A redistribution method for axisymmetric diusion
S.Shankar and A. F. Ghoniem
Department of Mechanical Engineering
MassachusettsInstituteof Technology
Cambridge, Massachusetts, 02139, U.S.A.
Abstract
Wedevelopgrid-freenumericalprocedures to computeaxisymmetric incompressibleows.
In particular, we formulate a grid-free `redistribution method' to handle diusion processes
accurately on a disordered collection of computational elements. We validate the numerical
proceduresbycomputingtheStokesowandtheNavier-Stokesowofavortexring.
1 Introduction
Vortexmethodssimulateuidowsbyfollowingacollectionofcomputationalelements(`vortex
elements')whichtransportofvorticity,andotherconservedscalarssuchasenergyandchemical
species[4, 6,7]. Often,those quantities aretransported dueto convectionand diusion. The
convectionprocess is handled accuratelyin agrid-freemanner by movingthe computational
points accordingto the local velocityeld. The diusion process must alsobe implemented
accuratelyusinggrid-freeprocedurestomaintaintheadvantagesofaLagrangiancomputation.
Inthispaperweformulategrid-freemethodstocomputetheevolutionofthevorticityand
temperature for axisymmetric incompressible ows in free-space. A fractional-step method
[2,5]isusedtosolvethegoverningequations. Inthiscontext,thediusiveeectsaregoverned
bytheunsteadyStokesequations. TocomputetheStokesequationsaccuratelyonadisordered
distributionofcomputationalelementsisourprimaryobjective.
WepresentthegoverningequationsinSection2. InSection3weformulateavortexmethod
to solve those equations. InSection 4we briey describe the extension of the redistribution
method[11]tosolvetheStokesequationsforthevorticityandtemperatureinanaxisymmetric
domain; details of theformulation aregivenin [9]. InSection 5 we discuss thepropertiesof
the method. The numerical procedures are validated in Section 6 by computing the Stokes
ow andtheNavier-Stokesowof avortexring. Conclusionsand further work arediscussed
inSection 7.
2 Governing equations
Weconsiderthefree-spaceowofahomogenous,incompressibleuidofconstantdensityinan
axisymmetricdomain;weassumethatthekinematicviscosity,,andthethermaldiusivity,,
areconstant. Letrandzbetheradialandaxialdirectionsoftheowinther zplanesothat
1<r;z <1. Thelocation ofapointin thisplane isdenoted by~x = (r;z) = r^r + zz;^
the symbols ^r and z^are the unit vectors. Let t be the time. The velocity eld is denoted
by ~u(~r;t) = (u
r ;u
z ) = u
r
(~r;t)r^+ u
z
(~r;t)z.^ The vorticity eld, which is the curl of the
velocity, is denoted by !(~r;t); it pointsin the direction z^r^into the plane. LetT(~r;t) be
constant, the temperature is proportionalto the internal energy of theuid. The governing
equationsforthevorticityandtemperatureare[1,3]:
1 r @(ru r ) @r + @u z @z
= 0 ; (1)
D! Dt = !u r r + @ 2 ! @r 2 + 1 r @! @r ! r 2 + @ 2 ! @z 2 ; (2) ! @u r @z @u z @r ; (3) DT Dt = @ 2 T @r 2 + 1 r @T @r + @ 2 T @z 2 ; (4) where D Dt @ @t + u r @ @r + u z @ @z (5)
isthematerialderivative.
Tosolvetheaboveequations, theinitial vorticityand thetemperaturedistributions must
bespecied. Theboundaryconditionsalongtheaxisofsymmetryare discussedin Section4.
Inthenextsectionweformulate avortexmethod tosolvetheaboveequationsnumerically.
3 Vortex method
Thevortexmethodpresentedhereisbasedonafractional-stepalgorithm[2,5],thatseparates
theinviscidandviscousprocessesateachtime-step. Todothis,theconservationequations(2)
and (4)areapproximatedbythefollowingtwosteps:
Inviscidstep:
d ~
X
dt
= ~u ; (6)
d
!
dt
= 0 ; (7)
d
T
dt
= 0 : (8)
Viscous step:
d ~
X
dt
= 0 ; (9)
@! @t = @ 2 ! @r 2 + 1 r @! @r ! r 2 + @ 2 ! @z 2 ; (10) @T @t = @ 2 T @r 2 + 1 r @T @r + @ 2 T @z 2 : (11)
Intheaboveequations, ~
X isthelocationofauidelement,~uisitsvelocity,
!
Z Z
!drdz (12)
isthecirculation,and
T
Z Z
istheinternalenergyperunitmassdividedbythespecicheat. Theintegralsin(12)and(13)
aretakenovertheareaoftheuidelementinthehalf-plane0 r. FortheStokesequations,
thecirculationofanelementdecaysintime, whiletheenergyisstillconserved.
Equations(7)and(8)areobtainedbyintegratingoverthehalf-planetheequations(2)and
(4)withoutthediusion terms. Thevelocityisobtainedfrom thevorticity([1],eqn. 2.4.10).
Tosolve(6) through(11)using avortexmethod,theinitial vorticityandtemperatureare
rstrepresentedby computationalelements. Todothis, wedivide theowregionintosmall
cells,andassignanelementtoeach cell. Themathematicalrepresentationisthen
!(~x ;0) = N
X
i=1 !
i
(0)Æ(~x ~x
i
(0)) ; (14)
T(~x;0) = M
X
i=1 T
i
(0)Æ(~x ~x
i
(0)) ; (15)
where,iistheindexfortheelement,~x
i
(t)isitspositionattimet,
!
i
(t)isitscirculation,
T
i (t)
isitsinternalenergy,andÆ()istheDiracdeltafunction.Accordingtoequations(12)and(13),
theinitial circulationof anelementis takento be thevorticityat theelementlocation times
theareaofthecell;theinitialinternalenergyistheproductofthetemperatureattheelement
location, the radial location of the element and the cell area. Each computational element,
however, need not carry both the circulation and energy, although saving in computational
timemaybeachievediftheydid.
Duringtheinviscidstep,wemoveeachcomputationalelementaccordingtothelocalvelocity
eld(6)whilekeepingitscirculationandenergyunchanged(7,8). Thevelocityofeachelement
iscomputedbysummingupallthevelocitiesinducedatitslocationbyallthevorticitycarrying
elementsincludingitself([1],eqns. 2.2.11&7.2.13). Thevelocityisdesingularized[5,13]using
asmallsmoothingparameterofsize p
0:5t ,wheretisthetime-stepin thecomputation.
Attheendoftheinviscidstep,weperformtheviscoussteptoaccountfordiusion. During
theviscousstepwedonotmovethecomputationalelements(9),butchangetheircirculations
(10) and theenergies (11)using theredistribution method. Weformulate the redistribution
methodin thenextsectiontosolvetheStokesequations(10)and(11).
We reverse the sequence of the inviscid and viscous steps at everytime-step to improve
accuracy[2].
4 Redistribution method
Theeectofdiusionissimplytospreadoutthecirculationandenergyofeachcomputational
element. Theredistributionmethodisbasedonthisidea: itspreadsthecirculationandenergy
ofan elementto itsneighboring elementslocated within achosenradius,R ,accordingto the
Stokes equations. This radius is proportionalto the diusion length scale h
=
p
t ; the
diusivitybeingeitherthekinematicviscosityorthermaldiusivity,andtisthetime-step.
Inourcomputations,theaveragedistancebetweenthecomputationalelementsis takentobe
p
8h
;andtheneighborhoodradiusRistakentobe4h
. Thesetwoparametersarechosenso
thatasolutionto thesystemofequations,derivedbelow,becomespossible[11].
More precisely, leti be anelement located at(r
i ;z
i
),whose circulation (orenergy) needs
to bediused, and j =1;:::;m beits neighboring elements locatedwithin the radiusR . Let
f n
ij
bethefractionofthecirculation(orenergy)movedfromitoitsneighborj toadvancethe
vorticity(or temperature)from time leveln to thenext timeleveln+1. Then the vorticity
eldat timeleveln,
! n
= X
i n
i
Æ(~x ~x
i
) (16)
! n+1 = X i X j f n ij n i
Æ(~x ~x
j
) (17)
at thenexttimelevel.
Nowthequestionishowdowendthosefractionsf n
ij
. Todothat, wewillensurethatthe
Fouriermodesof the numerical solution arecorrectly damped [11]. This is accomplishedby
matching theHankel-Fouriertransform[12]ofthethenumericalsolution(17)givenby
^ ! n+1 = X i X j f n ij n i r j J 1 (k r r j )e
ikzzj
(18)
with thatoftheexactsolution
^ ! n+1 e = X i n i r i J 1 (k r r i )e
ikzzi
e k
2
t
(19)
todesiredorderofaccuracyO(h M
),asdescribedbelow;andM isapositiveinteger. Theexact
solution(19)isobtainedbysolvingtheStokesequation(10)withinitialvorticity(16). Inthe
aboveequations,J
1
()istheBesselfunction ofrstkindandorder one;andthewavenumber
is denotedby ~
k=k
r ^ r+k
z ^
z, anditsmagnitudebyk. Forthediusion ofinternalenergy,the
aboveprocedureisthesameexceptthattheradialmodesarenowBesselfunctionsofrstkind
and orderzero.
Wenext expand(18) and (19)using Taylor'sseries around~x
i
for small times andensure
thatthetruncationerrorsvanishtodesiredaccuracy;thedistancesbetweenthevorticeswithin
theneighborhoodarescaledbyh
. Thisresultsinasystemoflinearequationsforthefractions
f n ij : X j f n ij
= 1 ; (20)
X j f n ij r ij = " r i =h ; (21) X j f n ij z ij
= 0 ; (22)
X j f n ij r ij r ij
= 2 ; (23)
X j f n ij r ij z ij
= 0 ; (24)
X j f n ij z ij z ij
= 2 ; (25)
where ij ( j i )=h
. Theequations(20){(25) areO(h
) approximationto theStokes
equations(10)and(11). Higher-orderspatialaccuracycanbeachievedbyincludingadditional
redistributionequationsthatareobtainedbyretaininghigher-ordertermsintheTaylor'sseries
expansion.
The value of theparameter " in (21)is 1 for thediusion of circulation, and 1for the
diusion of energy. When " = 0, however, the same equations represent the diusion of
circulationandenergyin Cartesiantwo-dimensionalows. Theaboveredistributionequations
maintaintheconservationlawsoftheStokesequationsexactly.
The above system of linear equations is usually underdetermined, since the number of
elements within the neighborhood is often more than the number of equations. There is,
however, a restrictionon the values of the fractions f n
ij
. For stability, all fractions must be
nonnegetive. The linear system can be solved using any standardsimplex method program
from IMSLfor example. Whennononnegetivesolutioncanbefound, itisanindication that
until an acceptable solutionis obtained. This automatic addition of computationalelements
maintains the chosen resolution at all times, and further ensures that the vorticity diuses
correctlyintothesurroundingirrotationalow.
In our computations, we use elements only in the half-plane 0 r. Then the axis of
symmetry isa boundary ofthe computationaldomain, and appropriateboundaryconditions
mustbeapplied there. Todo so,weplace alayerof`mirror elements'onthe negetivesideof
r. Themirrorelementsaresimplyreectionsoftheelementsonthepositiveside,whoseradial
locationsarelessthantheneighborhoodradiusRmentionedabove.
Thevorticityisantisymmetricinr,andhencevanishesontheaxis. Thereforewhilediusing
thevorticityoftheelementsneartheaxis,theamountsofcirculationmovedtothemirrorsare
considered`lost'. Thisensuresthecorrectdecayofthecirculationthehalf-plane.
Thetemperatureis symmetricin r,andhencethereisnoheatux acrossaxis. Therefore,
theenergyreceivedbyamirrorelementisgivenbacktoitsmirrorlocatedonthepositiveside.
Thisconservestheenergyin thehalf-plane.
We mustalso point outanumericaldiÆculty caused bytheequation (21): therighthand
sidebecomessingularwheneverthe elementto bediused isverycloseto theaxis. Toavoid
this singularity,wekeepathin stripnear theaxisfreeofcomputational elements. Ofcourse,
convectioncouldstillmovesomeoftheelementsinsidethisstripduetonumericalerrors.
How-ever,thoseelementsaremovedbackontotheedgeofthestripandatthesametimemaintaining
theappropriate conservation laws. The numericalresults presentedin thenext sectionshow
thatthese proceduresindeedreproducethevorticityandtemperatureelds accurately.
Wenextdiscussthetheoreticalpropertiesofthemethod.
5 Properties of the redistribution method
In this section, we givea summary the properties of the method. Further discussion of the
followingproperties is given in [9, 11]. Theadvantages ofthe method achieved forcartesian
two-dimensionalowssuccessfullycarryovertotheaxisymmetriccaseaswell.
1. Theredistributionequationsspreadtheconservedquantitiesoveronlyanitenumberof
neighbors inside asmall area. This allowsthemethod to resolvesharp gradientsin the
oweldsaccurately[10].
2. The redistributionmethod does notrequireauniform distribution of computational
el-ements,or evenan ordering ofthe elements. Inparticular, evenif convectiondisorders
theinitial uniformity ofthe elements, theadaptive additionof new elements allowsthe
computationtocontinuewithoutlossofaccuracy.
3. Theredistributionmethoddoesnotuseanysmoothingfunctiontoperformdiusion;the
resultingdesireable properties, especially theresolution of short scales, arediscussed in
[11],Sec. 9.1.
4. Theconservationlawsof theStokesequationsareexactly maintainedusingonly anite
numberofelementswithin theneighborhood, eveniftheelementsaredisordered.
5. Thesignofauniformlypositiveornegativevorticityeldispreservedforanyniteorder
ofaccuracybythemethod, atleastfortheStokesequations([11],Sec. 9.5).
6. The redistribution method can, in principle, have any order of accuracy. At the same
time, thesplitting errorof thefractional-step algorithm must be considered for spatial
accuracybeyondfourth-order.
The advantages of the aboveproperties havebeen demonstrated for the computation of
two-dimensionalunsteadyseparatedowsathighReynoldsnumbers[10]. Inthenextsection
wepresentnumericalresultstoverify thatthose properties canbeachievedforaxisymmetric
6 Numerical results
Tovalidate themethod, we computedtheStokesowand theNavier-Stokesowof avortex
ring. Wepresentthenumericalresultsinthenexttwosubsections. Weveriedtheresultsusing
varioustime-stepsandcorrespondingspatialresolutions. Theresultspresentedherecorrespond
tothesmallesttime-stepusedinourconvergencestudy. Thecomputationswerecarriedoutin
single-precision onSGI/IRIX-6.1with anaveragespeedof 15Mopsandwith amachinezero
ofabout6:010 8
.
6.1 Stokes ow
TheredistributionmethodistestedontheStokesowduetoapointsourceofcirculationand
energy. Forthesetwocases,exactsolutionsareavailablethataresimplythefree-spaceGreen's
functions oftheStokesequations(10)and(11).
For the diusion of vorticity, an initial point source of unit circulation, representedby a
singlecomputationalelement,isplacedat(r;z)=(2:5;0:0). Thetime-stepis =0:004,with
the corresponding spatial resolution R 0:2530. The computation wascontinued until time
=1:0whenthecirculationinthehalf-planehaddecayedtoabouteightypercentoftheinitial
value; by thistime itwasclearthat thehandlingof theaxisboundaryconditionis accurate.
The numberofcomputational elements isabout 3600at this time. The impulseis conserved
with arelativeerrorless than10 6
at alltimes, andtheaxial center ofvorticityis conserved
witharelativeerrorlessthan10 5
. Attheendofthecomputation, therelativeerrorinmean
square axialexpansionisabout2:5510 3
, andtheerrorin thecirculationin thehalf-plane
is about1:1710 3
. Moreimportantly,the highwavenumbersthat aremost susceptibleto
dissipative errors are also correctly damped: the relative error in the maximumvorticity at
=0:50is about2:4710 4
, and at =1:00the errorisabout3:4710 4
. Ingure1(a)
thecomputedvorticitycontoursattime =0:50arecomparedwiththeexactsolution.
For the diusion of heat, an initial point source of unit internal energy, represented by
a single computational element, is placed at (r;z) = (2:5;0:0). The time-step and spatial
resolution is the same as that for the vorticity. The computation is performed until time
= 1:30, long enough to verify that the axis boundary condition works correctly; at this
time,themaximumtemperatureisalreadyattainedontheaxis. Thenumberofcomputational
elements is about 4100 at this time. The energy in the half-plane is conserved to machine
precision. Theaxialcenteroftemperatureisconservedwitharelativeerrorlessthan10 6
at
alltimes. Themeansquareradialandaxialexpansionsofthetemperatureeld growlinearly
withtimeaspredictedtheoretically,andtherelativeerrorinbothislessthan10 5
throughout
the computation. Further, the relativeerror in the maximum vorticity at = 0:70is about
1:6610 3
and at = 1:30the error is about 4:4310 4
. The highererror at the earlier
time is due to the high gradientsarisingfrom the transition of thetemperaturefrom a local
minimumontheaxistothemaximumvalueatthesourcelocation. Ingure1(b)thecomputed
temperaturecontoursattime =0:70arecomparedwiththeexactsolution.
Thenumericalerrorsintheaboveconservedquantitiesareprimarilyduetoround-oerrors,
andduetothetruncationoftheexponentiallysmallvorticityandtemperatureeldsdetermined
aprioribyacut-ovalueofcirculationandenergy. Thecirculationortheenergyofanelement
isnotdiusedifitfallsbelowthiscut-o;inthecomputations,acut-ovalueof10 6
isused.
Inthenextsubsectionweincludeconvectionaswelltoverifythatthenumericalprocedures
canhandletheowjustasaccuratelyasitdidfortheStokesow.
6.2 Navier-Stokes ow
Whenconvectionispresent,thecomputationalelementsbecomedisorderedduetothestraining
oftheow. Inthissectionweshowthatthemethodstillmaintainshighaccuracyunder these
conditions. As a nontrivial test case, we compute theNavier-Stokesow of the propagation
of avortexringin free-space. Atthesametime, wewantedto verifythat ourvortexmethod
wedidnotnd suitabledatain theliterature,wecomputedthisowusing anite-dierence
methodalso. Aconvergencestudyofthenite-dierencesolutionwasperformedbasedonthe
mesh-size, time-step, and domain size. Inour nite-dierence computations, we veried the
convergence,notonly of the various global momentsof vorticityand temperature elds, but
alsothepoinwisevaluesoftheeldsthemselvesatdierenttimes. Thebestresolutionusedas
many as5121024computational points distributed uniformly overthe domain 0r 8,
8z8;andatime-stepof0.0005.
The redistribution computation starts from a vortexring of small core located radiallya
unit distance from theaxisofsymmetry. Theringhasunit circulation and unit energy. The
Reynolds number is R e = 50 based on the initial circulation and ring radius. The Prandtl
numberisPr=1.
Theinitial vorticitydistributionoverthe coreofthevortexringisobtainedbyallowinga
pointsource to diuseoutasaStokesow (noconvection)untilthe scaledtime t=R e=
0:002. At this time, the vorticity distribution is almost a two-dimensional Gaussian with a
core radius of about 0:18; just outsidethe core, the vorticity is less than onepercent of the
maximum. Theinitialtemperaturedistributionisobtainedsimilarly.
Thegure2showsthedistributionof computationalelements. Noticethat thepointsare
disordered and denser near the center of the vortex due to the straining of the ow. New
elementsareautomaticallyaddedonlywheretheconservedquantitiesspreadout. Wereplaced
elementsveryclosetoeachother,locatedwithinaradius p
0:5,byasingleelement[10]. The
scaledtime-step is0.0005,andthecorrespondingspatialresolutionR0:0894. Thesmall
circle in gure1(c)indicates thesize ofthe spatialresolution. Thenumberof computational
elements initially is about 400, and growalmost linearly to roughly 5000 elements at scaled
time = 0:15. At this time, the circulation in the half-plane had reduced to about eighty
percentofitsinitialvalue.
Thegure3providesaclearevidenceoftheaccuracyofthemethod: itcorrectlyreproduces
thevorticityandtemperatureelds. Wealsoveriedthat theyare correctatseveral
interme-diate times. The vorticity is obtained by convolving the circulations with an innite-order
axisymmetricsmoothingfunction;thetemperatureis obtainedsimilarly. Notice ingure3(a)
thatthevorticityandtemperatureeldshaveevolvedverydierentlyalthoughtheywere
iden-tical initially; the maximumtemperature is attainedon theaxis,while thevorticity vanishes
there. Theimpulse of thevortexring is conserved with arelativeerror lessthan 2:010 5
at alltimes. Theinternal energyin thehalf-plane isconservedwitharelativeerrorlessthan
4:010 5
. Therelativeerrorin thecirculationislessthan2:1110 3
. Therelativeerrorin
thevelocityofthevortexring[8]isabout2:3510 4
attheendofthecomputation.
The accuracy of the above numerical results is excellent considering that the
computa-tional elementsare disorderedand that a relativelycoarse spatial resolution wasused in the
computations.
7 Conclusions
Wehavedevelopednumericalproceduresthataretrulymesh-freeforcomputingaxisymmetric
ows. Theaxisymmetric redistribution method simulates diusion accurately evenif
convec-tion causes strong distortions in the distribution of computational elements. The ability of
the method to satisfy various conservation lawsexactly by construction, togetherwith other
properties, provide reliableand accurate simulationsof more complexow elds. Infact, we
aredevelopingtheseproceduresforthedirectnumericalsimulationofjetdiusionames. The
redistribution method is an explicit time-stepping scheme, and we are working to obtain an
implicitformulationtorelaxstabilityrestrictions. Theprocedurescanalsobeusedtosimulate
manyreactingowsbasedoncouplingfunctions[14]. Tocomputeanevenwiderclassofows,
however,weneedto handlediusivities thatvary spatially;thatis alsonecessaryto compute
incompressible ows based on large eddy simulations. In this direction, our preliminary
nu-merical results for one-dimensional diusion are just as accurate as the constant diusivity
Acknowledgements
This work was supported by the US Department of Energy, Basic Energy Sciences, mics,
DE-F602-98ER25355. S. Shankar appreciates the helpful discussions with Professor L. van
Dommelen,DepartmentofMechanicalEngineering,FAMU-FSU CollegeofEngineering.
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0
2
4
6
r
-2
0
2
z
τ
= 0.70
(b)
0
2
4
6
r
-2
0
2
z
τ
= 0.50
(a)
Figure 1: Stokesowof pointsource: (a)Vorticitycontoursat =0.50: ! =0.010, 0.025, 0.045,
0.075, 0.110, 0.140. (b)Temperature contoursat =0:70: T =0.010, 0.020, 0.032, 0.048, 0.067,
0
1
2
3
4
5
r
-2
-1
0
1
2
3
z
τ
= 0.00
2(a)
0
1
2
3
4
5
r
-2
-1
0
1
2
3
z
τ
= 0.01
2(b)
0
1
2
3
4
5
r
-2
-1
0
1
2
3
z
τ
= 0.15
2(c)
0
1
2
3
r
0
1
2
3
z
τ
= 0.15
3(a)
0
1
2
3
r
0
1
2
3
z
3(b)
τ
= 0.15
0
1
2
3
r
0
1
2
3
z
3(c)
τ
= 0.15
Figure2: Computational elementsfortheowofa vortexringatR e=50 andPr=1. The small
circlein2(c) representsthespatialresolution.
Figure 3: (a)Solid lines are vorticity contours, ! = 0.08, 0.15, 0.25, 0.375; dotted linesare
tem-peraturecontours,T =0.08,0.15, 0.25,0.375, 0.475, 0.54. (b)&(c)arevorticityandtemperature
