Comprehensive assessment of newly developed slip jump boundary conditions in high speed rarefied gas flow simulations

Comprehensive assessment of newly developed slip jump boundary conditions in high speed rarefied gas flow simulations Aerospace Science and Technology 91 (2019) 656–668 Contents lists available at Sci[.]

aDivison of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

bFaculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

cHigh Performance Computing (HPC) Laboratory, Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, P.O Box

91775-1111, Mashhad, Iran

dFaculty of Mechanical Engineering, Industrial University of Ho Chi Minh City, Viet Nam

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

Article history:

Received 18 January 2019

Received in revised form 23 May 2019

Accepted 3 July 2019

Available online 9 July 2019

Keywords:

Rarefied gas flows

Slip-jump boundary conditions

Aoki et al conditions

Slip velocity

Surface gas temperature

In thispaper wenumericallyevaluate the recentlydeveloped Aoki etal.slipand jump conditionsin high-speed rarefiedgas flows forthe first time Theseslipand jump conditions are developed tobe employedwiththeNavier–Stokes–Fourierequations.TheywerederivedbasedontheBoltzmannequation withthefirstorderChapman–Enskogsolution,andtheanalysisoftheKnudsenlayer.Fouraerodynamic configurationsareselectedforacomprehensiveevaluationoftheseconditionssuchassharp-leading-edge flat plate, verticalplate, wedgeand circular cylinder incross-flow withthe Knudsen number varying from0.004to0.07,andargonastheworkinggas.ThesimulationresultsusingtheAokietal.boundary conditionsshow suitableagreementwiththeDSMCdataforslipvelocityandsurfacegastemperature TheaccuracyoftheseboundaryconditionsissuperiortotheconventionalMaxwell,Smoluchowskiand

Leboundaryconditions

©2019ElsevierMassonSAS.Allrightsreserved

1 Introduction

Rarefiedgasflowgenerallyhasfourdistinctregimes.Theyare

characterizedaccordingtotheirKnudsennumber,Kn,thatitis

de-finedastheratioofgasmeanfreepath,i.e., theaveragedistance

amoleculemovesbetweensuccessiveintermolecularcollisions,to

acharacteristiclengthofthevehiclebody.Thecontinuum regime

correspondstoverysmallKnnumber,Kn≤0.001.Theslipregime

withthetemperaturejumpandslipvelocityconditionsatthe

sur-face is indicated by the range 0.001 ≤ Kn ≤ 0.1 When the gas

character-izedasthetransitionandfreemolecularregimes,respectively.The

transition-continuum regime corresponds to 0.1 ≤ Kn ≤ 1, and

the free molecular regime to Kn ≥ 1.Two typical methods have

beenusedtosolvetherarefiedgasflowssuchasDirectSimulation

Monte-Carlo(DSMC)andComputationalFluidDynamics(CFD).The

DSMCmethodhassuccessfullysimulatedtherarefiedgasflowsfor

fourregimesaforementioned,butitscomputationaleffortisquite

expensiveatsmallKnudsen numberconditions.The CFDmethod

that solves the Navier–Stokes–Fourier (N–S–F) equations

accom-* Corresponding author.

E-mail addresses:letuanphuongnam@tdtu.edu.vn (N.T.P Le),

e.roohi@ferdowsi.um.ac.ir (E Roohi), tranngocthoai@iuh.edu.vn (T.N Tran).

panied withappropriate slipandjump boundary conditionsmay successfullysimulatetherarefiedgasflowsintheslipregimeand even beyond.The slipandjumpconditionsplay an essentialrole

intheaccuratepredictionofthesurfacequantities.Duringthelast decades, several slip and jump boundary conditions were devel-oped basedonthekinetictheoryofgases,theLangmuirisotherm adsorption,andcombinationoftheLangmuirisothermadsorption andkinetictheoryofgasesin[1 9] toworkwiththeN–S–F equa-tionstosimulatetherarefiedgasflows.However,theyhavenotyet predictedwellthesurfacequantitiesinrarefiedgassimulations

macroscopicequations.In[10,11] theslipandjumpboundary con-ditions havebeenrecentlyderived fromtheBoltzmannequations

on the basis of the first-order Chapman–Enskog solution of the Boltzmann equation, andthe analysisofthe Knudsen layer adja-cent to theboundary These conditionswere developedfor large densityand temperaturevariation toemploy withthe compress-ibleN–S–Fequations.Theywerederivedfortherarefiedgasflows appliedtomonatomicgasin[10],andpolyatomicgasin[11].They have been used andevaluated for the numerical analysis of the Taylor-vortexflow in[12].Inthispaper,we onlyfocusonthe re-visitandassessmentoftheslipandjumpboundaryconditionsfor themonatomicgasin[10]

AsfortheAokietal.slipandjumpconditionsderivedfor poly-atomicgasesin[11],weneedtodeterminethetermrelatedtothe

https://doi.org/10.1016/j.ast.2019.07.005

1270-9638/©2019 Elsevier Masson SAS All rights reserved.

Table 1

List of all CFD cases.

Sharp-leading-edge flat plate 0.0042 4 Argon

Circular cylinder in cross-flow 0.01; 0.05 10 Argon

Sharp-leading edge wedge 0.05 10 Argon

collision frequency of the gas molecules thattheycannotbecalculated

inCFD The surface quantities such as the pressure,temperature

andslipvelocityusingtheAokietal.slipandjumpconditionsare

comparedwiththose usingthe conventionalMaxwell slip

condi-tion,theSmoluchowskijumpcondition,andDSMCdata.Recently,

thetemperaturejumpboundaryconditionsweredevelopedbased

onthe kinetic theory of gases considering the viscous heat

gen-erationin[13,14].Thesejumpconditionsimprovedtheprediction

ofthesurfacegastemperaturesincomparingwiththoseusingthe

classicalSmoluchowskijumpcondition,andgavegoodagreement

withthe DSMC data So, the surface gas temperatures using the

Aokietal.jumpconditionarealsocomparedwiththoseusingthe

temperaturejumpcondition consideringthe viscous heat

genera-tion[14].Fouraerodynamicconfigurationsincludingthesharpand

blunt bodiesare selected to assessthe Aoki etal.slip andjump

conditionsin predicting thesurface quantities, andthey are

pre-sentedinTable1

2 Slip and jump boundary conditions in CFD and slip quantities

in DSMC

The temperatureand velocity of the rarefied gas ata surface

are not equal to the wall temperature andthe wall velocity,

re-spectively.This resultsin theslip andjump boundaryconditions

needtobederivedinsimulationsoftherarefiedgasflows.Inthis

section, we revisit the classical slipandjump conditions inCFD

TheconventionalMaxwellslipvelocitycondition,includingthe

ef-fectofthecurvatureandthermalcreep,canbeexpressedas[1]:

u+



2− σu

σu



λ ∇n(S·u)

=uw−



2− σu

σu



λ

μS· (n·
mc) −3

4

μ ρ

S· ∇T

wheretensor S=I-nn,wheren istheunitnormalvectordefined

aspositive in the directionpointing out ofthe flow domain,

re-moves normal components of any non-scalar field, e.g., velocity,

sothatsliponlyoccursinthedirectiontangential tothesurface;

thesymbol‘·’istheinner product;λisthemeanfree path; μis

theviscosity; ρ isthedensity;T isthetemperature;u istheslip

velocity; and uw is the wall velocity The tangential momentum

accommodationcoefficientdeterminestheproportionofmolecules

reflectedfromthesurface specularly(equalto1− σu)ordiffusely

(equalto σu),and0≤ σu≤1.Tensormc= μ (( ∇u)T−2

3Itr(∇u));

superscriptTstandsthe transpose;andtr isthe trace.The

right-hand side of equation (1) contains 3 terms associated with (in

order): thewall velocity, theso-calledcurvature effect,and

ther-malcreep The Maxwellian meanfree path isdefined asfollows

[15]:

ρ



π

whereR isthespecificgasconstant.Theviscosityiscalculatedby

theSutherlandlaw,

μ =AS T

1.5

wherethecoefficientsAS=1.93×10−6 Pa s K−1 2andTS=142 K forargon[3

In rarefied gas flows, the gas temperature ata surface is not equal to the wall temperature, and this difference is called the temperaturejump Theclassical Smoluchowskitemperaturejump condition isderived by the heat flux normal to the surface, and canbewritten[2]:

T+2σT

σT

2γ

where γ isthe specific heatratio; Pris thePrandtl number; Tw

isthewalltemperature,and σT isthethermalaccommodation co-efficientthatvariesfrom0to1.Perfectenergyexchangebetween thegasandthesolidsurfacecorrespondsto σT=1,andnoenergy exchangeto σT=0

Recently, the temperature jump conditions were derived by consideringviscousheatgenerationinheatfluxatthesurface[13,

14] The viscous heat generation was first introduced by Maslen

in[16] Themodified Patterson jumpcondition wasdeveloped in [14], so-called Le jump condition in the presentwork It predicted bettertemperaturesthanthenewtypeoftheSmoluchowskijump condition [13] inhigh-speedrarefiedgasflowsimulations There-fore,theLejumpcondition isadoptedforsimulations,andis ex-pressedasfollows[14]

T+1

2



2− σT

σT



γ

γ −1

Tw

T

λ

Pr∇nT

=Tw−1

2



2− σT

σT



1

c v( γ −1)

Tw T

λ

μ



S· (n·
) ·u

where c v is thespecific heat of thegas atthe constant volume; and
isthestresstensor.Thesecondtermintheright-handside

ofequation(5) presentstheviscousheatgenerationpart

Alternative slipand jump conditions were recently developed forrarefied gasflow basedonthe Boltzmannequationswiththe first-orderChapman–Enskog solutionoftheBoltzmannequations Moreover,theanalysisoftheKnudsenlayeradjacenttothe bound-ary, and kinetic corrections of the macroscopic quantities inside theKnudsenlayerwerealsoinvolvedinthederivationoftheslip andjumpconditions.Theywerederived formonatomicand poly-atomic gases in [10,11] with the accommodation coefficients of unity,theso-called Aoki et al slip and jump conditions inthepresent work.Theyarepresentedinequations(6) and(7) respectively,as [10]

u+



2

R av1

μw

ρ √

Tw∇n(S·u) =uw+ 4

5R aTI

kw

ρTw(S· ∇T), (6) and

T+ 2

5R



2

R aTII

kw

ρ √

Tw∇nT=Tw+ 1

R avII

μw ρ



( ∇u·n) ·n

where kw and μw are the thermal conductivityand viscosity at the wall temperature, respectively.For thehard-sphere molecule, the coefficients (avI, avII, aTI, aTII) in equations (6) and (7) have the values as follows[10] avI = 0.98733; avII = 0.36185; aTI =

0.33628;andaTII=1.24859

Thermal conductivity kw and viscosity μw are calculated as [10],

μw =0.17913618m

√

R

d2



and

Fig 1 Thegeometries, freestream conditions, and CFD numerical setups of cases a) sharp-leading-edge flat plate, b) vertical flat plate, c) circular cylinder, and d) sharp-leading-edge wedge.

kw=0.67783290mR

√

R

d2



wherem ismassofamolecule;andd isdiameterofmolecule

It is noticed that the classical Smoluchowski jump condition

(Eq (4))is independentof thevelocity of gasflow nearthe

sur-face,whilethe LeandAokietal.jumpconditions aredependent

thevelocityandgradientofthevelocityofgasflownearthe

sur-face.All slipandjump conditionsaforementioned(Equations (1

(4 (5 (6) and(7))are implementedinOpenFOAM[17] towork

withtheN–S–F equationsthat are numericallysolved usinga

fi-nite volumediscretizationandhigh-resolution centralschemesin

the solver rhoCentralFoam to simulate high-speed viscous flows

A calorically perfect gas for which p= ρR T is assumed in this

solver.Theimplementedapproachoftheslipandjumpconditions

inOpenFOAMwasdescribedin[3

Finally,the slip velocity andsurface gas translational

temper-aturein DSMCare treatedin thepost-process[18],andthey are

calculatedwiththeaccommodationcoefficientsofunityasfollows

[19]:

u=



((m/ n)u p)



T=



((m/ n) u2) −  (m/ n)u2

3k B

wherek B istheBoltzmannconstant;anduisthevelocity

mag-nitude The velocity normal to the surface, u n, and the velocity

parallel to the surface, u p, in equations (10) and (11) are taken

prior toandafterthecollision withthesurface Thesummations

include pre-collision and post-collision molecules Equation (11)

pointsoutthat thesurface gastranslationaltemperatureinDSMC

isaddressedasafunctionofvelocity

Table 2

Mesh cell sizes of all CFD cases.

3 Numerical setup

In OpenFOAM the model is often built in three-dimensional Our CFD andDSMC two-dimensional simulations are carriedout

by applyingthe condition“empty”to patchesthat donot consti-tute thesolutiondirection Thegeometries,freestreamconditions, andcomputational domainsofall casesarepresentedinFigs 1a,

1b, 1cand 1d.Thefreestreamconditionsof(p, T ,u)areappliedto theinletboundary,andaremaintainedthroughthecomputational process.ThezeroGradientconditionisappliedtootherboundaries

so that fluid is allowed to leave the computational domain This condition specifiesthatthenormalgradientsoftheflowvariables

(p, T ,u)vanishatthoseboundaries.Thesimulationsofthe circu-larcylinderincross-flowarecarriedoutforthecylinderforebody, andtheirfreestreamconditionsareadoptedin[20]

The hexahedral structured mesh is selected forall cases.The computationalmeshisconstructedtowraparoundtheobliqueand bow shocks The mesh independence is conducted to obtain the final mesh forall cases Thesmallest cell sizes nearthe surfaces

ofallCFDcasesarepresentedinTable2.Therectangularmeshis usedforthesharp-leading-edgeflatplateandverticalplatecases Themeshoftheflatplatecaseisquitesimple,andonlythatofthe verticalplatecaseispresentedhere.Typicalmeshesofthevertical plate, cylinderandwedge casesare found inFigs 2a, 2b and 2c, respectively

Various slipandjumpboundary conditionsare appliedto the surfaces.TheKnudsen-layer correction[10] wasnot implemented

inourCFDsimulations.Weusethreedifferentslipandjump

mod-Fig 2 Typical structured meshes of a) the vertical plate, b) circular cylinder (Kn=0.01) and c) wedge cases (every fifth line presented).

elsfortheCFDsimulationsinthepresentworkasfollows:1)the

classicalMaxwell-Smoluchowskiboundaryconditions(BCs),2)the

Aokiet al.slip andthe Le jump BCs,and 3) the Aokiet al slip

simu-lations of cases The solver dsmcFoam in OpenFOAM is used to

run the DSMC simulations The hard-sphere molecular model is

adoptedforall DSMC casesbecause thecoefficients (avI, avII,aTI,

aTII) oftheAokietal.slipandjumpconditionsare computedfor

thehard-sphere molecule A structured mesh isalso used inthe

DSMC simulations.The cell size is determined by the freestream

meanfree path, λ∞,that it is calculatedby the freestream

(pre-shock)gas flow conditions Acell size of approximately λ∞/3is

usedinallDSMCsimulations.The tangentialandthermal

accom-modationcoefficientsofunityareusedforallDSMCandCFD

sim-ulations

4 Simulation results

Thesimulationresultssuchasthesurfacepressure,surfacegas

temperature,slipvelocityareplottedagainstthecylinderangle,φ,

for the cylinder cases, andthe normalized distances, where dis-tances arenormalized by their lengths,for theflat plate,vertical plate and wedge cases Moreover, the temperature and velocity magnitude contours of the DSMC solutions and CFD simulations usingtheAokietal.conditions,arealsopresentedinthissection forallcases

4.1 Sharp-leading-edge flat plate case, Kn=0.0042

Thesimulationresultsofthesharp-leading-edgeflatplatecase arepresentedinFigs.3, 4, 5, 6, 7,and 8.Theyareplottedagainst

x L,where x runsfromthetip tothetailoftheflatplate, andL

is the length of the flatplate Fig 3presents the distribution of the surface gas pressure over the flatplate Atthe leading edge, surface gas pressures of all CFD runs obtain the peak values of 1) 26.5 PafortheMaxwell–Smoluchowski conditions,2)31.06 Pa for the Aoki etal and Le conditions, 3) 28.1 Pa for the Aoki et

al.conditions, and4) 13.51 Pafor theDSMC data Past thepeak location, all CFDandDSMC simulation resultsgradually decrease

Fig 3 Distribution of surface gas pressure over the flat plate Fig 4 Distribution of KnGLL over the flat plate.

Fig 5 TheKn GLL contours of CFD and DSMC simulations, a) Maxwell–Smoluchowski condition, b) Aoki et al – Le conditions, c) Aoki et al conditions, d) DSMC, and e) KnGLL contours near the leading edge of all simulations.

overtheflatplatesurface.Thereisagoodagreementbetweenall

CFDand theDSMC data inthe range x L≥0.1,while there isa

largedifferencebetweenall CFDsimulationresultsandtheDSMC

dataforx L≤0.1.Inordertoexplainthisone,thelocalgradient length (GLL) Knudsen numberis computedbased onthe density gradientofthegasflowinCFDasfollows,

Fig 6 Distribution of surface gas temperature over the flat plate.

Fig 7 Distribution of the slip velocity over the flat plate.

KnGLL= λ ∇ ρ 

where ∇ ρ  is magnitude of the density gradient The KnGLL is calculatedinDSMCasfollows,

KnGLL= λDSMC∇ ρ 

wherethetermλDSMC iscomputedas[20,21]

λDSMC=2(5−2ω )(7−2ω )

15



m

2πk B T



μDSMC ρ



,

whereμDSMC = μref



T

Tref

ω

where subscript ref denotes the reference values, Tref = 273 K,

ωisthemacroscopicviscositytemperatureexponent(ω =0.5for thehard-spheremolecularmodel),and μrefiscalculatedasfollows [20,21]:

μref= 15

√

mπk B Tref

2πd2ref(5−2ω )(7−2ω ) . (15)

ThevaluesofKnGLL overtheflatplatesurfaceareshowninFig.4 They obtain the peak values at the leading edge with the peak valuesof1)0.475forsimulationwiththeMaxwell–Smoluchowski conditions,2)0.55forsimulationwiththeAokietal.andLe condi-tions,3)0.48forsimulationwiththeAokietal.conditions,and4) 0.57fortheDSMCdata.Itisnotedthatthecontinuumbreakdown occurs whenever KnGLL > 0.05 [20,21] Considering large KnGLL value at the leading edge, the nonequilibrium effects are quite high there, andit is not expected that justimprovement in slip andjumpboundaryconditionsresultsinasuitable agreement be-tweentheCFDandBoltzmannsolutions.Inotherwords,theshear stress, which is described simply in conjunction with the linear

constitutive relation of gradients ofvelocity, could not represent thetruebehaviorofthegasviscouseffectsandboundarylayerat highlynon-equilibriumconditionexperiencedattheleading edge Thus,thereshouldbeadifferencebetweenthekineticand contin-uum descriptions of the boundarylayer in additionto the shock

Fig 9 Temperature and velocity magnitude contours for the vertical plate, a) temperature and b) velocity.

wavestructure[21].Boundarylayerandshockthicknessare

differ-entfromtheCFDandDSMCpredictions.TheDSMCpredictionfor

shockismorediffusivethantheCFDone, whichisquitesharper

Moreover,froma mathematicalview point,the modelofinviscid

gasneartheleadingedgeisasingularityfordifferentialoperation

Thisisalsotrueforviscousgasflowsbuttheeffectisweaker.This

modeldifferentiatesthegasflowvariationsandthat whythereis

apeakvaluethere.TheBoltzmannsolution(DSMC)hasanintegral

collision operator and integrates this singularity by the collision

operator.Inotherwords,itaveragesandsmearsinsomesensethe

solution,thisisthereasonthatDSMCshockwaveisusuallythicker

thanN–S–Fsolution.TheKnGLLcontoursofCFDandDSMC

simu-lationsarefoundinFigs.5a, 5b, 5c,and 5 toseehowslipmodels

changetheshapeoftheshockattheleadingedge.Magnitudesof

KnGLL arehigherattheshockwaveandintersectionofshockand

boundarylayer There are thecurvedshocks atthe leadingedge,

shownin Fig.5e.Theshockproducingby theCFDsimulation

us-ingtheAokietal.andLeconditionsismorecurvedthanthetwo

otherCFDshocksandDSMCattheleading edge.Thisisexpected

byobservingthepeakvaluesofthesurfacepressuresatthe

lead-ingedge that oneusingthe Aokietal.andLe conditionsobtains

thehighestvalue

Distribution oftheCFDandDSMCsurface gastemperaturesis

plottedinFig.6.ThesimulationresultsusingtheLe jump

condi-tionandtheDSMCdataobtainhighervaluesneartheleadingedge

whilethose usingthe Aokietal.andSmoluchowski jump

condi-tions obtainlower values.Theformationofa shock, andthe

cor-respondingtemperaturejumpacrossitoccursslightlydownstream

oftheplatetip,yieldingtheresultsinFig.6thattemperature

in-creases from the freestream temperature to a level downstream

of the shock [22] The results using the Le jump condition are

closetotheDSMCdataattheleadingedge.Thismaybeexplained

bythe significanteffectoftheviscous heat generationthere.The

simulationresults ofthe Aokiet al.andthe Smoluchowskijump

conditions underpredict temperature near the leading edge, and

generallygive good agreement withtheDSMC data over theflat

plateforx L≥0.1

The slip velocities are presented in Fig 7 All DSMC and

CFD simulation results predict the peak values near the leading

edge They are 1) 344.2 m/s for simulation with the Maxwell– Smoluchowski conditions, 2) 317.23 m/s for simulation with the Aoki et al.and Le conditions,3) 376.34 m/s forsimulation with the Aokietal.conditions, and4) 332.36 m/sfortheDSMC data Thereafter, all of them gradually decrease in x L≤0.1 In the range 0.1≤x L≤1, all CFD simulation results stay nearly con-stant overtheflatplatewhilethoseofDSMCstaynearly constant untilx L=0.9.AllCFDsimulationresultsgenerallyagreewiththe DSMC datain0.1≤x L≤1.Thedifference betweentheCFDand DSMC data forx L≥0.9 may be affected by theflow separation nearthetrailingedge.TheDSMCmethodcancapturetheflow sep-arationwhiletheCFDsimulationresultsmaynot.Finally,Figs 8a and 8b presentthetemperatureandvelocity magnitudecontours for the CFD and DSMC simulations, respectively The DSMC and CFDsolutionsperformdifferentlyintheboundarylayerandshock wave regions.Attheleading edge,aboundarylayerisdeveloped, andacurvedshockisformedbytheviscouseffects,the compres-sionacrosstheshock,andtheshock-boundarylayerinteraction

4.2 Vertical flat plate case, Kn=0.07

Abowshockisformedinfrontoftheverticalflatplate,andis symmetricalwithrespecttothestagnationlinedepictedinFig.1b

AstrongbowshockinfrontoftheverticalplateisfoundinFig.9

that representsthe temperatureandvelocity magnitudecontours

of the CFDand DSMC simulations The shockstand-off distances predictedbybothsolutionsarenearlythesame.Intheshockwave region, there is a high-temperature region and the velocity de-creasesduetotheshockcompression.There isagoodagreement betweentheCFDandDSMCtemperaturecontoursinthefront,and theyaredifferenceintheregionbehindtheverticalplate.The sur-facequantities(p, T ,u)alongthefrontsurfaceoftheverticalflat platearepresentedinFigs.10, 11and 12aresymmetricalwith re-spectto thestagnationline.Consideringthesurface gaspressure, there is a suitable agreementbetween all CFDsimulation results andtheDSMCdata,asshowninFig.10.TheSmoluchowskijump conditionpredictshighertemperaturesthanthoseusingtheLeand Aokietal.jumpconditions,presentedinFig.11.Thesimulation

re-Fig 10 Distribution of surface gas pressure along the vertical plate surface.

Fig 11 Distribution of surface gas temperature along the vertical plate surface.

sultsusingtheAokietal.jumpconditionarelowest,andcloseto

theDSMCdata

Thedistributionofslipvelocityalongthefrontsurfaceisshown

inFig.12 The magnitudesofthe slipvelocity are largestat two

boundaries of the plate ( y/ =0 and y L=1), and obtain the

smallestvalue atthe location y L=0.5 The slip velocities

pre-dictedbytheAokietal.slipconditionintwoCFDsimulationswith

theAokiet al.–Le conditionsandthe Aokietal conditionsare

closetogether,andshowagoodagreementwiththeDSMCdata

4.3 Circular cylinder in cross-flow case, Kn=0.01

ThesimulationresultsofthecylinderofthecaseKn=0.01are

presentedinFigs 13, 14, and15 AllCFD andDSMC simulations

givethepeakpressureatthestagnationpoint(=0-deg.)which

areclosetogether.Past thestagnationpoint, thesurface pressure

isgraduallyreducedalongthecylindersurfacefrom =0-deg.to

 =90-deg.ThereisgoodagreementbetweenallCFDsimulations

andDSMCdata,seeninFig.13

Fig.14presentsthesurfacegastemperaturesalongthecylinder

surface.TheSmoluchowski jumpboundarycondition overpredicts

thetemperaturealong thecylindersurface Thetemperatures

us-ingtheAokietal.andtheLejumpconditionsareclosetogetherin

0-deg.≤ ≤40-deg.TheLe jumpcondition predicts higher

tem-peratures than the Aoki et al jump condition in 40-deg.≤ ≤

90-deg.Thismay be explainedthat the viscosity inthe Le jump

condition is calculated at the gas temperature near the surface

Fig 12 Distribution of slip velocity along the vertical plate surface.

Fig 13 Distribution of surface gas pressure along the cylinder surface, Kn=0.01.

Fig 14 Distributionof surface gas temperature along the cylinder surface, Kn= 0.01.

whilethat iscomputedatthewall temperatureintheAokietal jumpcondition Thisleads tothe viscosityintheLe jump condi-tion to be higherthan that in the Aokietal jumpcondition In overall, the temperature using the Aoki et al jump condition is closetotheDSMCdatawhilethatoftheLejumpconditionisnot The averageerrors betweenthem andtheDSMC dataare 14.33%

Fig 15 Distribution of slip velocity along the cylinder surface, Kn=0.01.

Fig 16 Distribution of surface gas pressure along the cylinder surface, Kn=0.05.

forthesimulationwiththe Lejump condition,and10.7%forthe

simulationwiththeAokietal.jumpcondition

Slipvelocitiesalong thecylindersurface areshownin Fig.15

At the stagnation point, they are approximatethe value of zero

Thereafter,theygraduallyincreasealongthesurfacecylinderfrom

 =0-deg.to  =90-deg., andobtain thepeak valuesatthe

lo-cation =90-deg.The Maxwellslipboundarycondition predicts

higherslip velocitiesthan the Aoki etal.slip condition The slip

velocitiesusing theAokiet al.slipcondition intwo CFD

simula-tionswiththeAokietal.–LeconditionsandAokietal.conditions

areclosetotheDSMCdata,andtheiraverageerrorsincomparing

theDSMCdataare13.33%and15.13%,respectively

4.4 Circular cylinder in cross-flow case, Kn=0.05

Consideringthe surface gas pressurein Fig.16, all CFD

simu-lation resultsgive suitable agreement withthe DSMC data They

obtain thepeak valuesatthe stagnation point (=0-deg.), and

thereafterthey graduallydecrease alongthecylindersurface The

surface gas temperatures are presented in Fig 17 All CFD and

DSMC simulation results obtain the lowest value at the

stagna-tion point, andthen increase along the cylindersurface All CFD

jump conditionsoverpredict the temperaturesalong thecylinder

surface.ThetemperaturesusingtheLeandAokietal.jump

condi-tionsareclosetogetherin0-deg.≤ ≤45-deg.Thetemperatures

usingtheAokietal.jumpconditionarelowerthanthosegivenby

theLe jump condition in45-deg.≤ ≤90-deg.This maybe

ex-Fig 17 Distribution of surface gas temperature along the cylinder surface, Kn= 0.05.

Fig 18 Distribution of slip velocity along the cylinder surface, Kn=0.05. plainedthat1)theviscositycalculatingatthewalltemperaturein theAokietal.islowerthanthatcomputingatthegastemperature nearthesurfaceintheLejumpcondition,and2)theslipvelocities predictedbythesimulationwiththeAokietal.andLeconditions arehigherthanthosegivenbythesimulationwithAokietal con-ditionsin45-deg.≤ ≤90-deg.Inoverall,temperaturesusingthe Aokietal.jumpconditiongiverelativelygoodagreementwiththe DSMCdataalongthecylindersurface

Fig 18presents thedistribution ofthe slipvelocity along the cylinder surface.All CFDandDSMC simulation resultsobtain ap-proximately the value of zero at the stagnation point, and then they graduallyincrease alongthe cylindersurface.Theslip veloc-ityusing theAokietal.slip condition iscloseto theDSMC data whilethoseusingtheMaxwellslipsolutionisnot

For the completeness of the circular cylinder case, the tem-perature andvelocity magnitudecontours predictedby both CFD

Kn=0.05 cases,respectively.Thetemperaturecontoursrepresent the typical flow features found in a blunt body flow as a fairly shockwavestandingofffromthebody,andahightemperature re-gionfollowingtheshock.Thetranslationalenergyisconvertedinto thermal energywith thedecrease in velocity dueto shock com-pression.Athermalboundarylayerthatgraduallythickensaround thecylinder[20].OverallagreementbetweenCFDandDSMC tem-perature contours is generallygoodfor bothof cases,withsome small differences in the shockstructure, shown in Figs 19a and

Fig 19 Temperature and velocity magnitude contours of the cylinder, Kn=0.01, a) temperature and b) velocity.

20a.The velocitymagnitudecontoursofCFDandDSMC solutions

give good agreement for the case Kn = 0.01, seen in Fig 19b,

butthere is the smalldifference betweenthose forthe case Kn

=0.05,seeninFig.20b.Theshockstand-offdistancepredictedby

bothmethodsisnearlythesameasisthethermalboundarylayer

thicknessatthestagnationpointforbothofcases[20] WhenKn

increasestheshockstand-offdistancealsoincreases

4.5 Wedge case, Kn=0.05

The results of the surface quantities (p, T , u) are plotted as

a function of the distance, S, along the wedge surface

normal-ized by the length L. Similar to the distribution of the surface

pressureoftheflat platecase,Fig.21 showsthat thesurface gas

pressuresare highestnearthe leading-edge withthe peakvalues

of1)24.1 Pa forthesimulationwiththe Maxwell–Smoluchowski

conditions,2) 24 Pa for thesimulation withthe Aokiet al.– Le

conditions, 3) 23.25 Pa for the simulation with the Aoki et al

conditions, and 4) 16.5 Pa for the DSMC data Thereafter, they

gradually decrease along thewedge surface The simulation with the Maxwell–Smoluchowski conditions overpredicts the surface gas pressurealong the wedge surface The resultsof the simula-tionwiththeAokietal.conditionsareclosetotheDSMCdatafor

S L≥0.2

Fig.22 showsthe distributionof thesurface gastemperature Theyobtainthepeakvaluesneartheleading-edgewiththevalues

of1)2764 KfortheSmoluchowskijumpcondition, 2)2331 Kfor theLe jumpcondition, 3) 2587 KfortheAokiet al.jump condi-tion,and4)2578 KfortheDSMCdata.Past theleading-edge,the gastemperaturesgraduallydecreasealongthewedgesurface.The Smoluchowskicondition overpredictsthesurface gastemperature along the wedgesurface The Le andAoki etal.jump conditions include the termsof velocity and gradientof the velocity of gas flows.TheslipvelocityinsimulationwiththeAokietal.–Le con-ditionislowerthanthatgivenbythesimulationwiththeAokiet

al.conditions,seeninFig.23.Thismayresultinthetemperatures usingthe Lejumpcondition are lower thanthoseusing theAoki