MCS 8 C¸ es¸me, Izmi , Tu key, Sep embe 8-13, 2013
COMPRESSIBILITY AND FINITE-RATE CHEMISTRY INFLUENCE
ON REACTIVE SHEAR LAYERS DEVELOPMENT
Ped o J. Ma ´
ınez Fe e , Guillaume Lehnasch and A naud Mu a
[email p o ec ed]
Ins i u Pp ime - UPR 3346 - CNRS - ENSMA - Uni e si ´
e de Poi ie s
BP 40109, 86961 Fu u oscope, F ance
Abs ac
The shea laye s g ow h a e is known o be al e ed by comp essibili y and hea elease e ec s.
On he one hand, o inc easing le els o comp essibili y, he e is indeed a dec ease in p essu e
luc ua ions leading o a educ ion in p essu e-s ain e ms, which is esponsible o he dec ease
in g ow h a e. On he o he hand, p e ious nume ical simula ions showed ha he main in lu-
ence o he he mal expansion is o educe he magni ude o he o ici y wi hin he o ex co es
and eac ion zones, while he ba oclinic o que gene a es al e na e egions o posi i ely and
nega i ely signed o ici y a he b aids and icini y o o ex co es. The combina ion o he
wo e ec s a o s he di usion o he o ici y ield as well as i s a enua ion wi hin he o ex
co e, which esul s in lowe g ow h a es o shea ed laye s. Such insigh s we e mos ly gained
om he conside a ion o empo ally de eloping mixing laye analyses ha ela e he longi u-
dinal spa ial coo dina e o ime ia he in oduc ion o an a i icial a e age eloci y ac oss he
laye . In his case, he low becomes symme ic wi h espec o i s de elopmen axis, and i does
no dis inguish be ween he luids om he wo s eams wi h di e en eloci ies. The analysis
o bo h comp essibili y and hea elease is e isi ed he e by conside ing nume ical simula ions
o spa ially-de eloping high speed eac i e shea laye s o di e en con ec i e Mach numbe
alues including he in luence o ini e a e chemis y e ec s. The e olu ion o he o ici y
hickness g ow h a e is analysed in de ail in he ligh o he u bulen kine ic ene gy and ens o-
phy budge s. I he use o global single s ep chemis y end o suppo p e ious indings, ha
we e ob ained om empo ally de elopping mixing laye simula ions, he conclusions p esen ly
d awn om he conside a ion o ini e a e chemis y e ec s h ough he use o de ailed chem-
ical kine ics a e di e en . Finally, hese in es iga ions a e u he ex ended o he conside a ion
o shocked ( eac i e) mixing laye s, a si ua ion signi ican ly less documen ed in he li e a u e.
In oduc ion
Nume ical simula ions o high-speed eac i e shea laye s al eady concen a ed a la ge
amoun o esea ch e o s, see o ins ance [1, 2], and he in e es in such eac i e low con-
di ions has been, o a la ge ex en , mo i a ed by he de elopmen o ai b ea hing ehicles able
o c uise a hype sonic speeds. Mos o he a o emen ioned wo ks deal wi h empo ally de-
eloping u bulen eac i e mixing laye s and o en conside ed ei he single-s ep chemis y o
in ini ely as chemis y. Nume ical simula ions o spa ially-de eloping high speed mixing lay-
e s we e much seldom epo ed in he li e a u e and mainly de o ed o ine si ua ions [3, 4].
The amoun o compu e esou ces needed o ob ain an equi alen esolu ion is indeed g ea e
o spa ially g owing laye s bu hey display some impo an di e ences wi h empo al simu-
la ions. In spa ially-de eloping mixing laye s, e en s ha occu downs eam can in luence he
low ield ups eam, whe eas in empo ally de eloping laye s, no e en can a ec he low a p e-
ious imes. Mo eo e , in con as wi h empo ally de eloping laye s, en ainmen a es o luid
om he wo s eams a e no necessa ily he same. In he p esen wo k, di ec nume ical simu-
la ions o spa ially-de eloping shea laye s a e pe o med o eac i e condi ions, and including
a de ailed desc ip ion o molecula anspo also. Figu e 1, which epo s a nume ical Schlie en
supe imposed wi h empe a u e iso-lines a a con ec i e Mach numbe alue Mc=0.8, p o-
ides a ypical example o he esul s ha may be ob ained om he wo-dimensional nume ical
simula ion o a mixing laye de eloping be ween i ia ed ai and hyd ogen mix u e s eams 1.
Figu e 1: Tempe a u e iso-con ou s supe imposed on nume ical Schlie en o he p essu e ield.
T anspo equa ions and nume ical me hods
The p esen nume ical s udy is ca ied ou wi h a mul icomponen comp essible and eac-
i e low sol e . Special a en ion is paid o he compe i ion ha may exis be ween molec-
ula di usion and chemical kine ics as well as complex low ield s uc u es ha may ea u e
shock and expansion wa es. We he e o e conside he comp essible Na ie -S okes equa-
ions w i en o a eac i e mul icomponen mix u e cha ac e ized by he conse a i e ec o
q=�ρ,ρu1,ρu2,ρu3,ρe ,ρY1,...,ρYα,...,ρYNsp � , whe e ρdeno es he densi y, uia e he
componen s o he eloci y ec o , pis he p essu e, e is he o al ene gy, Yαis he mass ac ion
o he α h species (α=1,...,Nsp) wi h Nsp he o al numbe o chemical species.
The co esponding sys em o conse a i e equa ions is supplemen ed wi h he ideal-gas
equa ion o s a e p=ρRT/W, wi h R he uni e sal gas cons an and W he molecula mass
o he mix u e ob ained om W−1=�Nsp
α=1 Yα/Wα. The speci ic hea capaci ies a cons an
p essu e and en halpies o each species a e exp essed as polynomial o ms o he empe a u e
in ol ing a se ies o coe icien s de e mined om he JANAF da abases. The j-componen o
he hea lux is e alua ed om Jj=�Nsp
α=1 ρYαVα,j (hα+RT˜χα/Wα)−λ∂T/∂xj, whe e λis
he he mal conduc i i y o he mix u e and ˜χαis he escaled he mal di usion a ios o he α h
species, which is de ined in such manne ha DαβXβ˜χβ=θα, wi h αand β∈[1, ..., Nsp]. In
he p e ious exp ession, θαand Dαβ deno e he he mal di usion ec o and di usion ma ix,
espec i ely, while Xβdeno es he β h species mola ac ion. The j-componen o he molecu-
la di usion lux is ep esen ed by ρYαVα,j =−�Nsp
β=1 ρ˜
Dαβ (dβ,j +(Xβ˜χβ/T)∂T/∂xj), wi h
dβ,j =∂Xβ/∂xj+(Xβ−Yβ)∂ln P/∂xj, whe e ˜
Dαβ is he lux di usion ma ix o med by
he lux di usion componen s YαDαβ. All he anspo coe icien s men ioned abo e, i.e., ˜
Dαβ,
˜χβas well as he olume and shea iscosi y κand µa e e alua ed using he gene al pu pose
o an lib a y EGLIB [5].
The sol e handles de ailed chemical kine ics as desc ibed h ough N elemen a y eac-
ion s eps in ol ing Nsp chemical species �Nsp
α=1 ν�
α,jMα��Nsp
α=1 ν��
α,jMα,j=1,...,N ,
whe e Mαis he chemical symbol o he α h species, while ν�
α,j and ν��
α,j deno e he s oichio-
me ic coe icien s. The esul ing chemical sou ce e ms a e gi en by ˙ωα=�N
j=1(�
α,j −ν�
α,j)qj
wi h qj=k j �Nsp
α=1 [Xα]ν�
α,j −k j �Nsp
α=1 [Xα]ν��
α,j , whe e k j and k j deno e he o wa d and
e e se a e cons an s o he j h elemen a y eac ion, espec i ely, and [Xα]is he mola con-
cen a ion o he α h species.
The ea men o he in iscid componen o he anspo equa ion o he conse a i e ec-
o elies on he se en h-o de accu a e Weigh ed Essen ially Non-Oscilla o y (WENO7) econ-
1No e ha (eddy) shockle s a e isible in Fig. 1, and hei p esence has been also e idenced in he h ee-
dimensional nume ical simula ions we conduc ed unde he same condi ions.
s uc ion o he cha ac e is ic luxes [6]. In p ac ice, he nume ical sol e uses a se en h-o de
accu a e cen e ed ini e di e ence scheme, and he applica ion o he WENO7 scheme is con-
di ioned o a smoo hness c i e ion which in ol es he local alues o he no malized spa ial
a ia ions o bo h p essu e and densi y [7]. The iscous and molecula di usion lux unc ions
a e de e mined using an eigh h-o de cen e ed di e ence scheme. The empo al in eg a ion is
pe o med using he S ang spli ing echnique: he non- eac i e pa o he equa ions is in e-
g a ed in ime hanks o an explici hi d-o de TVD Runge-Ku a algo i hm [8] while he chem-
ical p oduc ion a es a e calcula ed by sol ing a s i sys em o O dina y Di e en ial Equa ions
(ODEs) wi h he gene al pu pose sol e VODE [9]. Fu he de ails abou he nume ical me hods
as well as an exhaus i e e i ica ion o he sol e a e p o ided in e e ences [7, 10].
Table 1: Inle s eam condi ions e ained o ine and eac i e nume ical simula ions.
Global chemis y De ailed chemis y
Fuel s eam Oxidize s eam Fuel s eam Oxidize s eam
P(Pa) 94232.25 94232.25 94232.25 94232.25
T(K) 545.0 1475.0 545.0 1475.0
ρ(kg/m3)0.354 0.203 0.354 0.203
YH2(−)0.05 0.0 0.05 0.0
YO2(−)0.0 0.278 0.0 0.278
YH2O(−)0.0 0.17 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 5.10 ×10−6
YH2O2(−)- - 0.0 2.50 ×10−7
YN2(−)0.95 0.552 0.95 0.55
Flow con igu a ion and nume ical se up
The pa ame e s cha ac e izing each inle s eam a e de e mined om p e ious expe imen al
and nume ical s udies, see o ins ance e e ences [11, 12, 13] which a e ela ed o supe sonic
combus ion amje (Sc amje ) condi ions. In hese s udies, hyd ogen is he e e ence uel be-
cause o i s high le el o ene gy pe uni weigh and sho e igni ion delays. Fo ins ance, in he
expe imen al s udy conduc ed by Mille e al. [11], he uel s eam is composed o hyd ogen
dilu ed wi h ni ogen (o helium) while he oxidize s eam co esponds o i ia ed ai ob ained
om combus ion ope a ed be ween hyd ogen and ai in a p e-chambe . The le els o hea e-
lease eached in his expe imen we e no la ge enough o induce signi ican changes in he la ge
scale de elopmen o he eac i e shea laye , which emains e y simila o hose p e iously
epo ed o ine cases a he same le el o comp essibili y. As he p esen s udy aims a in es-
iga ing such hea elease e ec s in high speed shea lows, he e ained inle low condi ions,
p esen ed in Table 1, sligh ly di e om hose p e iously conside ed in [11]. We will deno e
by GC he simula ions based on he global single s ep chemis y while DC will e e o simu-
la ions pe o med wi h a de ailed chemis y ep esen a ion. Condi ions a chemical equilib ium
a e de e mined om a p elimina y s udy in o de o se le he adical mass ac ion alues in he
i ia ed ai s eam, see Table 1, he in luence o which is a om being negligible [11].
F om his p elimina y s udy, we also de e mined he adiaba ic empe a u e a equilib ium as
well as he au o-igni ion imes. Figu e 2(a) epo s he empe a u e a equilib ium as a unc ion
o he mix u e ac ion z, i.e., a passi e scala de ined o be uni y in he uel inle s eam and
ze o in he oxidize inle s eam. I is compa ed o he alues associa ed wi h o he inle s eams
condi ions ha we e p e iously conside ed in he li e a u e: Mille e al. [11], Cheng e al. [12]
and Seka e al. [13]. The co esponding da a can be used o ob ain a ough es ima e o he
maximum le els o empe a u e ha may be ob ained wi hin he shea laye and associa ed hea
elease. On he one hand, in he con igu a ion o Mille e al. [11], he adiaba ic empe a u e a
equilib ium emains lowe han he i ia ed ai s eam empe a u e. On he o he hand, in he
con igu a ions o Cheng e al. [12] and Seka e al. [13], hea elease is clea ly mo e p onounced.
Finally, ou con igu a ion appea s as an in e media e case ea u ing a maximum empe a u e o
2400 K in he icini y o he s oichiome y (zs =0.41) and, he e o e, le els o hea elease a e
expec ed o p oduce signi ican changes in he la ge scale de elopmen o he shea laye .
T(K)
0
500
1000
1500
2000
2500
3000
0 0.2 0.4 0.6 0.8 1
P esen mix u e
Seka e al. (1990)
Cheng e al. (1994)
Mille e al. (1998)
z(−)
(a)
(µs)
10
100
1000
0 0.2 0.4 0.6 0.8 1
DC, pu e ai
DC, i ia ed ai
GC
z(−)
(b)
Figu e 2: Adiaba ic empe a u e le el a equilib ium (a), and sel -igni ion ime scale (b), o
di e en alues o he mix u e ac ion.
Figu e 2(b) p o ides he sel -igni ion delays o he mix u e conside ed in his wo k. Resul s
associa ed wi h global chemis y (GC) ha e been ob ained by using he single s ep global eac-
ion o Ma ino e al. [14] while hose associa ed wi h de ailed chemis y (DC) a e based on he
chemical eac ion mechanism o O’Conai e e al. [15] ea u ing 21 elemen a y eac ions s eps
and 10 species. As expec ed, he sho es igni ion ime alues a e ob ained om he global
eac ion mechanism. Howe e , he compa ison o he esul s ob ained wi h ei he a pu e ai
o a i ia ed ai oxidizing s eam con i ms ha he p esence o chemical adicals signi ican ly
sho ens he sel -igni ion delay, and also enla ges he lammable domain.
The inle mix u es ha ha e been conside ed abo e, see Table 1, will be used o speci y he
bounda y condi ions o spa ially-de eloping nume ical simula ions o bo h ine and eac i e
mixing laye s. The co esponding nume ical simula ions will be conduc ed o di e en le els
o comp essibili y cha ac e ized by con ec i e Mach numbe alues Mc=0.35,Mc=0.70
and Mc=1.10. In he case Mc=1.1, wo-dimensional nume ical simula ions did no lead
o he des abiliza ion o he shea laye . This con i ms he conclusion o linea and nonlin-
ea s abili y analyses, which indeed show ha , as he con ec i e Mach numbe inc eases, he
mos ampli ied modes a e no longe wo-dimensional and become h ee-dimensional wi h pe -
u ba ions p opaga ing wi h an angle θ ela i e o he (x1,x
2)plane. This beha io has been
also assessed o comp essible eac i e mixing laye s [16]. In p ac ice, despi e he g ow h in
compu ing pe o mance, pa ame ic DNS in es iga ions wi h de ailed anspo and chemical
kine ics desc ip ion s ill emain limi ed by compu a ional esou ces. The p esen s udy is he e-
o e es ic ed o wo-dimensional nume ical simula ions, which ha e al eady been shown o
accu a ely po ay he cha ac e is ic la ge-scale ollup and o ex-pai ing p ocesses. Implica-
ions o chemis y-induced hea elease on hese p ocesses a e hus expec ed o be meaning ully
add essed wi h such nume ical simula ions. The main conclusions ob ained he ein indeed con-
ce n he la ge-scale de elopmen o shea laye s, i.e., unde he p incipal in luence o he la ge
eddies esul ing om he Kel in-Helmhol z p ima y ins abili y. Such cha ac e is ics a e clea ly
less sensi i e o h ee-dimensional small-scale dynamics. Finally, i is no ewo hy ha he in-
sigh s gained om hese wo-dimensional nume ical simula ions ha e been con i med om he
esul s o wo dis inc h ee-dimensional nume ical simula ions o e e ence, which ha e been
conduc ed o Mc=0.35 unde ine condi ions, and Mc=0.70 wi h de ailed chemis y unde
eac i e condi ions.
In all cases o be in es iga ed, he wo inle s eams emain supe sonic. The eloci y
in he oxidize inle s eam is inc eased o each he desi ed alue o he con ec i e Mach
numbe , while he eloci y in he uel inle s eam is kep ba ely la ge han he speed o
sound. The ini ial o ici y hickness, δω,0, is adjus ed o main ain he same alue o he ini ial
Reynolds numbe Reω,0=640 o all he simula ions. Nume ical simula ions a e conduc ed
in a compu a ional domain wi h dimensions L1×L2=350δω,0×80δω,0 o Mc=0.35 and
L1×L2=350δω,0×122δω,0 o Mc=0.70. These compu a ional geome ies a e espec i ely
disc e ized wi h N1×N2= 1739×369 poin s and N1×N2= 1739×403 poin s. Mesh s e ching
is applied in bo h di ec ions o space. In he ans e se di ec ion, he g id size ∆x2=0.168δω,0
is main ained cons an om he cen e o he domain o x2=±20δω,0. A sligh s e ching is
also applied along he longi udinal di ec ion om he beginning o he compu a ional domain,
whe e ∆x1=0.40δω,0, o x1=150δω,0whe e a cons an g id size ∆x1=0.168δω,0is se .
In any case, he maximum g id size alues con o m wi h hose e ained in he di ec nume ical
simula ions conduc ed by Fu and Li [3] and he ex en o he compu ional domain is su icien
o allow o he shea laye de elopmen [4]. Finally, he simula ions a e ini ialized wi h an hy-
pe bolic angen p o ile o he s eamwise eloci y componen , empe a u e and species mass
ac ions [1]. A Di ichle bounda y condi ion is applied a he inle while non- e lec ing condi-
ions a e se a he ou le , bo om and op o he domain. A andom pe u ba ion is supe imposed
on he ans e se eloci y componen wi hin a gaussian spo cen e ed a (x1,x
2)=(4δω,0,0) o
igge he des abiliza ion o he mixing laye .
Spa ially-de eloping shea laye s analysis
Figu e 3 shows he ins an aneous s uc u e o he ine and eac i e shea laye s o wo
con ec i e Mach numbe alues. The op inle s eam is associa ed wi h uel injec ion and he
bo om wi h oxidize injec ion. The s uc u e co esponding o he ine case a Mc=0.35
(I-2D-0.35) shows a ce ain le el o cohe ence wi h h ee (app oxima ely) equidis an ellip ical-
shaped o ices in he domain x2/δω,0∈[200,350]. As he con ec i e Mach numbe alue is
inc eased, he low ield s uc u e becomes mo e dis o ed and he cohe ence e idenced in he
p e ious case is no longe obse ed. Fo he eac i e cases, wo di e en low s uc u es can be
obse ed depending on he e ained eac ion mechanism. Wi h he de ailed eac ion mechanism
o O’Conai e e al. [15], he low ield s uc u e emains simila o he one o he ine cases.
Hea elease le els a e p esen on he uel lean side, delinea ing he high empe a u e egions.
These egions a e con inuous a Mc=0.35 (DC-2D-0.35) and become isola ed spo s a Mc=
0.70, sugges ing ha he in luence o hea elease is less impo an in his case. Wi h he global
eac ion mechanism o Ma ino e al. [14], he shea laye is mo e se e ely a ec ed by hea
eleased, wha e e he le el o comp essibili y. Low densi y luid in he mixing egion esul s
in a shi o he uns able mode o lowe wa e numbe s (longe wa eleng hs). Indeed, when
single s ep chemis y is e ained, he la ges pa o hea elease occu s in he ea ly de elopmen
o he shea laye , which explains why he e a e only ew aces o hea subsequen ly eleased
u he downs eam, whe e almos all hyd ogen and oxygen ha e al eady been con e ed in o
combus ion p oduc s.
Table 2 epo s some quan i a i e con i ma ions o he quali a i e obse a ions discussed
abo e. The o ici y hickness δωis de ined as
δω=∆U
|∂�u1� /∂x2|max
,(1)
whe e ∆U=U1−U2. Ope a o �·� e e s o Reynolds a e age while he same ope a o wi h
index , i.e., �·� , deno es Fa e a e age. P imes and double p imes indica e he co esponding
luc ua ions. I can be seen ha he o ici y hickness g ow h a e, gi en by η−1dδω/dx1wi h
η=(U1−U2)/(U1+U2), is g ea e in ine cases. Signi ican ly lowe alues o he g ow h
a e a e ob ained wi h global chemis y, which ea u es he sho es igni ion delays and also he
la ges hea elease le els. Wi h a de ailed chemical kine ics desc ip ion, he same endency is
obse ed a Mc=0.35. Howe e , he g ow h a e alue eached a Mc=0.70 is e y close o
he one p e iously ound unde ine condi ions, see Table 2, which is con i med by he low ield
s uc u es displayed in Figs. 3(b) and 3(d). Hea elease le els become indeed lowe a highe
le els o comp essibili y when using de ailed kine ics, whe eas he opposi e end is obse ed
wi h global single-s ep chemis y. The esul s ob ained he e wi h de ailed chemical kine ics
con as wi h hose issued om se e al nume ical simula ions conduc ed in simila condi ions
unde he as chemis y assump ion which did no e idence he same e ec s.
(a) I-2D-0.35 (ine , Mc=0.35). (b) I-2D-0.70 (ine , Mc=0.70).
(c) DC-2D-0.35 (de ailed chemis y, Mc=0.35). (d) DC-2D-0.70 (de ailed chemis y, Mc=0.70).
(e) GC-2D-0.35 (global chemis y, Mc=0.35). ( ) DC-2D-0.70 (global chemis y, Mc=0.70).
Figu e 3: Fields o he no malized empe a u e c=(T−Tmin)/(Tmax −Tmin)supe imposed
wi h iso-lines o hea elease o he ine and eac i e spa ially-de eloping shea laye s.
Figu e 4 p o ides he no malized g ow h a e alues. Resul s ob ained o ine condi ions
a e ound compa able wi h s anda d da a a ailable om he li e a u e [16, 17, 18, 19]. He e,
he e e ence g ow h a e δ�
ihas been ob ained om he p elimina y simula ion o an ine and
incomp essible ai -ai mixing laye , which is no p esen ed he e jus o he sake o conciseness.
The alues o he Reynolds numbe , Reω=Reω,0δω/δω,0, eached a he loca ion x1=300δω,0,
whe e mos o he da a pos -p ocessing will be pe o med, a e also compa able o hose associ-
a ed wi h p e ious nume ical in es iga ions conduc ed in simila condi ions [1, 2, 20].
Table 2: Resul s ob ained om ine
and eac i e spa ially-de eloping shea
laye s. The Reynolds numbe is com-
pu ed a x1=300δω,0.
Cas η−1dδω/dx1Reω
I-2D-0.35 0.143 6421
I-2D-0.70 0.133 8247
DC-2D-0.35 0.103 5158
DC-2D-0.70 0.129 7884
GC-2D-0.35 0.049 3735
GC-2D-0.70 0.031 4638
δ�/δ�
i(−)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Dimo akis (1991)
Debbischop & Bonne (1993)
Hall e al. (1993)
Day e al. (1998)
Langley exp. cu e
Fu & Li (2006)
I-2D
CD-2D
CG-2D
Mc(−)
Figu e 4: No malized g ow h a es s. Mc.
Figu e 5 illus a es he budge o he u bulen kine ic ene gy (TKE) o each simula ion as
a unc ion o x2/δω. The TKE anspo equa ion is gi en by
D(�ρ�k)
D =−�ρ��Rlk
∂�ul�
∂xk�−�ρ��1
�ρ��τ�
lk
∂u��
l
∂xk
��
−1
2
∂
∂xk
�ρu��
lu��
lu��
k+2p�u�
lδlk −2τ�
lku��
l�+�p�∂u��
k
∂xk
�+�u��
l��∂�τlk�
∂xk
−∂�p�
∂xl�.
(2)
The i s con ibu ion in he RHS o he abo e equa ion is he p oduc ion e m �ρ�P. I
is ollowed by he dissipa ion a e �ρ��, he anspo e m T, he p essu e dila a ion con ibu-
ion Π, and he mass lux coupling e m Σ[20]. As shown in Fig. 5, o he p esen se o
wo-dimensional nume ical simula ions o ine and eac i e mixing laye s, he mos impo an
con ibu ions o he TKE budge a e he p oduc ion and anspo e ms, �ρ�Pand T. Thei
le els ge educed as he con ec i e Mach numbe Mcis inc eased o hea elease is p esen , a
beha io simila o he one obse ed o he o ici y hickness g ow h a e. This endency is
e en mo e p onounced o he simula ion conduc ed wi h global chemis y a Mc=0.70.
The in luence o comp essibili y and hea elease on he shea laye g ow h a e is u he
in es iga ed by s udying he ens ophy anspo equa ion
D�ωiωi�
D =2�ωiωj
∂ui
∂xj
�−2�ωiωi
∂uj
∂uj
�+2�eijkωi
1
ρ2
∂ρ
∂xj
∂p
∂xk
�+2�eijkωi
∂
∂xj�1
ρ
∂τkm
∂xm��.(3)
The ou e ms appea ing in he RHS o Eq.(3) desc ibe, espec i ely, o ex s e ching, olu-
me ic expansion (o comp ession), ba oclinic o que and iscous e ec s. The dila a ion and
ba oclinic mechanisms a e s ic ly ze o in cons an -densi y lows. Volume ic dila a ion may
occu when he densi y dec eases due o hea elease and, since he e m ∂uj/∂xjis posi i e o
expanding lows, he quan i y −ωiωi∂uj/∂xjis associa ed wi h a possible ens ophy educ ion
mechanism. Finally, he ba oclinic o que ep esen s he gene a ion (o des uc ion) o o ici y
induced by he di e en ial luid accele a ions ha esul om non-aligned p essu e g adien s
and densi y g adien s.
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
-3 -2 -1 0 1 2 3
〈
ρ
〉P
〈
ρ
〉
ε
T
Π
Σ
x2/δω(−)
(a) I-2D-0.35.
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
-3 -2 -1 0 1 2 3
〈
ρ
〉P
〈
ρ
〉
ε
T
Π
Σ
x2/δω(−)
(b) DC-2D-0.35.
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
-3 -2 -1 0 1 2 3
〈
ρ
〉P
〈
ρ
〉
ε
T
Π
Σ
x2/δω(−)
(c) GC-2D-0.35.
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
-3 -2 -1 0 1 2 3
〈
ρ
〉P
〈
ρ
〉
ε
T
Π
Σ
x2/δω(−)
(d) I-2D-0.70.
-0.002
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
-3 -2 -1 0 1 2 3
〈
ρ
〉P
〈
ρ
〉
ε
T
Π
Σ
x2/δω(−)
(e) DC-2D-0.70.
-0.00025
-0.0002
-0.00015
-0.0001
-5e-05
0
5e-05
0.0001
0.00015
0.0002
0.00025
0.0003
-3 -2 -1 0 1 2 3
〈
ρ
〉P
〈
ρ
〉
ε
T
Π
Σ
x2/δω(−)
( ) GC-2D-0.70.
Figu e 5: TKE c oss-s eam budge e alua ed a x1=300δω,0 o ine and eac i e spa ially-
de eloping mixing laye s. Quan i ies a e made non dimensional wi h ∆Uand δω.
Figu e 6 displays he las h ee e ms o he RHS o Eq.(3) o Mc=0.35 and Mc=0.70.
Wha e e he le el o comp essibili y, ba oclinic o que, which con ibu es o he p oduc ion o
ens ophy, and iscous di usion, which con ibu es o i s des uc ion, a e he mos impo an
e ms in he ens ophy budge o ine and eac i e mixing laye s 2. The dila a ion e m plays
a less impo an ole in he des uc ion o ens ophy whe eas he o ex s e ching mechanism
is no ep esen ed in such wo-dimensional nume ical simula ions. A Mc=0.35, he hea
elease dec eases he ampli udes o dila a ion, ba oclinic o que and iscous di usion e ms.
A Mc=0.70, he e is no clea educ ion o hese e ms in he eac i e case using de ailed
kine ics compa ed o he ine case. This also explains why he g ow h a es a e e y simila
o his wo con igu a ions (I-2D-0.70 and DC-2D-0.70). Mo eo e , he le els o ens ophy
p oduc ion (ba oclinic o que) when using de ailed chemis y a e highe when comp essibili y
inc eases, see Figs. 6(b) and 6(e), hus enhancing he shea laye g ow h a e. Ne e heless, he
use o a global eac ion mechanism a Mc=0.70 esul s in he lowes alues o he p oduc-
ion/des uc ion e ms in ol ed in he ens ophy budge . Finally, i is also wo h no ing ha ,
in eac i e cases a Mc=0.35, hese e ms display an asymme y o nega i e alues o he
ans e se coo dina e. The co esponding loca ions a ea is associa ed wi h he uel lean side,
whe e combus ion occu s, see Figs. 3(c)–3( ).
2No e ha his will no be necessa ily he case o h ee-dimensional nume ical simula ions, whe e o ex
s e ching may compensa e he iscous di usion e m.
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
-3 -2 -1 0 1 2 3
I-2D-035
DC-2D-035
GC-2D-035
x2/δω(−)
(a) Dila a ion, Mc=0.35.
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-3 -2 -1 0 1 2 3
I-2D-035
DC-2D-035
GC-2D-035
x2/δω(−)
(b) Ba oclinic o que, Mc=0.35.
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
-3 -2 -1 0 1 2 3
I-2D-035
DC-2D-035
GC-2D-035
x2/δω(−)
(c) Viscous di usion, Mc=0.35.
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
-3 -2 -1 0 1 2 3
I-2D-070
DC-2D-070
GC-2D-070
x2/δω(−)
(d) Dila a ion, Mc=0.70.
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
-3 -2 -1 0 1 2 3
I-2D-070
DC-2D-070
GC-2D-070
x2/δω(−)
(e) Ba oclinic o que, Mc=0.70.
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
-3 -2 -1 0 1 2 3
I-2D-070
DC-2D-070
GC-2D-070
x2/δω(−)
( ) Viscous di usion, Mc=0.70.
Figu e 6: Ens ophy budge o he ine and eac i e na u ally de eloping mixing laye s, com-
pu ed a x1=300δω,0. Quan i ies a e made non dimensional wi h ∆Uand δω.
Shocked spa ially-de eloping shea laye s
The abo e esul s con i med ha mixing be ween uel and ai s eams unde highly com-
p essible condi ions is inhe en ly slow. Howe e , since shock wa es (SW) a e p esen inside he
in ake and combus o o Sc amje engines, shock wa e impingemen may p o ide an a ac i e
me hod o enhance mixing. In his las sec ion, we epo nume ical simula ions o bo h ine
and eac i e shea laye s in e ac ing wi h s eady oblique shock wa es.
U1
U2
x1
x2
1
2
3
4
5
6
0
Figu e 7: Ske ch o he SW mixing laye in e ac ion geome y. 0: spli e pla e; 1: SW gene -
a o ; 2: inciden SW; 3: shea laye ; 4and 6: e lec ed SW; 5: ansmi ed SW.
Figu e 7 p o ides a schema ic iew o he low con igu a ion. The uel ( op) and oxidize
(bo om) inle s eam p ope ies emain s ic ly he same as hose conside ed in he abo e cases
and, he e o e, alues ga he ed in Table 1 s ill hold. The ini ial Reynolds numbe alue is kep
o 640 and he con ec i e Mach numbe is se o Mc=0.48. The compu a ional domain size is
L1×L2=275δω,0×120δω,0and is uni o mly disc e ized in space using N1×N2= 1638×717
