Ten h In e na ional Con e ence on
Compu a ional Fluid Dynamics (ICCFD10),
Ba celona, Spain, July 9-13, 2018
ICCFD10-2018-0086
On he ole o bulk iscosi y
in comp essible eac i e shea laye de elopmen s
Radouan Boukha ane∗, Ped o J. Ma ínez Fe e ∗, A naud Mu a∗
∗PPRIME UPR 3346 CNRS, ENSMA, 86961 Fu u oscope Chasseneuil Cedex, F ance
Vincen Gio angigli∗∗
∗∗ CMAP UMR 7641 CNRS, Ecole Poly echnique, 91128 Palaiseau Cedex, F ance
Co esponding au ho : a naud.m[email p o ec ed]
Abs ac Despi e 150 yea s o esea ch a e he e e ence wo k o S okes, i should be
acknowledged ha some con usion s ill emains in he li e a u e ega ding he impo ance
o bulk iscosi y e ec s in lows o bo h academic and p ac ical in e es s. On he one hand,
i can be eadily shown ha he neglec ion o bulk iscosi y (i.e., κ= 0) is s ic ly exac
o mono-a omic gases. The co esponding bulk iscosi y e ec s a e also unlikely o al e
he low ield dynamics p o ided ha he a io o he shea iscosi y µ o he bulk iscosi y
κ emains su icien ly small. On he o he hand, o polya omic gases, he sca e ed a ail-
able expe imen al and nume ical da a show ha i is ce ainly no ze o and ac ually o en
a om negligible [13]. The e o e, since he a io κ/µ can display signi ican a ia ions
and may each e y la ge aluesa, i emains unclea o wha ex en he neglec ion o κ
holds [3]. The pu pose o he p esen s udy is hus o analyze he mechanisms h ough
which bulk iscosi y and associa ed p ocesses may al e a canonical u bulen low. In his
con ex , we pe o m di ec nume ical simula ions (DNS) o spa ially-de eloping comp ess-
ible non- eac i e and eac i e hyd ogen-ai shea laye s in e ac ing wi h an oblique shock
wa e. The co esponding low ield is o special in e es o a ious eac i e high-speed low
applica ions, e.g., Sc amje s. The co esponding compu a ions ei he neglec he in luence
o bulk iscosi y (κ= 0) o ake i in o conside a ion by e alua ing i s alue using he
EGlib lib a y [15]. The quali a i e inspec ion o he esul s ob ained o wo-dimensional
cases in ei he he p esence o he absence o bulk iscosi y e ec s shows ha he local
and ins an aneous s uc u e o he mixing laye may be signi ican ly al e ed when aking
bulk iscosi y in o accoun . This con as s wi h some mean s a is ical quan i ies, e.g., he
o ici y hickness g ow h a e, which do no exhibi any signi ican sensi i i y o he bulk
iscosi y. Ens ophy, Reynolds s ess componen s, and u bulen kine ic ene gy (TKE)
budge s a e hen e alua ed om h ee-dimensional eac i e simula ions. Sligh modi ica-
ions a e pu in o e idence on he ene gy ans e and dissipa ion con ibu ions. F om he
ob ained esul s, one may expec ha e ined la ge-eddy simula ions (LES) may be a he
sensi i e o he conside a ion o bulk iscosi y, while Reynolds-a e aged Na ie -S okes
(RANS) simula ions, which a e based on s a is ical a e ages, a e no . The il e ing o he
p esen da ase may p o ide u he insigh s so as o assess (o no ) such a conclusion.
Keywo ds: Bulk Viscosi y, Shea Laye , Di ec Nume ical Simula ion, Molecula T anspo
aI can exceed hi y o dihyd ogen.
1
1 In oduc ion
The bulk (o olume) iscosi y κ, which can ela ed o he second (o dila a ional) iscosi y
coe icien λ, is ela ed o he ib a ional and o a ional ene gy o he molecules. F om he
mac oscopic iewpoin , i cha ac e izes he esis ance o dila a ion o an in ini esimal bulk
elemen a cons an shape [2]. I is s ic ly ze o only o dilu e monoa omic gases and his
heo e ical esul is o en used o disca d i , ega dless o he na u e o in e nal s uc u e o he
luid as well as he low ield condi ions. Howe e , acous ic abso p ion measu emen s pe o med
a oom empe a u e ha e shown ha he a io o he olume o he shea iscosi y κ/µ may
be up o hi y o dihyd ogen [10], and ecen analyses o eac i e mul icomponen high-speed
lows ha e con i med ha i is no jus i ied o neglec i , excep o he sake o simplici y [3].
The dila a ional iscosi y is impo an in desc ibing sound a enua ion in gaseous media, and
he abso p ion o sound ene gy in o he luid depends i sel on he sound equency, i.e., he
a e o luid expansion and comp ession. Fo polya omic gases, he a ailable measu emen s
o κ, which emains qui e seldom due o he complexi y o i s de e mina ion, show ha i
is ce ainly no ze o and ac ually a om negligible. I is also no ewo hy ha heo e ical
analyses do show ha κ/η is a leas o he o de o uni y. The e o e, since he a io κ/µ can
display signi ican a ia ions and may each e y la ge alues, i is unclea o wha ex en he
S ockes hypo hesis (i.e., λ=−2µ/3o κ= 0) may hold o comp essible and u bulen lows
o gases ea u ing a a io κ/µ g ea e han uni y.
In ei he an expansion o a con ac ion o he gas mix u e, he wo k done by he p es-
su e modi ies immedia ely he ansla ional ene gy o he molecules, while a ce ain ime-lag
is needed o he ansla ional and in e nal ene gy o e-equilib a e h ough inelas ic colli-
sions [8]. This can be desc ibed h ough a sys em o wo coupled pa ial di e en ial equa ions
w i en o he in e nal and ansla ional empe a u es, wi h a p essu e-dila a ion e m ha
ac s as a sou ce e m in he ansla ional empe a u e budge . The olume (o bulk) iscos-
i y is associa ed o his elaxa ion phenomenon and i is e alua ed om his in e nal ene gy
elaxa ion ime-lag. The e alua ion o his p ope y o a mix u e o polya omic gases is a
om being an easy ask since he kine ic heo y o gases does no yield an explici exp es-
sion o his anspo coe icien , bu ins ead linea sys ems ha mus be sol ed [14]. The
co esponding sys ems a e de i ed om polynomial expansions o he species’ pe u bed dis-
ibu ion unc ions. The bulk iscosi y is ob ained he e using he lib a y EGlib de eloped by
E n and Gio angigli [13,15]. I is e alua ed as a linea combina ion o he pu e species olume
iscosi ies, which equi e he e alua ion o a ious collision in eg als [14].
The impac o bulk iscosi y e ec s has been p e iously analysed in se e al si ua ions in-
cluding shock-hyd ogen bubble in e ac ions [3], u bulen lames [17], comp essible bounda y
laye s [11], shock-bounda y laye in e ac ion [1], and plana shock-wa e [9]. All hese s udies
con i m ha he bulk iscosi y e ec s may be signi ican . The pu pose o he p esen wo k is
o assess i s in luence in ega d o bo h he ins an aneous and s a is ical ea u es o canonical
comp essible u bulen mul icomponen lows. Using di ec nume ical simula ion (DNS), we
in es iga e he impac o he bulk iscosi y coe icien κon he spa ial de elopmen o eac i e
and non- eac i e comp essible mixing laye s in e ac ing wi h an oblique shock wa e. Such a
canonical low ield is ypical o he shock-mixing laye in e ac ions ha ake place in com-
p essible lows o p ac ical in e es . Fo ins ance, supe sonic je s a high nozzle-p essu e a io
(NPR) gi e ise o a complex cellula s uc u es, whe e shocks and expansions wa es in e ac
wi h he u bulen ou e shea laye [7]. I is also encoun e ed in Sc amje in akes and combus-
o s, whe e shock wa es in e ac wi h he shea laye s issued om he injec ion sys ems. On
2
he one hand, i is clea ha he occu ence o shock wa es in supe sonic combus o s induces
p essu e losses ha canno be a oided bu , on he o he hand, he esul ing shock in e ac ions
wi h mixing laye s con ibu e o scala dissipa ion (i.e., mixing) a es enhancemen [4], and
may a o combus ion s abiliza ion in high-speed lows.
The p esen manusc ip is o ganized as ollows: he ma hema ical model is p esen ed in he
nex sec ion (i.e., §2), which also includes a sho desc ip ion o he nume ical me hods. The
de ails o he compu a ional se up a e subsequen ly p o ided in sec ion §3. Sec ion §4ga he s
all he esul s issued om (i) wo-dimensional nume ical simula ions o bo h ine (§4.1) and
eac i e (§4.2) cases, and (ii) he h ee-dimensional case, which is analysed in §4.3. Finally,
some concluding ema ks and pe spec i es o u u e wo ks a e p esen ed in sec ion §5.
2 Ma hema ical desc ip ion and compu a ional model
In his wo k, he in-house massi ely pa allel DNS sol e CREAMS is used. I sol es he uns eady,
h ee-dimensional se o comp essible Na ie -S okes equa ions o mul icomponen eac i e
mix u es [23]:
∂ (ρ) + ∇·(ρu)=0,(1a)
∂ (ρu) + ∇·(ρu⊗u) = ∇·σ,(1b)
∂ (ρE ) + ∇·(ρuE ) = ∇·(σ·u−J),(1c)
∂ (ρYα) + ∇·(ρuYα) = −∇·(ρVαYα) + ρ˙ωα,(1d)
whe e deno es he ime, ∇is he spa ial de i a i e ope a o , uis he low eloci y, ρis
he densi y, E =e+u·u/2is he o al speci ic ene gy (ob ained as he sum o he in e nal
speci ic ene gy, e, and kine ic ene gy), Yαis he mass ac ion o chemical species α(wi h
α∈S={1,...,Nsp}), Vαis he di usion eloci y o species α,J he hea is lux ec o and
˙ωα ep esen s he chemical p oduc ion a e o species α. The in ege Nsp deno es he numbe
o chemical species.
The abo e se o conse a ion equa ions (1) equi es o be comple ed by cons i u i e laws.
In his espec , he ideal gas mix u e equa ion o s a e (EoS), P=ρRT/Wwi h R he
uni e sal gas cons an , is used o ela e he p essu e P o he empe a u e T. In his exp ession,
he quan i y Wdeno es he mola weigh o he mul icomponen mix u e, which is ob ained
as he sum o he molecula mass o each indi idual species W−1=PNsp
α=1 Yα/Wα. Wi hin
he amewo k o he kine ic heo y o dilu e polya omic gas mix u es, he molecula di usion
eloci y ec o Vα,α∈S, hea lux ec o J, and second-o de s ess enso σa e exp essed
as ollows:
ρVαYα=−X
β∈S
ρYαDα,β (dβ+χβXβ∇( log T) ) ,(2a)
J=X
α∈S
ρVαYαhα+RTχα
Wα−λT∇T, (2b)
σ=−PI+τ=−PI+µ(∇u+∇u|) + λ(∇·u)I,(2c)
whe e Dα,β,(α, β )∈S2, a e he mul icomponen di usion coe icien s, dα,α∈S, he
species di usion d i ing o ces, χα,α∈S, he escaled he mal di usion a ios, Xα,α∈S,
he species mole ac ions, hα he en halpy pe uni mass o he α- h species, and λT he
he mal conduc i i y. The di usion d i ing o ce dαo he α- h species is gi en by dα=
3
∇Xα+ ( Xα−Yα)∇( log P). The quan i y µdeno es he shea iscosi y and λdeno es he
second (o dila a ion) iscosi y coe icien .
The bulk iscosi y coe icien κappea s explici ly in he exp ession o he iscous s ess
enso τ. A ela ionship be ween he bulk iscosi y κand iscosi y coe icien s µand λcan be
deduced om he exp ession o he o al p essu e, which can be e alua ed as he componen
o he sphe ical enso based on he ace o he o al s ess enso σ:
− (σ)
3=−
i=3
X
i=1
σii
3=P−λ+2
3µ∇·u=P−κ∇·u(3)
The second e m in he igh -hand-side o he abo e exp ession is he dila ional con ibu ion,
which de ines he bulk iscosi y as κ=λ+2µ/3. As men ioned abo e, he S okes’ hypo hesis,
s a ing ha λ=−2µ/3(and hence κ= 0), is o en e ained as a simpli ying assump ion.
Many e o s ha e been de o ed o he de i a ion o ela ionships be ween he bulk iscosi y
and undamen al luid p ope ies [21,29]. I we conside a single polya omic gas wi h a unique
in e nal ene gy mode, he in e nal ene gy elaxa ion ime τin can be ela ed o he bulk
iscosi y [8,6]:
κ=PR/c2
·cin τin ,(4)
whe e cin deno es he in e nal hea capaci y and c he speci ic hea a cons an olume.
When he e a e se e al in e nal ene gy modes and/o se e al species p esen in he mix u e,
he abo e simple exp ession is eplaced by he solu ion o a linea sys em [12]. Wi hin he
Monchick and Mason app oxima ion [26], neglec ing complex collisions cha ac e ized by mo e
han one quan um jump, he educed sys em is diagonal and yields κ[3]:
κ=PR/c2
·X
k∈P
Xkcin
kτin
k,(5)
whe e P= 1,· · · , npis he polya omic species indexing se . The a e age elaxa ion ime o
in e nal ene gy o he k- h species τin
kis hen exp essed as:
cin
k/τin
k=X
m∈N
cm
k/τm
k,(6)
whe e τm
kdeno es he a e age elaxa ion ime o in e nal ene gy mode m o he k- h species,
and Nis he in e nal ene gy mode indexing se .
The CREAMS sol e is coupled wi h he EGlib lib a y o es ima e anspo coe icien s om
he kine ic heo y o gases [16]. In his lib a y, he op imized sub ou ines EGSKma e used o
e alua e he bulk iscosi y. The in ege m∈J2,6Kassocia ed o he sub ou ine name e e s o
e ained le el o app oxima ion. The highe he alue o m, he mo e expensi e he algo i hm
bu also he mo e accu a e he bulk iscosi y exp ession. Following he wo k o Bille e al.
[3], he alue m= 3 is e ained o he pu pose o he p esen s udy. The shea iscosi y and
di usion eloci ies a e e alua ed wi h he ou ines EGFE3 and EGFYV, espec i ely. EGFLCT3 is
used o de e mine he he mal conduc i i y λTand escaled he mal di usion a ios χα.
The abo e sys em (1) is disc e ized on a Ca esian g id. A se en h-o de accu a e WENO
scheme is used o app oxima e in iscid luxes, while an eigh h-o de accu a e cen e ed di e -
ence scheme is e ained o app oxima e iscous and di usi e con ibu ions. Time in eg a ion
is pe o med wi h a hi d-o de accu a e TVD Runge-Ku a scheme. The s i ness associa ed
4
o he wide ange o ime scales in ol ed in he desc ip ion o he chemical sys em is add essed
using he Sundials CVODE sol e [20]. A s anda d spli ing ope a o echnique, simila o he
one p e iously e ained in e e ence [32], is used.
β
U1
U2
U1
U2
x2
x1
@R
Mixing laye
T ansmi ed shock
Shock gene a o
@I
Compu a ional domain
@I
Inciden shock
@R
Re lec ed shock
Figu e 1: Ske ch o he wo-
dimensional shock–mixing laye
in e ac ion geome y. The com-
pu a ional domain dimensions
a e Lx1×Lx2= 275.0×120.0
in inle o ici y hickness uni s
(i.e., δω,0). I is uni o mly dis-
c e ized using Nx1×Nx2=
1640 ×720 g id poin s.
3 P oblem s a emen and compu a ional se up
We s udy he in e ac ion o an oblique shock wi h a spa ially-de eloping shea laye . The
uppe s eam co esponds o he uel inle , i.e., a mix u e con aining hyd ogen, and he bo om
inle s eam o i ia ed ai . Bo h wo- and h ee-dimensional compu a ions a e pe o med.
Figu e 1 p o ides a ypical ske ch o he co esponding compu a ional geome y and Table 1
ga he s he alues o he main pa ame e s ele an o he p esen nume ical simula ion. The
low ini ializa ion is simila o he one e ained in e e ence [23]. Assuming equal ee-s eam
speci ic hea capaci y a ios, he con ec i e Mach numbe may be e alua ed om Mc= (U1−
U2)/(a1+a2), whe e a1and a2deno es he sonic speeds o s eams 1 (oxidize inle s eam)
and 2 ( uel inle s eam) espec i ely. Fo he p esen se o compu a ions, i is equal o
Mc= 0.48.
Fuel Oxidize
T(K) 545.0 1475.0
u1(m/s) U2U1
u2(m/s) 0.0 0.0
u3(m/s) 0.0 0.0
ρ(kg/m3)0.354 0.203
YH2(−)0.05 0.0
YO2(−)0.0 0.278
YH2O(−)0.0 0.17
YH(−)0.0 5.60 ·10−7
YO(−)0.0 1.55 ·10−4
YOH (−)0.0 1.83 ·10−3
YHO2(−)0.0 2.50 ·10−7
YN2(−)0.95 0.55
Table 1: Pa ame e s o he
shock–mixing laye in e ac-
ion case.
The mixing laye low is impinged by an oblique shock wa e ha is issued om he oxidize
inle s eam (1) a he bo om bounda y. The oblique shock wa e angle is β= 33◦, see Figu e 1.
The geome ical pa ame e s ele an o he p esen se o nume ical simula ions a e p o ided
in Table 2. The quan i ies Lx1,Lx2, and Lx3deno e he compu a ional domain leng hs in
each di ec ion no malized by he ini ial o ici y hickness δω,0, while Nx1,Nx2, and Nx3a e
5
he co esponding numbe s o g id poin s. In he wo-dimensional compu a ions, only he x1-
and x2-di ec ions a e conside ed.
Table 2: Compu a ional mesh desc ip ion.
Lx1Lx2Lx3Nx1Nx2Nx3δω,0(m)
280 130 15 1640 750 180 1.44e−4
The low is ini ialized wi h a hype bolic angen p o ile o he s eamwise eloci y compo-
nen , while he o he eloci y componen s a e se a ze o. Species mass ac ions and densi y
a e also se acco ding o he ollowing gene al exp ession:
ϕ(x1, x2, x3) = ϕ1+ϕ2
2+ϕ1−ϕ2
2 anh 2x2
δω,0,(7)
whe e ϕdeno es any o he low a iables men ioned abo e (i.e., species mass ac ion o
s eamwise eloci y componen ). The alue o he Reynolds numbe Reω, based on he ini-
ial o ici y hickness and inle eloci y di e ence ∆U = U1−U2is Reδω= 640. Di ichle
bounda y condi ions a e applied a he wo supe sonic inle s, pe ec ly non- e lec ing bound-
a y condi ions a e se a he ou low, and pe iodic bounda y condi ions a e se led along he
x3-di ec ion. A slip bounda y condi ion is imposed a he op, while he bo om bounda y
condi ion is se by using Rankine-Hugonio ela ions, gene alized o a mul icomponen mix-
u e [25]. In o de o igge low ansi ion, a sligh whi e noise luc ua ion is supe imposed
o he ans e se eloci y componen along he line (x1, x2) = (4δω,0,0). The alue o he CFL
numbe is se o 0.75. Reac i e low simula ions a e conduc ed wi h he de ailed mechanism
o O’Conai e e al. [27]. I consis s o nine chemical species (H2, O2, H2O, H, O, OH, HO2,
H2O2, and N2) and 21 elemen a y eac ion s eps. The concen a ions o hese species a he
inle ha e been de e mined om equilib ium condi ions so as o each a o able sel -igni ion
condi ions wi hin he ex ension o he compu a ional domain.
u1(m/s) u2(m/s) T(K) P(Pa) ρ(kg/m3)
Fuel 1634.0 0.0 1475.0 94232.25 0.354
Oxidize 1526.0 156.7 1582.6 129951.6 0.421
Bo om 973.0 0.0 545.0 94232.25 0.203
Table 3: Flow pa ame e s o he
shock–mixing laye in e ac ion.
Th oughou his manusc ip , he Reynolds and Fa e a e ages o any quan i y ϕa e de-
no ed by ϕand eϕ, while he co esponding s a is ical luc ua ions a e deno ed by single and
double p imes, i.e., ϕ0and ϕ00, espec i ely. A e aging is pe o med o e bo h ans e se
di ec ions o s a is ical homogenei y (x1and x2) and ime .
4 Analysis o compu a ional esul s
4.1 Ine wo-dimensional mixing laye
Figu e 2 displays he ins an aneous ield o he a io κ/µ compu ed o he p esen low con-
di ions. F om his igu e, i is no ewo hy ha (i) his a io eaches alues signi ican ly la ge
han uni y, (ii) i exhibi s impo an spa ial a ia ions, he mos signi ican o which a e e-
la ed o mix u e composi ion. This con as s wi h i s sensi i i y o p essu e a ia ions (i.e.,
6
shock wa es), which seems o emain a he mode a e. Conside ing he alues o κ/µ, as
well as he ampli ude o i s a ia ions, one may expec some ema kable e ec s o he bulk
iscosi y on his ine low ield. I is he objec i e o his p elimina y sec ion o s udy o wha
ex en he bulk iscosi y may in luence he ins an aneous and s a is ical cha ac e is ics o he
wo-dimensional mixing laye de elopmen .
Figu e 2: Ins an aneous ield o he a io κ/µ in he case κ6= 0 a ∆U/δω,0= 75.0.
Ins an aneous low isualiza ions a e e y e ealing o some local ea u es o he shea
laye , which a e il e ed ou once ields o c oss-s eam p o iles o a e aged quan i ies a e con-
side ed ins ead. Fo ins ance, quan i a i e compa isons o he onse o he s eamwise o ices
o ma ion can be ob ained om he ins an aneous ields o he dimensionless magni ude o he
densi y g adien , i.e., “nume ical Schlie en”, epo ed in Figu e 3.
(a) κ6= 0 (b) κ= 0
Figu e 3: Ins an aneous ield o he nume ical densi y-based Schlie en a ∆U/δω,0= 75.0.
In his igu e, i is ema kable ha , in he absence o bulk iscosi y, he no malized ab-
scissa a which he o ex oll-up p ocesses ake place is app oxima ely x1/δω,0= 103.0,
while in he si ua ion ea u ing non-ze o alue o he bulk iscosi y, i can be es ima ed o be
x1/δω,0= 116.0. This can be explained by he same a gumen as he one in oked by Bille
e al. [3] in hei s udy o shock/hyd ogen bubble in e ac ion. The shea laye is also a di u-
sion laye associa ed o a densi y g adien , he absence o bulk iscosi y makes he ba oclinic
e m ∇P×∇ρ/ρ2g ea e a he shock / mixing laye in e ac ion loca ion, hus a o ing
he bi h o eloci y luc ua ions. When olume iscosi y is aken in o accoun , he shock is
much smoo he in ag eemen wi h he physical heo y o shock wa e in e nal s uc u e. As a
consequence, he ba oclinic p oduc ion e m is lowe han in absence o olume iscosi y. The
in e ac ion be ween he mixing laye and he shock wa e can be u he assessed by conside -
ing he Rich mye -Meshko ins abili y de elopmen , he key poin o which is he ba oclinic
7
e ec [5]. The Rich mye -Meshko ins abili y indeed akes place when wo luids ha ing di -
e en densi ies a e impulsi ely accele a ed, simila ly o wha occu s when he shock wa e
impinges he mixing laye in he p esen s udy. This p ocess may be analysed by conside ing
he anspo equa ion o he ens ophy Ω = |ω|2/2.
Such a anspo equa ion is eadily ob ained by (i) aking he cu l o he momen um
anspo equa ion and subsequen ly (ii) mul iplying each e m o he esul ing equa ion by
he o ici y ec o i sel :
DΩ
D =∂ Ω + u·∇Ω
=1
2ω·(∇u+∇u|)·ω
| {z }
E
−Ω∇·u
| {z }
D
+ω·∇P×∇ρ
ρ2
| {z }
B
+ω·∇×∇·τ
ρ
| {z }
V
(8)
The p oduc ion / des uc ion e ms on he igh hand side (RHS) o (8) a e associa ed o
o ex-s e ching, dila a ion, ba oclinic o que, and iscous dissipa ion. The ba oclinic con-
ibu ion ∇P×∇ρplays a signi ican ole in he ens ophy and o ici y p oduc ion and i
appea s as one o he main sou ces o o ici y in supe sonic lows [31].
(a) κ6= 0
(b) κ= 0
Figu e 4: Ins an aneous ield o he mixing zone colo ed by k∇P×∇ρk.
Figu e 4 displays he ins an aneous ield o he magni ude o ∇P×∇ρwi hin a egion
es ic ed o mix u e ac ion alues such ha ξ∈[0.05,0.95]. This passi e scala ξis de ined
on he basis o conse ed elemen al (i.e., a omic) mass ac ions [28]. The mass ac ion o
chemical elemen γ, deno ed aγ, is eadily deduced om he chemical species mass ac ions:
aγ=
Nsp
X
α=1
YαNα,γAγ
Wα
,(9)
whe e Aγis he a omic weigh associa ed o elemen γand Nα,γ deno es he numbe o γ
a oms p esen in each molecule o chemical species α. Fo a wo- eeding inle sys em such as
8
he one conside ed he e1, he mix u e ac ion is hen ob ained by summing o e all elemen al
mass ac ions and no malizing he esul
ξ=Pγ|aγ−aγ,O|
Pγ|aγ,F−aγ,O|,(10)
whe e aγ,Oand aγ,Fdeno e he mass ac ions o a om γin he oxidize and uel inle s eams,
espec i ely.
Figu e 4 shows ha he e m ∇P×∇ρdisplays la ge alues a he pe iphe y o he mixing
laye whe e he p essu e g adien and he densi y g adien a e signi ican ly misaligned, hus
p omo ing he de elopmen o he mixing laye . The misalignmen o he p essu e g adien
– ha is imposed by he shock wa e – and local densi y g adien – ha is associa ed o he
mixing laye – se es as a basis o o ici y gene a ion h ough he ba oclinic e m. I may
con ibu e signi ican ly o mixing enhancemen .
0.2 0.3 0.4 0.5 0.6 0.7 0.8
20
40
60
80
100
|| P× ||
0.2 0.3 0.4 0.5 0.6 0.7 0.8
20
40
60
80
100
0.0
20.8
41.7
62.5
83.4
104.2
125.0
145.9
166.7
187.6
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8
20
40
60
80
100
20
40
60
80
100
20
60
100
140
180
ξ ξ
k∇P×∇ρk
Figu e 5: JPDF o ξand k∇P×∇ρk o κ6=(le ) and κ= 0 ( igh ).
To quan i y he o ici y p oduc ion ha is induced by he ba oclinic e m in he p es-
ence o in he absence bulk iscosi y, Figu e 5 epo s he join p obabili y densi y unc ion
(JPDF) o he passi e scala ξand ba oclinic e m k∇P×∇ρkob ained o bo h cases. This
igu e e eals ha he p oduc ion o o ici y by ba oclinic e ec s is concen a ed a ound he
s oichiome ic mix u e ac ion alue ξs = 0.43. I also con i ms ha he neglec ion o bulk
iscosi y ends o enhance la ge alues o his p oduc ion e m.
One o he undamen al s a is ical quan i ies ha cha ac e izes he mixing laye de elop-
men is i s no malized g ow h a e [30]. Al hough he de ini ion o his g ow h a e is no
unique, i s mos s anda d exp ession elies on he o ici y hickness de ini ion:
δω(x1) = U1−U2
∂eu1/∂x2|max
(11)
Figu e 6 displays he spa ial e olu ion o no malized o ici y hickness o bo h cases. I
e eals ha he in e ac ion o he e lec ed shock wa e wi h he mixing zone changes signi i-
can ly he mixing laye g ow h a e (i.e., he slope) and ha he e olu ion o he mixing laye
1O he si ua ions ea u ing mo e han wo inle s ha e been ecen ly add essed by Gome e al. [18].
9
Table 4: Values o (U1/U2)dδω/dx1ob ained om he h ee-dimensional mixing laye compu-
a ions.
1s egion 2nd egion 3 d egion
κ= 0 0.023 0.223 0.142
κ6= 0 0.022 0.192 0.137
P/Pin, a he same loca ions. I is no ewo hy ha he minimum le els achie ed by he
a e age p essu e a e la ge o he case κ6= 0. This inding is in line wi h he p e ious esul s
o Gonzalez and Emanuel [19] conce ning he high sensi i i y o he p essu e ield o he S okes
hypo hesis and he s ong modi ica ion o he p essu e dis ibu ion ob ained o la ge alues
o he a io κ/µ.
0 5 10 15 20 25
−0.4
−0.2
0.0
0.2
0.4
0.6
0.8
x2/δω,0
(eu−U2)/∆U
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(a)
0 5 10 15 20 25
0.0
0.2
0.4
0.6
0.8
1.0
x2/δω,0
e
Z
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(b)
Figu e 17: Compa ison be ween he no malized eloci y and passi e scala p o iles ob ained
o bo h case κ6= 0 and κ= 0 plo ed a h ee abscissae. The solid lines co espond o κ6= 0
and he lines wi h symbol (++) o κ= 0.
6 8 10 12 14
1.34
1.35
1.36
1.37
x2/δω,0
e
P/Pin
x1/δω,0= 140
(a)
8 10 12 14 16
1.84
1.85
1.86
1.87
1.88
x2/δω,0
e
P/Pin
x1/δω,0= 170
(b)
6 8 10 12 14 16 18
1.86
1.87
1.88
1.89
x2/δω,0
e
P/Pin
x1/δω,0= 200
(c)
Figu e 18: Longi udinal p o iles o he mean p essu e no malized by i s alue a he inle Pin,
same symbols as hose used in Figu e 17.
Since i has been shown ha he i s -o de s a is ical momen s do no display a signi ican
sensi i i y o he bulk iscosi y, a close look is now aken a some second-o de momen s ha
cha ac e ize he eloci y and passi e scala luc ua ions. Figu e 19 epo s he longi udinal
e olu ion o he maximum alues o he no malized TKE. I is no ewo hy ha he h ee-
dimensional cha ac e o he p esen se simula ion sligh ly modi ies he conclusion ha we e
16
p e iously d awn om he wo-dimensional compu a ions: he in luence o bulk iscosi y is
no iceable. Up o he abscissa x1/δω,0≈150.0, neglec ing he bulk iscosi y only leads o a
e y sligh o e es ima e compa ed o he case whe e he e ec s o κa e conside ed. The egion
ha ex ends om x1/δω,0= 150.0un il he in e ac ion wi h he e lec ed shock is cha ac e ized
by a signi ican change o beha io and he alues ob ained wi h κ6= 0 a e la ge han hose
ob ained wi h κ= 0. This egion is cha ac e ized by s ong p essu e wa e e lec ion om he
uppe limi o he compu a ional domain. A e he in e ac ion wi h he e lec ed shock wa e
(and up o he abscissa x1/δω,0= 275.0), he maxima o he TKE ob ained wi hou aking in o
accoun he bulk iscosi y e ec s a e again unde es ima ed compa ed o hose issued om he
compu a ions pe o med wi h κ6= 0 and his end is sligh ly modi ied u he downs eam as
he end o he compu a ional domain is app oached. I can be concluded ha , in he absence
o he second shock wa e in e ac ion and associa ed pa asi ic p essu e wa es issued om he
op o he compu a ional domain, he TKE le els would be unde es ima ed i he e ec s o κ
we e no aken in o accoun . In an a emp o be e unde s and he beha io o he TKE,
he analysis o he main e ms in ol ed in i s anspo equa ion is now ca ied ou .
The anspo equa ion o he u bulen kine ic ene gy Kis gi en by
∂ (ρK) + ∇·(ρe
uK) = P+ε+T+ Π + Σ,(12)
In his equa ion, Pis he p oduc ion e m, εis he dissipa ion e m, Tdeno es he u bulen
anspo e m, Πis he p essu e-s ain e m, and inally Σ he mass lux e m. The budge
(12) is deduced om he anspo equa ion o he Reynolds enso componen s:
∂(ρRij)
∂ +∂(ρeukRij)
∂xk
=Pij +εij +Tij + Πij + Σij,(13)
wi h
Pij =−ρRik
∂euj
∂xk
+Rjk
∂eui
∂xk,
εij =−τ0
ik
∂u00
j
∂xk
−τ0
jk
∂u00
i
∂xk
,
Tij =−∂
∂xkρu00
iu00
ju00
k+P0u00
iδjk +P0u00
jδik −τ0
jku00
i−τ0
iku00
j,
Πij =P0∂u00
i
∂xj
+P0∂u00
j
∂xi
,
Σij =u00
i
∂τjk
∂xk
+u00
j
∂τik
∂xk−u00
i
∂P
∂xj
+u00
j
∂P
∂xi,
(14a)
(14b)
(14c)
(14d)
(14e)
The analysis o he main e ms in ol ed in he TKE anspo equa ion is ca ied ou a wo
dis inc loca ions o in e he impac o he olume iscosi y. Figu e 20 shows ha he mos
impo an con ibu ions a e associa ed o he p oduc ion and dissipa ion e ms. Thei ampli-
ude is ound o be sligh ly smalle when κis no aken in o accoun . The u bulen anspo
e m is posi i e a he pe iphe y o he mixing laye while i ends o be nega i e wi hin he
mixing laye . This quan i y, which is la ge in he case ea u ing κ6= 0, emo es ene gy om
egions cha ac e ized by la ge luc ua ions le els o deposi i in egions cha ac e ized by lowe
le els o TKE. Figu e 20 also shows ha he con ibu ions due o p essu e-s ain and mass
lux e ms emain negligible compa ed o he o he s, o bo h cases.
17
0 50 100 150 200 250
0
2
4
6·10−2
x1/δω,0
max ( ρK)/ρ0∆U2
κ6= 0
κ= 0
Figu e 19: Spa ial e olu ion
o he longi udinal maxima o
no malized u bulen kine ic
ene gy ρK.
4 6 8 10 12 14
0.0
5.0
·10−4
x2/δω,0
x1/δω,0= 130
P(14a)
ε(14b)
(a)
4 6 8 10 12 14 16 18
−4.0
−2.0
0.0
2.0
4.0
6.0
·10−4
x2/δω,0
x1/δω,0= 200
P(14a)
ε(14b)
(b)
4 6 8 10 12 14
−4.0
−2.0
0.0
2.0
·10−4
x2/δω,0
x1/δω,0= 130
T(14c)
Π(14d)
Σ(14e)
(c)
4 6 8 10 12 14 16 18
−2.0
−1.0
0.0
1.0
2.0
·10−4
x2/δω,0
x1/δω,0= 200
T(14c)
Π(14d)
Σ(14e)
(d)
Figu e 20: TKE budge wi h all e ms no malized by ∆U3/δω,0, same symbols as hose e ained
in Figu e 17.
Figu e 21 epo s he dis ibu ion o he Reynolds s ess componen s as well as he a i-
ance o he passi e scala and he scala o eloci y co ela ions o bo h cases κ6= 0 and
κ= 0. The h ee s eamwise posi ions unde conside a ion a e ep esen a i e o he a ia ions
obse ed on he TKE p o ile epo ed in Figu e 19. The p o iles o he Reynolds s ess com-
ponen s show ha he maxima o i s diagonal componen s ollow he ends epo ed in Fig-
u e 19.Figu e 21( ), which displays he longi udinal e olu ion o he scala lux componen
]
u00
1ξ00/(u1,RMSξRMS), e eals ha he maximum alue o he co ela ion be ween he longi udinal
eloci y luc ua ion and he scala luc ua ion is sligh ly unde es ima ed when he e ec s o
bulk iscosi y a e no conside ed.
Figu e 22 epo s he a iance o he mass ac ions o chemical species p esen in he
18
0 5 10 15 20 25
0.00
0.05
0.10
0.15
0.20
x2/δω,0
q]
u00
1u00
1/∆U
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(a)
0 5 10 15 20 25
0.00
0.05
0.10
0.15
x2/δω,0
q]
u00
2u00
2/∆U
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(b)
0 5 10 15 20 25
0.00
0.05
0.10
0.15
x2/δω,0
q]
u00
3u00
3/∆U
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(c)
0 5 10 15 20 25
0.00
0.05
0.10
x2/δω,0
q]
u00
1u00
2/∆U
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(d)
0 5 10 15 20 25
0.00
2.00
4.00
6.00
·10−2
x2/δω,0
g
ξ00ξ00
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(e)
0 5 10 15 20 25
−2.00
−1.00
0.00
·10−2
x2/δω,0
]
u00
1ξ00/(u1,RMSξRMS)
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
( )
Figu e 21: P o iles o he Reynolds s ess enso componen s no malized by ∆U oge he wi h
he mix u e ac ion a iance g
ξ00ξ00 and longi udinal componen o he scala lux
]
u00
1ξ00 a h ee
abscissae, same symbols as hose e ained in Figu e 17.
mix u e. The hyd ogen, which is cha ac e ized by he highes a io κ/µ is he one ha
displays he la ges di e ences (up o app oxima ely en pe cen ) be ween he wo cases, i.e.,
κ= 0 and κ6= 0. The di e ences obse ed a he h ee loca ions conce n bo h he shape
and maximum le els, which depend on he species unde conside a ion. Indeed, i is ound
ha he dis ibu ion o he p o iles o all chemical species is sligh ly wide – indica ing ha
he luid is inco po a ed mo e ma kedly – when he e ec s o bulk iscosi y a e aken in o
accoun , which leads o a educ ion o luc ua ions a ound he a e aged alue. A simila e ec
is obse ed when he con ec i e Mach numbe alues a e inc eased [22,24].
Fo he p esen h ee-dimensional simula ion, i is in e es ing o conside he e olu ion o
19
6 8 10 12 14 16
0.0
0.2
0.4
0.6
0.8
1.0
1.2·10−4
x2/δω,0
^
Y00
H2Y00
H2
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(a)
6 8 10 12 14 16
0.0
2.0
4.0
6.0·10−3
x2/δω,0
^
Y00
O2Y00
O2
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(b)
6 8 10 12 14 16
0.0
0.5
1.0
1.5
2.0
·10−3
x2/δω,0
^
Y00
H2OY00
H2O
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(c)
Figu e 22: P o iles o he a iances o chemical species mass ac ions a h ee abscissae, same
symbols as hose e ained in Figu e 17.
highe momen s ( hi d- and ou h-o de ) o eloci y and passi e scala luc ua ions. Once
p ope ly no malized, hese quan i ies p o ide he skewness and ku osis o he s a is ical dis-
ibu ion, i.e., he p obabili y densi y unc ions (PDF). They a e de ined by S=µ3/σ3and
F=µ4/σ4, espec i ely, wi h µ3and µ4 he hi d and ou h cen e ed momen s, σbeing he
s anda d de ia ion. The skewness ac o measu es he symme y o he luc ua ions a ound he
mean whils he la ness ac o cha ac e izes i he PDF ends o be peaked o no . Thei al-
ues can be compa ed o hose associa ed o a Gaussian dis ibu ion wi h SG= 0 and FG= 3.
Figu e 23 displays he ans e se p o iles o Su1,Su2,Fu1, and Fu2ob ained in h ee planes
o κ= 0 and κ6= 0. In he ee s eams (i.e., ou side he mixing laye ), he skewness and
la ness coe icien s end o 0.0and 3.0, espec i ely, which e lec s a quasi-Gaussian beha iou
o he esidual u bulence ou side he mixing zone. The conside a ion o bulk iscosi y e ec s
a o s his end. A sha p a ia ion o hese wo coe icien s is obse ed when app oaching he
mixing zone bounda ies. These changes a e associa ed o he in e mi ency be ween u bulen
“pu s” (mixed luid) in he ees eam as well as luid incu sions in o he mixing laye ( luid
en ainmen ). The sign o he a ia ion o he asymme y coe icien depends upon he s eam
om which he mixing laye bounda y is app oached, ei he om he as -cold s eam side
o om he slow-ho s eam. On he one hand, on he high-speed and low- empe a u e side,
in e mi en e en s, which dis up he as and cold uni o m low, co espond o ho e and
slowe condi ions and gi e ise o Su1<0and Su2<0. On he o he hand, on he low speed
and high empe a u e side, such e en s co espond o colde and as e luid pa icles, and we
20
ha e Sξ<0,Su1>0, and Su2>0. These in e mi en e en s in luence he la ness coe i-
cien s as ab up ly as he skewness coe icien s, see Figu e 23. The e o e, he di e en p o iles
app oach an an isymme ic shape o he skewness coe icien s and a symme ical shape o
he la ness coe icien s. The posi ions whe e he asymme y coe icien s Su1and Su2cancel
ou co espond o he posi ions o he ex ema on hei espec i e a iance p o ile. The bulk
iscosi y has he e ec o ampli ying he ampli ude o he peaks obse ed o he asymme-
y and la ness coe icien s, which may be explained by he s abilizing ac ion o molecula
p ocesses ha induces a delayed de elopmen o he mixing laye and a subsequen delayed
ac ion o molecula p ocesses. In addi ion, om he inne pa o he mixing laye owa ds i s
bounda ies (inle s eam condi ions), i has o be no iced ha he in e mi en zone appea s
ea ly in he p esence o bulk iscosi y e ec s.
0 5 10 15 20 25
−2
−1
0
1
x2/δω,0
Su1
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(a)
0 5 10 15 20 25
−2
−1
0
1
2
x2/δω,0
Su2
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(b)
0 5 10 15 20 25
2
4
6
8
10
12
14
x2/δω,0
Fu1
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(c)
0 5 10 15 20 25
5
10
15
20
x2/δω,0
Fu2
x1/δω,0= 140
x1/δω,0= 170
x1/δω,0= 200
(d)
Figu e 23: Skewness and la ness coe icien s issued om he s a is ics o he wo eloci y
componen s u1and u2, same symbols as hose e ained in Figu e 17.
Figu e 24 epo s he longi udinal e olu ion o he ens ophy maximum Ωno malized by
(∆U/δω,0)3. These wo p o iles display h ee dis inc egions. Be o e he in e ac ion o he
second e lec ed shock wi h he mixing laye , he maximum alue o he ens ophy ob ained
wi h κ= 0 seems o be unde es ima ed in compa ison o ha ob ained wi h κ6= 0 while
beyond he second in e ac ion, he wo alues become qui e compa able.
Two speci ic loca ions ha a e ypical o each egion a e now conside ed o s udy he o igin
o obse able di e ences in he ligh o Eq. (8). I has o be no ed ha he mos impo an
con ibu ions o he ens ophy budge a e associa ed o o ex s e ching and iscous di usion.
The e is an indi ec e ec o bulk iscosi y ha ends o p omo e he s e ching e m and, as
a di ec ou come o he inc eased molecula e ec s, he iscous di usion e m is also sligh ly
21
0 50 100 150 200 250
0
0.2
0.4
0.6
0.8
x1/δω,0
max ( Ω ) /( ∆U/δω,0)2
κ6= 0
κ= 0
Figu e 24: Longi udinal e olu-
ion o he no malized ens ophy
maxima.
la ge when κ6= 0. Finally, he ba oclinic e m is sligh ly la ge in he absence o bulk
iscosi y, while he dila a ion con ibu ion does no seem o be signi ican ly modi ied by he
conside a ion o bulk iscosi y e ec s.
4 6 8 10 12 14
−0.05
0.00
0.05
0.10
0.15
x2/δω,0
x1/δω,0= 130
E(8)
V(8)
(a)
4 6 8 10 12 14 16 18
0.00
0.05
0.10
x2/δω,0
x1/δω,0= 200
E(8)
V(8)
(b)
4 6 8 10 12 14
−1.50
−1.00
−0.50
0.00
0.50 ·10−2
x2/δω,0
x1/δω,0= 130
B(8)
D(8)
(c)
4 6 8 10 12 14 16 18
−6.00
−4.00
−2.00
0.00
2.00
·10−3
x2/δω,0
x1/δω,0= 200
B(8)
D(8)
(d)
Figu e 25: Ens ophy budge no malized by (∆U/δω,0)3, same symbols as hose e ained
in Figu e 17.
5 Summa y and conclusions
In he p esen manusc ip , wo- and h ee-dimensional nume ical simula ions o spa ially-
de eloping comp essible mixing laye s impac ed by an oblique shock wa e a e conduc ed o
a con ec i e Mach numbe Mc= 0.48. The emphasis is placed on he possible in luence o
he bulk iscosi y on he mixing p ocesses. Thus, a mix u e o hyd ogen and ai is conside ed
22
in condi ions ha a e ep esen a i e o expe imen al benchma ks ele an o high-speed low
combus ion. In a i s s ep o he analysis, wo-dimensional compu a ions o ine and eac i e
mixing laye s a e compu ed. A signi ican impac o he bulk iscosi y is obse ed on he
ins an aneous low ields while a e aged quan i ies do no exhibi any ema kable modi ica ion.
I is also wo h no ing ha he eac i e cases only display sligh di e ences wi h espec o
ine cases: his is especially ue o he longi udinal e olu ions o he o ici y hickness and
u bulen kine ic ene gy. Th ee-dimensional simula ions o ine mixing laye s a e subsequen ly
conduc ed. The in luence o he bulk iscosi y is mo e isible in hese h ee-dimensional
cases: i ends o educe he mixing laye g ow h a e compa ed o he case whe e i is no
aken in o accoun . The compa ison is also pe o med in e ms o highe -o de s a is ical
momen s. This las pa o he analysis shows ha he bulk iscosi y e ec s end o ampli y
he eloci y g adien s a he bounda ies o he mixing laye , and consequen ly a o he e u n
o equilib ium. F om he abo e syn hesis o he ob ained esul s, one may expec ha e ined
la ge-eddy simula ions (LES) may be a he sensi i e o he conside a ion o bulk iscosi y,
while Reynolds-a e aged Na ie -S okes (RANS) simula ions, which a e based on s a is ical
a e ages, a e no . Finally, om he p esen se o esul s, i is ecommended o ake he bulk
iscosi y e ec s in o accoun especially when highly- esol ed la ge-eddy simula ions (LES) a e
conside ed.
Acknowledgmen s
The compu a ions we e pe o med using he High Pe o mance Compu ing esou ces om
he mésocen e de calcul poi e in and om genci unde alloca ions x20142a0912 and
x20142b7251. The i s au ho also bene i ed om in e es ing discussions wi h Aimad E - aiy.
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