Vo ex Pa icle In ensi ied La ge Eddy Simula ionN. Ko ne and S. Sama bakhsh
VI In e na ional Con e ence on Pa icle-based Me hods – Fundamen als and Applica ions
PARTICLES 2019
E. O˜na e, M. Bischoff, D.R.J. Owen, P. W igge s & T. Zohdi (Eds)
VORTEX PARTICLE INTENSIFIED LARGE EDDY
SIMULATION - VπLES
S. Sama bakhsh1and N. Ko ne 2
Chai o modeling and simula ion
Uni e si y o Ros ock
Ros ock, Ge many
1e-mail: sina.sama bakhsh@uni- os ock.de
2e-mail: nikolai.ko ne @uni- os ock.de
Key wo ds: LES, subg id model, meshless me hod, ee je , hyb id Eule ian/Lag angian
me hod
Abs ac . This pape p esen s a no el La ge Eddy Simula ion app oach wi h a di ec
esolu ion o he subg id mo ion o fine concen a ed o ices. The me hod, p oposed
fi s by [10], is based on combina ion o a g id based and he g id ee compu a ional
o ex pa icle (VPM) me hods. The la ge scale flow s uc u es a e simula ed on he g id
whe eas he concen a ed s uc u es a e modeled using VPM. Due o his combina ion he
ad an ages o bo h me hods a e s eng hened whe eas he disad an ages a e diminished.
The p ocedu e o he sepa a ion o small concen a ed o ices om he la ge scale ones
is based on LES fil e ing idea. The flow dynamics is go e ned by wo coupled anspo
equa ions aking wo-way in e ac ion be ween la ge and fine s uc u es in o accoun . The
fine s uc u es a e mapped back o he g id i hei size g ows due o diffusion. Algo i h-
mic aspec s specific o h ee dimensional flow simula ions a e discussed. Validi y and
ad an ages o he new app oach a e illus a ed o a well ied benchma k es o he
decaying homogeneous iso opic u bulence using he expe imen al da a o [4] and ee
u bulen je flow using expe imen s o [8, 2, 15].
1 INTRODUCTION
Insufficien esolu ion o fine o ex s uc u es in u bulen flows is one o he key
p oblems in Compu a ional Fluid Dynamics (CFD). The mos ad anced and popula
echnique o esol e mul i scale flow s uc u es is he La ge Eddy Simula ion (LES) which
is based on he idea o scale decomposi ion in o la ge and small ones. While he la ge
eddies a e di ec ly esol ed on he g id, he effec o small o ices is aken in o accoun
h ough a subg id s ess (SGS) model.
The subg id mo ion is no esol ed in LES bu a he i is modelled using diffe en
unc ional and s uc u al app oaches. Howe e , he e a e many p oblems which equi e
di ec ep esen a ion o he subg id mo ion o simula e, o ins ance, mixing o pa icle
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dynamics in u bulen flows. In ou p e ious pape s (see [9], [10], [11], [16] ]), we p oposed
a simula ion echnique esembling LES wi h an effo o di ec ly ep oduce he subg id
mo ion a leas in he s a is ical sense. I is sugges ed o apply a hyb id g id and pa icle
based me hod, u ilizing a combina ion o he fini e olume and compu a ional o ex
pa icle (VPM) (see [5]) me hods. The la ge scale field is ep esen ed on he g id like in
LES, whe eas he small scale one (subg id field) is calcula ed using he VPM. The new
me hod called VπLES is a pu ely Lag angian one o small s uc u es and pu ely g id
based one o la ge scale s uc u es.
The me hod is based on he decomposi ion o he eloci y uand o ici y fields ωin o
he dis ibu ed la ge scale (uppe index ’g’) and concen a ed small scale (uppe index
’ ’) fields:
u(x, )=ug(x, )+u (x, ),ω(x, )=ωg(x, )+ω (x, ) (1)
The fine o ex de ec ion p ocedu e u ilizes he La ge Eddy Simula ion (LES) fil a ion
applied o he g id based eloci y field ug:
ug(x, )=
∞

−∞
ug(s, )F(x−s)ds(2)
whe e F(x−s) is a ce ain fil e unc ion. The small scale eloci y field u�calcula ed as
he diffe ence be ween he o iginal and fil e ed fields
u�(x, )=ug(x, )−ug(x, ) (3)
should be app oxima ed by o ex pa icles in egions o concen a ed o ices which a e
de ec ed using any o ex iden ifica ion c i e ia, o ins ance, λci ([1]). The cells wi h
λi>λ
ci,min con ain he o ices which in p inciple can be con e ed o o ex pa icles.
Such cells a e ma ked as ac i e ones using he λi,ac i e field:
λi,ac i e =1,i λci >λ
ci,min
0,o he wise (4)
whe e λci,min is a ce ain small alue in oduced in o de o limi he numbe o pa icles.
To keep he equi ed compu a ional esou ces on an accep able le el, only small o ices
wi h size p opo ional o he local cell size Δ a e o be con e ed o single o ex pa icles.
Neighbo ing cells which all ha e λi,ac i e = 1 o m la ge o ices. Fo hem i is supposed
ha he la ge o ices wi h scales o a ew Δ can accu a ely be ep esen ed on he g id.
The e o e he nex ask is o de ec cells wi h fine o ices. Acco ding o he algo i hm,
all neighbo ing cells o he i− h cell a e checked o he condi ion λci >λ
ci,min. I all
neighbo s ulfill his condi ion, we iden i y a cell clus e which emains on he g id and
all i s cells become non- ac i e λi,ac i e = 0. Only o ices in cells wi h λi,ac i e = 1 a e o
be eplaced by o ex pa icles.
A each cell wi h λi,ac i e = 1 he new o ex pa icle is in oduced a he cell cen e
i he pe missible numbe o o ex pa icles pe cell Np is no exceeded. O he wise,
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he new o ex eplaces he cell’s weakes one. The numbe Np was in oduced o keep
he o al numbe a a easonable le el. This es ic ion is con o m wi h he concep ha
he la ges con ibu ion o he subg id kine ic ene gy is made by a small ac ion o he
s onges o ices. The adius o he new o ex is se as σ=βV ol1/3
i, whe e βis he
o e lapping a io which is aken as β= 2 and V oliis he olume o he i- h cell. A
ho ough analysis o he influence o Np and λci,min o he je case is gi en in [11] in Sec.
4.3.5. The o ex pa icle s eng h is calcula ed as
α=V oliω =V oli(∇×u) (5)
The eloci y u (x, ), induced by he o ex pa icles, is calcula ed a g id poin s xusing
he Bio -Sa a law
u (x, )= 1
4π∇×FlowV olume
ω (ξ, )
|x−ξ|dV (ξ)
and sub ac ed om he g id eloci y ug,new =ug−u . Thus, he o al eloci y a g id
poin s ug,new +u =ug emains cons an a e he o ex pa icle gene a ion p ocedu e.
2 GOVERNING EQUATION
The e olu ion o o ex pa icles and la ge scale flow ep esen ed on he g id is desc ibed
by a sys em o wo coupled anspo equa ions de i ed in [9, 10, 11] o incomp essible
iso he mal flows:
∂ug
∂ +(ug·∇)ug=−1
ρ∇pg+νΔug+u ×ωg(6)
dω
d =(ω ·∇)(u +ug)+νΔω +∇×[u ×ωg−u ×ωg],(7)
The sum o he cu l o he fi s equa ion and he second equa ion e ie es he o iginal
Na ie S okes equa ion w i en in he o m o he o ici y anspo equa ion. The fi s
equa ion (6) is coupled wi h he second one (7) h ough he addi ional e m u ×ωg
whe eas he coupling o he second equa ion wi h he fi s one is due o he e ms (ug·
∇)ω ,(ω ·∇)ugand ∇×[u ×ωg−u ×ωg. The equa ions (6) and (7) a e sol ed
sequen ially. The fi s equa ion is sol ed on he g id whe eas he second one uses he
g id ee Vo ex Pa icle Me hod (VPM) [5]. The physical meaning o he coupling e m
u ×ωgis explained in [10].
The o ex pa icle displacemen is calcula ed om he ajec o y equa ion
d i
d =ug
i+u
i,(8)
whe e iis he pa icle numbe . Compu a ion o he eloci y induced by o ex pa icles
u is pe o med wi h he di ec summa ion o he Bio-Sa a law aking in o accoun
one o wo laye s o neighbo ing cells. Only induc ion o he neighbo ing poin s is aken
in o accoun because he eloci y u is much less han ugand i is mos ly de e mined
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by in e ac ion o neighbo ing pa icles lying a a sho dis ance. The jus ifica ion o his
simplifica ion p esen ed in [11] is ha he co ela ion be ween neighbo ing small scale
o ices is weak and hey a e well sepa a ed. This simplifica ion can be conside ed as a
kind o model which esul s in a e y as compu a ional p ocedu e o a local cha ac e
sui able o pa allel calcula ions.
The eloci y induced by a o ex pa icle can be calcula ed om he o mula
u
p=1
4π
α×ξ
ξ3(1 −e−ξ3/σ3) (9)
p oposed by [14]. He e αand σa e he s eng h and he adius o a o ex pa icle, which
a e defined below. The eloci y induced by a se o pa icles u is calcula ed as he sum
o u
p. I should be no ed ha he echnique p esen ed he e is independen o any specific
choice o o ex pa icles. Pa icula ly, a se o unc ions in oduced in [12] can be used
wi hin he p esen me hod.
2.1 Nume ical solu ion o he equa ions (7) and (8) using he VPM
Ins abili y o nume ical solu ion o he equa ion (7) caused by he s e ching e m
(ω·∇)uis he mos impo an p oblem o he VPM along wi h he compu a ion o he
eloci y u . In g id based me hods wi h low and mode a e o de schemes, he ac ion
o he s e ching is effec i ely coun e balanced by he nume ical iscosi y which is e y
low in Lang angian o ex pa icle me hods. Theo e ically, a s able VPM solu ion can
be ob ained by inc easing he accu acy o he s e ching and diffusion simula ion which
can be a ained by a high numbe o o ex pa icles and high empo al esolu ion. Bo h
make he me hod imp ac ical a leas o high Reynolds numbe s. A e many effo s he
au ho s se led on he algo i hm which was o iginally p oposed by [7] and modified in
[11]. This algo i hm consis s o he ollowing subs eps:
•Calcula ion o he change o he o ici y s eng h magni ude
d|ω |
d =d√ω ·ω
d =ω
|ω |·dω
d (10)
whe e dω
d is calcula ed om (7) wi hou he iscous diffusion e m. The e m (ω ·∇)u
is calcula ed aking in o accoun adjacen o ex pa icles loca ed only wi hin one o wo
laye s o neighbo ing cells.
•Calcula ion o he pa icle leng h om he equa ion o he elemen a y sec ion dl ans-
po ed in in iscid flow
dl
d =l
|ω |
d|ω |
d (11)
•Calcula ion o he pa icle co e adius om he equa ion desc ibing he anspo o an
elemen a y ube wi h leng h land adius σin in iscid flow
dσ
d =−σ
2l
dl
d (12)
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•Conside a ion o he iscosi y influence using he co e sp eading me hod (CSM) (see
[5]). The pa icle co e adius is inc eased by Δσ:
Δσ=√4νΔ (13)
•Calcula ion o he new pa icle o ien a ion
ω∗=ω ( )+dω
d Δ (14)
•Calcula ion o he pa icle s eng h magni ude om [7]
|α( +Δ )|=|α( )|σ( +Δ )
σ( )(15)
•Calcula ion o he new s eng h ec o
α( +Δ )=|α( +Δ )|ω∗
|ω∗|(16)
In he o iginal e sion p oposed by [7] he nex s ep should be he edis ibu ion o
pa icles whose leng h has doubled. Acco ding o ou expe ience he edis ibu ion esul s
in an a alanche-like inc ease o he o ex pa icles numbe in a eas o s ong s e ching.
To p e en his [7] p oposed a special elimina ion p ocedu e based on a knowledge o
a h eshold o he dissipa ion a e which is difficul o se in a gene al flow case. To
de elop a obus code, o ob ain a s able solu ion and o keep he pa icle numbe in a
easonable ange we a oid he edis ibu ion p ocedu e in ou compu a ions. Thus, he
smalles o ices a e emo ed. This educes he ange o scales ha mus be esol ed in
a nume ical calcula ion. Such a educ ion, as poin ed ou by [3], is an immanen pa o
e e y u bulence model.
The e is a pe manen exchange be ween he small o ices and la ge scale ones ep e-
sen ed on he g id. La ge scale o ices become small due o s e ching and a e con e ed
o pa icles. I pa icles g ow due o iscosi y and flow s agna ion and exceed some size
hey a e mapped back o he g id. The simple Eule me hod is used o he in eg a ion
o he diffe en ial equa ions. The flowcha o he whole algo i hm is p esen ed in [11].
The p esen me hod has he same e o sou ces as e e y LES model [6]. Two commen s
should be made on he fil e ing e o s. Fi s , he applican s unde s and ha he models
elying on he small scales compa able wi h cell sizes can suffe om he fil e aliasing
e o s inhe en ly p esen ed in each nume ical me hod. Fo ins ance, such e o s mos
s ongly affec dynamic ype models which ely hea ily on he smalles scales o de e mine
SGS p ope ies [6].Second, since ou algo i hm does no use commu a ion o diffe encing
and fil e ing ope a o s which is he big difficul y in LES o malism and ep esen s he
second pa o he fil e ing e o s, he commu a ion e o is no p esen .
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3 RESULT AND DISCUSSION
3.1 Summa y o p e ious esul
Valida ion and e ifica ion is pe o med o wall ee flows including decaying iso opic
u bulence (DIT) [4] and ee u bulen je [8, 2] es cases. Resul s e ealed ha he
effec o he e m u ×ωgin he equa ion (7) is simila o ha o a LES subg id model.
The e m u ×ωgbeha es as an ene gy d ain ans e ing he ene gy o he g id based
mo ion in o he fine scale ene gy. A coa se esolu ions, i ac s as a diffusi e La ge Eddy
Simula ion subg id model esul ing in a LES-like beha io o he whole me hod.
The addi ional e m u ×ωgis au oma ically swi ched off when he esolu ion in-
c eases, he p esen me hod is consis en and con e ges o he Di ec Nume ical Simula-
ion. The ene gy back sca e ing is also cap u ed by he p esen me hod. As men ioned in
[16], he in ensifica ion o he u bulen kine ic ene gy due o back sca e ing is p o ed o
be e y impo an o p ope ly ep oduce he je b eakdown and ansi ion o u bulence
close o he nozzle wi hou any a ificial u bulence o cing a he nozzle. The Reynolds
s esses o he eloci y field induced by pa icles possess he p onounced aniso opy which
is space dependen . I was also shown ha he model o he je case can be sufficien ly
educed by neglec ing he inne in e ac ion be ween pa icles. This esul s in a d as ic
educ ion o he compu a ional ime. Some addi ional esul on he model educ ion and
aniso opy Reynolds s esses a e gi en in he nex sec ion. The esul a e ob ain o he
ee je a he Reynolds numbe Re = 104. Mo e de ails abou g id p ope ies and se up
o he simula ion a e ho oughly desc ibed in [16].
3.2 Model educ ion
As shown in [11] he equa ions 7 and 8 can sufficien ly be educed by neglec ing inne
in e ac ion be ween o ex pa icles wi hou a significan loss o he simula ion accu acy.
The educed equa ions ake he o m:
∂ω
∂ +(ug·∇)ωg=(ω ·∇)ug+νΔω ,(17)
and he .h.s o he ajec o y equa ion (18) con ains only he g id based eloci y
d i
d =ug
i(18)
As seen in [11], he influence o inne in e ac ions on spa ially a e aged kine ic ene gy
and scala dissipa ion a e is ela i ely weak and can be neglec ed in he calcula ion.
The eby he compu a ions can be done sufficien ly as e .This model is u he e e ed o
as he passi e o ices model. Wi hin he nex simplifica ion s ep he influence o he g id
based solu ion on he e olu ion o o ex pa icles s eng hs is neglec ed. The equa ions
desc ibing he o ex pa icle e olu ion ake he simples o m:
∂ω
∂ +(ug·∇)ωg=νΔω ,(19)
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(a) influence o igno ing inne in e ac ion be ween
o ex pa icles
(b) influence o conside ing o ex pa icles only in
he cu e n ime s ep
Figu e 1: Influence o he in e ac ion be ween o ex pa icles and g id on he e olu ion o u ms
d i
d =ug
i(20)
Figu e 1-a demons a es esul s o he .m.s. o he axial eloci y ob ained using he
ull model equa ions 7 and 8, passi e o ices model equa ions 17 and 18 and he model
wi hou influence o he g id based flow on o ex pa icles s eng hs equa ions 19 and
20. The diffe ence be ween esul s is negligible poin ing ou ha o ex pa icles se e
jus as igge s o in ensifie s o u bulence and hei inne in e ac ion doesn’ con ibu e
sufficien ly o he flow e olu ion. Hence he name o he me hod is he LES in ensified
by he o ex pa icles o VπLES .
In he nex s ep o model educ ion only he influence o o ex pa icles gene a ed in
he cu en ime s ep a e conside ed and o ex pa icles gene a ed in he p e ious ime
s ep we e mapped back o he g id. In his case he ime o compu a ions is less han
ull model since we only deal wi h a o ex pa icles in he cu en ime s ep. As can be
seen in Figu e 1-b he esul shows ha he o ex pa icles igge he u bulence and
ha e a good ag eemen wi h expe imen in he nea je exi egion while in a field egion
decay o kine ic ene gy is no physical. I can be in e p e ed in his way ha he ene gy
d ain o fine scale mo ion om la ge scale mo ion is no high enough and accumula ion
o kine ic ene gy on he g id flow mo ion happens. Concluding, no only new gene a ed
o ices bu also he whole se o o ices including hose gene a ed ups eam in p e ious
ime s eps ha e a significan influence on he u bulence de elopmen downs eam. Wi h
o he wo ds, he fine scale o ices model (7) and (8) can be educes bu no neglec ed.
3.3 Aniso opy o fine scale s uc u es induced by o ex pa icles
Since a de e minis ic p edic ion o a u bulen flow as men ioned in he defini ion o he
u bulen mo ion by Lesieu [13] is p ac ically impossible, he ask o e e y SGS model is
o ep oduce he subg id mo ion only in he s a is ical sense. The ollowing ea u es o he
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(a) o al eloci y (ug+u )(b) fine scale eloci y u
Figu e 2: Dis ibu ion o he diagonal Reynolds s esses componen s o he o al and fine scale eloci y
along he je axis
subg id mo ion should be cap u ed by a p ope subg id model: non-equilib ium effec s
including lamina - u bulen zone, ene gy backsca e and aniso opy o fine scale mo ion.
In his subsec ion he aniso opy o eloci ies induced by fine o ices is discussed.
The o al flow shows a well p onounced aniso opy wi h he dominance o he axial
fluc ua ions on he je cen e line (see Fig. 2-a). Reynolds s esses R
ii o he eloci y field
u shows also a clea aniso opy which is space-dependen . On he cen e line a x/D > 5
wo diagonal s esses a e equal o each o he R
22 ≈R
33 and domina e o e R
11(see
Fig. 2-b). To explain his effec , we conside s ochas ic dis ibu ion o he s a is ically
independen axis-symme ic o ex pa icles on he cen e line wi h s eng hs aligned wi h
he x-axis. They induce eloci ies u
x= 0 and u
y�= 0, u
z�= 0. Due o axis symme ical
cha ac e o each o ex he spa ially a e aged squa es o eloci ies u
yand u
za e equal,
i.e. (u
y)2= (u
z)2. P ecession o o ices a ound hei spins causes he appea ance o
he longi udinal eloci ies u
xwhich a e much smalle han u
yand u
z. Thus, he je
axis a ea a x/D > 5 is popula ed by o ex pa icles wi h axes p edominan ly o ien ed
along he je p opaga ion o mean flow di ec ion. A x/D < 5 on he cen e line and
/D =0.25 he fine scale u bulence is nea ly iso opic R
11 ≈R
22 ≈R
33 in he beginning
o he je de elopmen (see Fig. 3-a), i.e. his a ea is popula ed wi h o ex pa icles
wi h o ien a ions uni o mly dis ibu ed a ound a sphe e. Fu he downs eam he same
aniso opy akes place as ha on he je axis. A he je bounda y he fine scale u bulence
becomes aniso opic wi h a clea dominance o he adial fluc ua ions R
22 >R
11 ≈R
33
(see Fig. 3-b), i.e. he domina ing fluc ua ions a e in he di ec ion o he domina ing la ge
scale en ainmen mo ion. Figu es 4 shows he p.d. o o ex axes o ien a ion whe e Lx,
Lyand Lza e defined as Lx =αx
|α|,Ly =αy
|α|,Lz =αz
|α|.
4 CONCLUSION
The pape p esen s alida ion and e ifica ion s udy o a no el VπLES me hod which is
based on he decomposi ion o he flow s uc u es in la ge scale ones, esol ed on he g id,
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(a) (b)
Figu e 3: Dis ibu ion o he diagonal Reynolds s esses componen s o he fine scale eloci y u along
he line /D =0.25 and /D =0.5
(a) (b)
Figu e 4: p.d. o o ex axes o ien a ion line /D =0
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