0 A LO   and C φ = I, 0, C LO .

ON-SKY RESULTS

Single Conjugate AO control on CANARY

The primary mirror of the WHT has a diameter of 4.2m. The Shack-Hartmann has a 14×14 lenslet array leading to 288 x and y measurements. The control commands are applied to a tip-tilt mirror and to a deformable mirror (coupling of 0.45) that has 243 active actuators. The science camera collects photons in the H spectral band. The control frequency was set to 200 Hz.

During this observing run, we have experienced very different atmospheric conditions that could broaden our views on the impact that the data-driven model for the low-orders can have on performance. Note that no vibrations were observed on the data we report here.

We have also performed simulations using on-sky AO telemetry data. The replay mode allows to compare controllers whose performance was not evaluated on sky. In this case, the performance index is the residual phase variance evaluated as the sum of the first 197 modes, which corresponds to the number of mirror modes.

LqgZer-TT refers to the controller in 4 that uses the model based on priors (5) with a tip-tilt data-driven model computed from the mean slopes. LqgZer-LOn LO refers to the new controller that relies on the Kalman filter (8) and that includes temporal models identified from telemetry data for the first n LO modes. LqgZer-LO2 thus differs from the LqgZer-TT from the use of tip and tilt that are reconstructed from data instead of using mean slopes.

First case: large wind

We report on the experiments from the night of July 17-18th, 2019, between 3h08min and 4h21min (local time). The guide star has a magnitude of 5.2 in H band. Main layers at 0, 2 and 8 km were measured with a wind speed of respectively 12, 15 and 13 m/s using the stereo-SCIDAR. [START_REF] Osborn | Characterising atmospheric turbulence using SCIDAR techniques[END_REF] Figure 2-left shows the residual phase variance as a function of n LO . At radial order 1, we indicate with a blue cross the performance on the integrator. As expected, all tip-tilt data-driven models improve performance with respect to the integrator. Also, increasing the number of modes n LO in (8) significantly lowers the residual phase variance. The plot on the right-hand-side will be commented in next section.

Second case: low wind

This paragraph reports on the experiments from the night of July 20-21st, from 4h31min to 6h. The guide star has magnitude 5.8 in H band. The wind speed at ground level was no more than 3m/s.

An LQG controller featuring a model for the temporal dynamics of 9 modes identified from telemetry data outperforms the integrator, as shown in Figure 3-left. In Figure 3-right, we see that increasing n LO from 5 to 9 improves the performance while it is less clear for n LO from 9 to 14 due to the large variability of the atmospheric conditions even for two values having similar r 0 .

Figure 2-right confirms what was observed on-sky: there is a significant improvement with a data-driven model on the 9 first modes rather than on the first 5 ones but increasing n LO up to 14 has less impact. This trend is very different from the conclusion drawn when the wind speed was larger.

CONCLUSION

In this paper we have proposed to augment a prior-based disturbance model with a data-driven model to better represent the actual temporal dynamics of the perturbation. This controller was tested on-sky at the WHT using the CANARY demonstrator in order to assess the impact on performance of modelling from data more modes than only the tip and tilt. A gain of performance was evidenced even though no vibrations were detected on these modes. The number of modes n LO that yields the lower residual phase variance depends very much on the atmospheric conditions, such as the wind speed of the different layers. If the wind speed is large, it was shown that increasing n LO up to 65 increases significantly the performance whereas it is not necessary to consider so many modes for a lower wind speed scenario, where 9 modes were enough. A more detailed analysis is to appear in a forthcoming publication.

Figure 1 .

 1 Figure 1. Schematic that illustrates how the temporal dynamics of the Zernike modes are represented, using both or either the prior-based and the data-based model.

Figure 2 .

 2 Figure 2. (Replay) Residual phase variance as a function of nLO for a case with large wind speed (left) and low wind speed (right).

Figure 3 .

 3 Figure 3. (On-sky) Strehl ratio as a function of the Fried parameter r0 for (left) the experiments from 4h31 with a comparison between integrator and LqgZer-LO9 and (right) the experiments from 4h51 the same night with a comparison between LqgZer-LO5, LqgZer-LO9 and LqgZer-LO14. One experimental point corresponds to 20 seconds of data recording.

E-mail: baptiste.sinquin@institutoptique.fr, caroline.kulcsar@institutoptique.fr

Acknowledgements

LQG tools for CANARY have been supported by the Agence Nationale de la Recherche (ANR) French program CHAPERSOA ANR-09-BLAN-0162-01 and ANR-10-LABX-0039-PALM. CANARY was supported by Agence Nationale de la Recherche (ANR) program 06-BLAN-0191, CNRS/INSU, Observatoire de Paris, and Université Paris Diderot Paris 7 in France, Science and Technology Facilities Council (Grants PP/E007651/1, ST/I002871/1, ST/K003569/1, ST/M007669/1 and ST/N002660/1), and the University of Durham in the UK and European Commission Framework Programme 7 (E-ELT Preparation Infrastructure Grant 211257 and OPTICON Research Infrastructures Grant 226604). Horizon 2020: This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under grant agreement number 730890. The William Herschel and Isaac Newton Telescopes are operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This work of James Osborn was supported by the Science and Technology Funding Council (UK) (ST/P000541/1), UKRI Future Leaders Fellowship (UK) (MR/S035338/1) and Horizon 2020.