asynchronous coupling of hybrid models for efficient simulation of multiscale systems

Coupling is performed asynchronously, with each model being assigned its own timestep size.This enables accurate longtimescale predictions to bemadeatthe computational costof theshort ti

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www.elsevier.com/locate/jcp

Duncan A Lockerbya, ∗ , Alexander Patronisa, Matthew K Borgb,

Jason M Reesec

aSchool of Engineering, University of Warwick, Coventry CV4 7AL, UK

bDepartment of Mechanical & Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK

cSchool of Engineering, University of Edinburgh, Edinburgh EH9 3JL, UK

a r t i c l e i n f o a b s t r a c t

Article history:

Received 6 August 2014

Received in revised form 16 December 2014

Accepted 20 December 2014

Available online 24 December 2014

Keywords:

Multiscale simulations

Unsteady micro/nano flows

Hybrid methods

Scale separation

Rarefied gas dynamics

Wepresent anewcouplingapproachforthetimeadvancementofmulti-physicsmodels

of multiscale systems This extendsthe method ofE et al (2009) [5] to dealwith an arbitrary number of models Coupling is performed asynchronously, with each model being assigned its own timestep size.This enables accurate longtimescale predictions

to bemadeatthe computational costof theshort timescale simulation.We proposea methodforselectingappropriatetimestepsizesbasedonthedegree ofscaleseparation thatexists betweenmodels.Anumber ofexampleapplicationsare used fortestingand benchmarking, including a comparison with experimental data of a thermally driven rarefiedgasflowinamicrocapillary.Themultiscalesimulationresultsareinveryclose agreementwiththeexperimentaldata,butareproducedalmost50,000timesfasterthan fromaconventionally-coupledsimulation

©2014TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC

BYlicense(http://creativecommons.org/licenses/by/4.0/)

1 Introduction

Amulti-physicsdescription ofa multiscalesystemisoftenreferredto asa ‘hybrid’model.Influid dynamics,a typical hybridcombines amolecular treatment(a‘micro’model)witha continuum-fluidone(a‘macro’ model),withtheaimof obtaining theaccuracy oftheformer withtheefficiencyofthe latter [1–4].Themicro andmacromodelsgenerallyhave characteristictimescalesthatareverydifferent,whichmeansthattime-accuratesimulationscanbeextremelychallenging: thesizeofthetimesteprequiredtomakethemicromodelstableandaccurateissosmallthatsimulationsoversignificant macro-scaletime periodsareintractable.Ifthesystemis‘scale-separated’,aphysical(asdistinctfromnumerical) approxi-mationcanbemadethatenablesthecoupledmodelstoadvanceatdifferentrates(asynchronously)withnegligiblepenalty

onmacro-scaleaccuracy.Eetal. [5]werethefirsttointroduceandimplementthisconceptinatime-steppingmethodfor coupledsystems,referredtointheclassificationofLockerbyetal. [6]asacontinuousasynchronous(CA)scheme (‘contin-uous’sincethemicroandmacromodelsadvancewithoutinterruption [5]).Inthispaperweextendthisideatomultiscale systemscomprisinganarbitrarynumberofcoupledmodels

* Corresponding author.

E-mail address:duncan.lockerby@warwick.ac.uk (D.A Lockerby).

http://dx.doi.org/10.1016/j.jcp.2014.12.035

0021-9991/©2014 The Authors Published by Elsevier Inc This is an open access article under the CC BY license

Fig 1 The continuous asynchronous (CA) coupling scheme extended to multi-model multiscale systems.

2 Extension to multi-model systems

WeconsideranN-modeltimescale-separatedsystem,wheretheithmodelhasacharacteristictimescaleT i,andindexing

isorderedsuchthat

Modeli=1 isthemicromodelandi=N isthemacromodel;modelsi=2 toi=N−1 are‘meso’models.Thedegreeof scaleseparation S i betweenmodeli and i+1 is

S i= T i+1

wherethetolerance Stolrequireseachdistinctmodelofthesystemtobescaleseparatedfromeveryothertosomedegree, for example, Stol= O(10) If this condition is not met, the two models are treated as one, and coupling is performed conventionally

Ingeneraleachmodelcanbeconsideredtohaveitsowntimevariable(t i),andberepresentedby

d X i

dt i = Fi



X(t i) 

where X i are the set of variables of the ith model, and Fi is some function of the complete system’s variables, X= {X1,X2, ,X N} It is important to make clearthe distinction between the characteristic timescale ofthe ith model in isolation(T i)andthetimescaleofitsvariableswithinthecoupledsystem;theyare,potentially,completelydifferent

Tosolvethissetofmodels,theindependenttimevariablesmustberelatedtoeachother.Ifalltimevariablesareequal (i.e.t=t1 N) thesystemisconventionallycoupled.However, wecan advancemodelsatdifferentrateswiththephysical modification

where g i is therate that the ith model advances relative to the micro model Thisapproximation provides a means to exchangefinetimescaleresolutionforlongtimescalepredictions,andtheextenttowhichitisvaliddependsonthedegree

ofscaleseparationbetweenmodels,i.e.onthemagnitudeof S i.Forcoupledmodels thatarehighly scaleseparated(S i>

Stol),thesmaller-scalemodelwillremainquasi-equilibrated tothedynamicsofthelarger-scalemodeldespitethephysical modification, and so behave similarly to as in the unmodifiedsystem The aim is thus to represent the scale-separated system(>Stol) withonethatisless,butstillsignificantly,scaleseparated(=Stol):thisishowacceptablevaluesofg i are determined(seebelowforthespecificprocedure).Detailedanalysesoftheerrorassociatedwiththisphysicalapproximation foratwo-modelsystemaregiveninEetal. [5]andLockerbyetal. [6]

Fig 1 provides an illustrationofa numericalimplementationofEq (4)usingdifferenttimestep sizes foreach model, whileexchangingvariablesasifthetimestepswereequivalent(thisisasynchronous coupling).Thetimestepoftheithmodel is

wheret1 isthemicromodeltimestep

The aimofthisasynchronous couplingschemeistomaximisethe totalsimulatedperiod.Thisisdone by maximising thetimestepineachmodelsubjecttothefollowingconstraints:

Fig 2 A coupled mass–spring system.

1 ThephysicalapproximationofEq.(4)representsaveryscale-separatedsystembyonethatisless,butstilltoadegree, scaleseparated(i.e.hasascaleseparationofStol).Thisplacesanupperlimitontheamountatimestep ofonemodel canbeincreasedrelativetoanother:

ti≤S i−1

Stolti−1.

2 Thenumerical accuracyissatisfactoryandstabilityguaranteedforeachindividualmodel:

ti≤ t i ,max,

wheret imaxisanestimationofthemaximumtimestepthatispermissibleforeachmodel

Basedontheseconstraintswecansetthetimestepofeachmodelrecursively:

ti=min



ti ,max;S i−1

Stol

ti−1



wheret1= t1,max

Wenowconsideraseriesofexamples.TheexamplesofSections 3–5areusedtoillustratetheeffectiveness,challenges, andshortcomingsofemployingthephysicalapproximationinEq.(4)andapplyingtimesteppingofEq.(6);theexampleof Section6providesademonstrationofanapplicationtohybridcontinuum-molecularmodelling.Note,thereare arangeof numericalmethodsthatcouldbeappliedtothenumerically-stiffsystemsofSections3–5,butwhichcannotbeappliedto thehybridexampleofSection6

3 Example 1: A simple mass–spring system

Aserialmass–spring systemwith N massesandN+1 springsisshownin Fig 2.Thegoverningequationsfortheith

modelare

d

dt



v i

x i



=

m i(ki(xi−1−x i) +k i+1(xi+1−x i))

v i



wherex i and v i arethedisplacement andvelocity oftheithmass(m i)andk i isthe ithspringconstant Acharacteristic timescaleforeachmodelcanbeobtainedfromitsnaturalperiodinisolation:

T i=2π



m i

k i+k i+1

InthisexampleN=5,thespringconstantsareequal,andthe(i+1)thmassis900×heavierthantheithmass,suchthat

Allmodelsarethussignificantlyscale-separated.1

Thecompletesystemofequationsissolvedusingthemidpointmethod(asecond-orderRunge–Kuttascheme),butwith timestepsforeachmodelchosenaccordingtoEq.(6),witht imax=T i/100.

Inthefirstcase,theinitialdisplacementsx i =0 arechosensuchthat,whenthemassesarereleasedfromrest,onlythe slowest eigenmode of thesystem isexcited. Fig 3(a) showsthe response ofthe lightest (m1) and heaviest (m5) masses governedbythemicroandmacromodels,respectively;theanalyticaleigenvaluesolution(thedashedline)is indistinguish-ablefromthe multiscalesolution. Fig 3(b)showsthesolution using Stol=5, whichplaces alessconservativerestriction

onthebalancebetweenaccuracyandefficiency.ForStol=10,themodeladvances81×fasterthanifstandard(synchronous) couplingwereused;forStol=5 itadvances1296×faster

The massesare now initiallydisplaced an equal distanceand thenreleased fromrest; thisexcitesall eigenmodes, to

a degree Fig 4 shows: (a) the response of a meso model (i=3) and(b) the response of the macromodel (i=5,the heaviestmass),forStol=5.Themacrodescriptionisaccurate,whileforthemesomodelonlythelowfrequencyeigenmode

isaccuratelycaptured.Thisexampleillustrates thefundamentaltrade-off requiredinmultiscaling:efficiencyinpredicting macrovariationscanbedramaticallyincreased,butattheexpenseofmicro/mesoscaleresolution

1 For clarity, we have chosen characteristic timescales of the models to increase withi sothat Eq (1) is satisfied without having to change index notation.

Fig 3 Normaliseddisplacement response of the lightest and heaviest masses(m1 andm5, respectively) to excitation of the slowest eigenmode The multiscale result (—) and an analytical eigenvalue solution (– –).

Fig 4 Normaliseddisplacement response of (a) the median and (b) the heaviest masses(m3 andm5, respectively) to an equal initial displacement The multiscale result (—) and an analytical eigenvalue solution (– –).

4 Example 2: A Lotka–Volterra system

TheLotka–Volterraequations,invariousforms,havebeenappliedtoanextremelydiverserangeofproblems,spanning economics [7], biology [8]and chemistry [9] Originating fromthe analysisof auto-catalytic chemical reactions [9], the equations arenowmostcommonlyusedto studythe populationdynamicsofcompetingbiologicalsystems,whichis the exampleweconsiderhere

Thepopulationgrowthrateofaspeciesina(sequential)foodchainisasfollows:

d y i

where y i isthepopulationsizeoftheithspecies, r iistheintrinsicdeathrate(intheabsenceofanypreyorpredator), p i

is thepopulationgrowthrateduetothe consumptionoflowerspeciesinthefoodchain,andq i isthedeathratedueto predationfromhigherspeciesinthefoodchain.Theintrinsicdeathrateofeachspeciesdefinesacharacteristictimescalefor that speciesmodel(r i=1/ i),andwhichclassifiesthemodelinthemacro-to-microhierarchy;thisrateincreasesmoving

upthefoodchain.Thus,theApexPredator,atthetopofthefoodchain,hasthehighestintrinsicdeathrate(intheabsence

ofprey),anditspopulationsize, y1,isgovernedbythemicromodel

Table 1 gives parameters for a food-chain example consistingof four species For the initial population sizes Y i the ecosystem is in equilibrium, and the numbers ofeach specieswill remain constant If, however, all plants are removed

(Y4=0),thepopulationsoftheremainingspecieswilleventuallyreducetoextinction. Fig 5showsthepopulationresponse

ofeachspeciesintheecosystem,aspredictedbyastandardnumericalsolution(thedashedline,usedasabenchmark)and themultiscaleapproachusingStol=10 (thesolidline).AsinSection3,timeintegrationisperformedusingasecond-order Runge–Kuttamethod,withtimestepsforeach modelchosenaccordingtoEq.(6),withti max=T i/5.Forthebenchmark solutiont i= t1,max

ThefirstobservationisthattheslowlyvaryingpopulationsizesoftheApexPredatorsandtheHerbivoresarepredicted veryaccuratelybythemultiscalescheme,whichis100×computationallyfasterthanthebenchmarksolution. Fig 6shows the macromodelpredictionusinglessconservativetolerances ontheminimumacceptablescale separation,i.e Stol=2.5 and S =5 (whichare1600× and400×fastertocomputethanthebenchmark,respectively).Asexpected,themultiscale

Table 1

Lotka–Volterra parameters for a 4-species food chain.

Trophic level Model i Y i(initial population) r i p i q i

Fig 5 Normalisedpopulation response to the instantaneous removal of Plants The multiscale result (—) and a conventional (benchmark) numerical solu-tion (– –).

Fig 6 NormalisedHerbivore population response to the instantaneous removal of Plants:Stol=10 (—), Stol=5 (– –),Stol=2.5 (–·–), and a benchmark numerical solution (•) Note, for clarity, the benchmark solution is not plotted at every timestep.

solutionconvergestothenumericalbenchmarkasStol isincreased;Stol=10 appearstoprovideaveryaccurateresultfor themacrovariable

However, compared to theother species, thePredators’ population decline israpid,and occursaftersome delay.The delayoccursbecause,initially,thePredators’preyand thePredators’predatorsarebothreducing–onlywhentheirpreyis significantlydiminishedisthereamajorreductioninPredatornumbers.Themultiscaleapproachdoesnotcapture,withany fidelity,theseshorterscalephenomena,andactuallyintroduceserroneousshorttimescaleoscillations.Thisagainhighlights that exploitingscale separationaffordsvery efficientpredictiononlarge timescales,butatthe expenseoffinertimescale resolution.Note,inthiscase,themicromodelpredictionisverygood,becausetheshortscaleresponseonlymanifestsitself

inthemesomodel’svariable

5 Example 3: A lubrication system

Inthissection weconsider air-layerlubricationofa liquidjournalbearing,asdepictedin Fig 7.The airlayer,despite beingthin,cansignificantlyreducetheoveralldragonthebearing,duetothelowerviscosityofairrelativetoliquids.This simpleair-layerlubricationconceptisexploitedinsuper-hydrophobiccoatings,whichhaveachemicalhydrophobicityand

Fig 7 Schematic of a liquid journal bearing with a lubricating air layer.

surfacetopologythat,whensubmergedinwater,combinetotrapairpocketsonthesurface.Suchcoatingshaveapplications

inmarinedragreduction [10,11]andforself-cleaningsurfaces [12].Inthecontextofmultiscalemodellingtheyarerelevant becauseoftheverydifferentscalesassociatedwiththeairlayer,externalwater,andthebody/vehicle

The bearingexampleofthissection consistsofasteelcylinder,ofradius R=10 cm,towhichisappliedanoscillatory torque, T.Thecylinderrotateswithin a fixedouter cylindercontaining water;the surfaceofthe innercylinderiscoated withan airlayerofthicknessair=1 μm; andtheannular thickness ofwateris wat=0.1 mm (i.e R wat air); see Fig 7

Thelow-speed,unsteady,incompressibleNavier–Stokesequationsprovidemodelsfortheairlayerandthewater(i.e.the microandmesomodels,i=1 andi=2,respectively):

∂vair

∂t1 = μair

ρair

∂2vair

and

∂vwat

∂t2 = μwat

ρwat

∂2vwat

where r is the radial coordinate fromthe cylinder centre, v is tangential velocity, μ is dynamic viscosity, ρ is density, andthesubscripts‘air’and‘wat’denotetherespectivefluids Giventhat R wat air,thecurvatureofthegeometry can beneglected.The microandmeso modelsarecoupledbytherequirementfortheshearstressandthevelocitytobe continuousattheair–waterinterface(assumingnoslip):

μair

dvair

dr

r=rint

= μwat

dvwat dr

r=rint

and

wheretheradialpositionoftheair–waterinterfaceisrint=R+air.Thewateratthewalloftheoutercylinderisstationary (i.e.thereisnoslip)

Newton’ssecondlawdeterminestheevolutionofthetangential velocityoftheinnercylindersurface (vcyl);thisisthe macromodel(i=3):

∂vcyl

∂t3 = R

I



2πL R2 μair∂vair

∂r

r=R

+ T (t3)



where L isthe lengthofthebearing(into thepage)and I isthemomentofinertia ofthecylinder.The appliedtorqueis

T =Asin(ωt3),where ωistheangularfrequencyand A istheamplitude.Themacromodeliscoupledtothemicromodel through shear stress inthe airlayer atthe cylinderwall (i.e.the termin parenthesisin Eq.(15)) and viano-slip atthe cylinder–airinterface:

See Appendix Aforvaluesofthephysicalparametersusedinthisexample

The characteristictimescalesestimatedforeach modelarethe viscoustimescale (forthetwo fluids),andafractionof thetorqueperiod(forthecylinder):

T1= ρair2air

μ ; T2= ρwat2wat

Fig 8 Tangentialvelocityv [m s−1 ] developing in macro timet3[s] for the cylinder wall and the air–water interface Response to an oscillatory cylinder torque.Stol=20 (•),Stol=10 (—), andStol=5 (– –) Note, for clarity, theStol=20 result is not plotted at every timestep.

In thisexample, the disparity intimescales is vast: T3/ 1∼109.For consistency withprevious examples, the midpoint method is used for time-advancement, with timestep sizes determined by Eq (6)(see Appendix A for tmax,) Spatial discretisationofthefluidmodels,i.e.ofEqs.(11)–(12),isperformedusingasecond-ordercentral-differenceapproximation; forthisillustrativeexample,only10gridpointsareusedineachfluidlayer(afinermeshdoesnotsubstantiallychangethe results)

Fig 8showsthe variationofthe velocityofthe cylinderwall,andthevelocity oftheair–water interface, withmacro time The velocity of the cylinder wall is almost 50% higher than that of the air–liquid interface (which would be the approximatevelocityofthecylinderwallifnoairlayerwerepresent);thedragcoefficientofthecylinderinwaterhasbeen reducedbyalmost50%duetothepresenceofthethinairlayer

Hereitisnotpracticaltobenchmarkthemultiscaleresultsagainstaconventionalnumericalsolution(i.e.onewithequal timestepsizes),becauseofthehighcomputationalcosttoobtainthelatter.Instead,andwhatmustbedoneinpractice,is

toshowtheindependenceofthemultiscaleresulttoincreasesin Stol.Thisisakintoagrid-dependencystudy–settinga larger Stol hastheeffectofreducingthedifferencebetweentimestepsizes

Fig 8showsresultsfor Stol=5,10, and 20;theresultsfor Stol=10 and 20 arebarelydistinguishable,indicatingthat

Stol=5 isafairprediction,andStol=10 isaveryaccurateone.Thecomputationalspeed-upaffordedbytheasynchronous timestep coupling is in this caseextremely high: ×9.5·107 (for Stol=5);×1.2·107 (for Stol=10); and×3·106 (for

Stol=20)

Nowweconsiderthesuddenapplicationofaconstanttorque,T =1,tothestationarysystem.Thiscasehighlightsthe potentialdifficultiesinidentifyingcharacteristictimescales.Themacromodel,Eq.(15),doesnothaveaninherenttimescale

intheabsenceofanoscillatorytorque.Inotherwords,inisolation(i.e.withoutairorwater),thecylinderwouldperpetually accelerate inresponseto theconstant torque.In thesecircumstancessome estimate ofthe timescaleof themodelwhen interacting with others, is needed Here we achieve this withan approximation of the acceleration and velocity of the cylinderwallintermsoftheair-layershearstress,andcombinethemtogetatimescale

Intheabsenceofanappliedtorque,theaccelerationofthecylinderwallswillbeproportionaltotheshearstressinthe airlayeratthewall(τwall),andinverselyproportionaltothemomentofinertiaofthecylinder(seeEq.(15)):

∂vcyl

∂t ∝L R3τwall

Ifwe assume a linearvelocity profile inthe airlayer, thevelocity of thecylinder wallwill be proportional tothe shear stressandtheair-layerthickness,butinverselyproportionaltothedynamicviscosity,i.e

vcyl∝ τwallair

DivisionofEq.(19)by (18)givesacharacteristictimescalethatwecanuseinoursimulation:

T3= airρcylR

where ρcyl isthedensityof thesteelcylinder.With theexception ofthismacrotimescale T3,andt3,max=T3/200,all otherparametersarethesameasabove

Fig 9showsthevelocityresponseofthecylinderwallandair–waterinterfacetothesuddenly-appliedconstanttorque Again,calculationsareperformedusingStol=5,10, and 20,withStol=10 providingaresultthatappearstobeinsensitive

tofurtherincreasesofS Thissolutionisachieved×6.5·106fasterthanasolutionusingequaltimesteps.Thecharacteristic

Fig 9 Tangentialvelocityv [m s−1 ] developing macro timet3[s]for the cylinder wall and the air–water interface Response to a suddenly-applied constant cylinder torque.Stol=20 (•),Stol=10 (—), andStol=5 (– –) Note, for clarity, theStol=20 result is not plotted at every timestep.

timescale predictedby Eq.(20), T3=78.5 s,is reasonable giventhe observedtimescales Evenso,if thispredictionhad been much different, the main consequence wouldbe that the Stol-independence thresholdwould be different, and the dependencystudymighthaverequiredadditionalsimulationstofindthatthreshold

6 Example 4: A Knudsen compressor

Finally, weconsider therarefiedgas flowbetweentwo reservoirs, heldatdifferent temperatures,connectedby athin cylindricalcapillary:asingle-stageKnudsencompressor,see Fig 10.Rarefactioneffectsinthecapillarytransportgasfrom the cold to the hot reservoir; thiscounter-intuitive phenomenon is thermal transpiration(sometimes known asthermal creep)andwasfirstobservedbyReynolds [13].Theconfigurationshownin Fig 10wasconstructedbyRojas-Cárdenasetal

[14] inordertostudythetransientbehaviourofthermaltranspirationinaclosedsystem;someoftheirexperimentaldata

ispresentedbelow

In terms ofsimulation, thissystemcannot be modelled usingstandard Navier–Stokes equationsand boundary condi-tions, sincethermaltranspirationisathermodynamic non-equilibrium phenomenon [15,16].Ontheother hand,anaccurate gas-kinetictreatmentwouldbecomputationalintractableovertheentiredomain.Totacklethis,wedecomposethesystem intothreecoupledmodels,applyingtheappropriatemodellingassumptionstoeach:thereservoirmodel(macro,i=3);the capillarymodel(meso,i=2);andthegas-kineticmodel(micro,i=1);see Fig 10.Themacromodeldefiningthereservoir pressuresisobtainedfrommassconservationandbyassuminganidealgas:

dp c

dt3 = − R θc

V c m˙(=0),

dp h

dt3 = − θh

θc

V c

V h

dp c

where p ispressure,Risthegasconstant,θ istemperature, V isthereservoirvolume,m is˙ themassflowratealongthe capillary,z isdistancealongthecapillaryfromthecoldtohotreservoir,andthesubscriptsc and h denotethecoldandhot reservoirs,respectively.Note,hereweassumethatthereisnosignificantchangeinmassofgaswithinthecapillary,though thiscaneasilybeaccountedforifnecessary

The meso modelforthehigh-aspect-ratio capillaryisobtainedfromthe continuityequation integratedoverthe cross section:

∂p

∂t2+ R θ

A

∂m˙

where A is the cross-sectional area of the capillary The meso model is coupled to the macro model by the boundary conditions: p=p c, m˙ = ˙m (=0) at z=0; and p=p h at z= , where is the length of the capillary The temperature variationalongthecapillaryisprescribed(usingthesamefitasinRojas-Cárdenasetal. [14])by

θ = θc+ (θh− θc) 

e α z−1

e α −1

where αisaconstant

Themicromodel,G,providesameanstoclosetheentiresystem,byrelatingmassflowratetopressureandtemperature:

∂m˙

∂t = G ∂θ

∂ ; ∂p

∂ ; X

Fig 10 Schematic of a multiscale simulation strategy for the single-stage Knudsen compressor experimental configuration of Rojas-Cárdenas et al.[14]

whereX containsinformationregardingthemolecularstructureofthegas,whichisrequiredtoaccuratelymodelthermal transpiration Here,themicro modelG isa spatially-distributedarray oflow-variance deviationalsimulation MonteCarlo (LVDSMC)subdomains(see Fig 10).LV-DSMCisaparticularlyaccurateandlownoisemethodforsimulatingsmalldeviations fromequilibriuminrarefiedgasflows [17,18].Usingmicroparticle-simulationsubdomainstorepresentpointsinthemeso domain is substantially more efficient than modelling the entire channel with a single particle simulation – thisis an applicationof the Internal Multiscale Method (IMM),and readersare referred to [19–21] fora detailed description.The simulatedparticles of each (streamwise periodic)subdomain are forcedby an effectivebody force, which representsthe equivalentpressureandtemperaturegradientoccurringatthatlocationinthemesomodel(thepressureandtemperature arealsoset).Themicromodelisthuscoupledtothemesomodelbythestreamwisepressuregradient,pressure,andmass flowrateateach ofthesubdomainlocations.Forthesimulationswe presenthere, 12subdomainsareused.Foraccuracy, thederivatives inz featuring in Eqs.(22)and (24)are evaluated fromaChebyshev polynomialinterpolation of p and m˙ fromsubdomainlocationscorrespondingtoChebyshev–Gauss–Lobattopoints

Viscousdevelopmentwithinthecross-sectionofthecapillarydefinesthecharacteristictimescaleofthemicromodel

T1= ρR

2

cap

where Rcap isthe capillaryradius and ρ and μare the average initial densityandviscosity ofthe gas.If we assume a quasi-steadyvelocityprofile(whichisonlyvalidfortT1),characteristictimescalesofthemesoandmacromodelcanbe estimatedfromEqs.(22)and (21),respectively:

T2= μ 2

and

T3= μ Vt

p R4

cap

wherep istheaverageinitialpressureandV t isthetotalvolumeofthecombinedreservoirs

TheexperimentsofRojas-Cárdenasetal. [14]wereperformedwithArgongas,aborosilicate(glass)capillaryofcircular cross-sectionwith =52.7±0.1 mm,Rcap=242.5±3 μm connectingtworeservoirsofvolume V c=19.81±0.54 cm3and

V h=14.85±0.40 cm3,heldatθc=301 K andθh=372 K Aheater appliedtothehotreservoirgeneratedatemperature distribution throughthe capillaryfittedby Eq (23), with α =84.82 m−1.Initially, thetwo reservoirs were held atfixed pressure(p=237.7 Pa),allowing thermaltranspirationflowtodevelop;thesystemwas thenclosed,andthepressurein thetworeservoirsallowedtoequilibrate.Note,inagasthatisnon-rarefiedtheinitialstatewouldbetheequilibriumone

WeranmultiscalesimulationsofthisexperimentalconfigurationusingEulertime-stepping,forsimplicity,withtimestep sizeschosenbyEq.(6).Ourcompletesimulationparametersareprovidedin Appendix B

Fig 11 Comparisonof experimental data and the multiscale solution, for transient development of a Knudsen compressor; a plot of reservoir pressures versus time Both reservoirs are held at a constant pressure until approximatelyt3=18 s, at which point the reservoirs are instantaneously closed to the environment Experimental data of Rojas-Cárdenas et al [14] (×) and the multiscale simulation (—).

Fig 12 Reservoir pressure versus time for increasing values of Stol : 10 (· · ·); 20 (– –); 30 (–·–); 40 (—).

Fig 11showsthetransientresponseofthepressureineachreservoirafterthesystemisinstantaneouslyclosed;thereis verycloseagreementbetweentheexperimentalmeasurementsandthemultiscalesimulation.Theasymmetryofthefigure around theinitial pressureis causedbythe differentreservoirvolumes(V c>V h).The multiscaleresultisobtainedwith

Stol=10, andrequired theuse oftwelve Intel Xeon X56502.66GHz coresfor 4.13hours (wall-clock time).If thetime stepsweresetequal(i.e.aconventionalsynchronouscoupling)thesimulationwouldhavetakenover20yearsonthesame hardware Infact, thesavingover conventionalmodellingis fargreater thanthis, ifwealso takeinto accountthesaving

Fig 12 showstheimpactofincreasing Stol;asimilarresultisobtainedinallcases,butwithlowernoiseathigherStol This highlights an important generallimitation of multiscalingwith stochastic models (e.g.LVDSMC) orinherently noisy methods (e.g Molecular Dynamics): fewer timesteps means lesssampling, and thus morenoise The trade-off inhybrid (continuum-particle)multiscalingcanthusbesummarised:

fine-scale accuracy↔noise & large-scale prediction.

Fig 10 Schematic of a multiscale simulation strategy for the single-stage Knudsen compressor experimental configuration of Rojas-Cárdenas et al.[14]... Internal Multiscale Method (IMM),and readersare referred to [19–21] fora detailed description.The simulatedparticles of each (streamwise periodic)subdomain are forcedby an effectivebody force,... thetworeservoirsallowedtoequilibrate.Note,inagasthatisnon-rarefiedtheinitialstatewouldbetheequilibriumone

WeranmultiscalesimulationsofthisexperimentalconfigurationusingEulertime-stepping,forsimplicity,withtimestep sizeschosenbyEq.(6).Ourcompletesimulationparametersareprovidedin