Simula ions o highly unde -expanded je s
R. Bu ay, P.J. Ma ´ınez Fe e , G. Lehnasch and A. Mu a
Ins i u Pp ime UPR 3346 CNRS, ISAE - ENSMA and Uni e si y o Poi ie s, FRANCE
1. In oduc ion
Highly esol ed nume ical simula ions o unde -
expanded je s a e conduc ed. Such high speed
je s may esul om he acciden al elease o high
p essu e flammable mix u es in o he quiescen a -
mosphe e, which poses impo an conce ns ela ed
o explosion haza ds. The na u e o he haza d
will depend on he s abili y o any je fi e esul -
ing om he unde -expanded elease o uel. Mo e
p ecisely, i will depend on whe he o no com-
bus ion can be sus ained in he icini y o he
elease o he e is a delay du ing which an ex-
plosi e cloud may o m. The desc ip ion o such
unde -expanded je s co e s also a b oad ange o
applica ions ela ed o spacec a p opulsion in-
cluding hype eloci y Sc amje s o ocke engines.
Fo ins ance, an unde expanded o ch je is used
o ini ia e combus ion in expande cycle engines.
The desc ip ion o scala mixing downs eam o
he Mach bo le hus appea s as an essen ial is-
sue, which is cen al o he p esen pape . I con-
s i u es a p elimina y s ep be o e a mo e de ailed
analysis o he effec s o hea elease, chemical ki-
ne ics and sel -igni ion on such comp essible je
s uc u es.
2. Nume ical me hods
The p esen s udy is ca ied ou wi h a nume i-
cal sol e able o desc ibe comp essible mul icom-
ponen eac i e mix u 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. The
ea men o he in iscid componen o he ans-
po 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) econs uc ion o he
cha ac e is ic fluxes. In p ac ice, he nume ical
sol e uses a se en h-o de accu a e cen e ed fi-
ni e diffe ence scheme, and he applica ion o he
WENO7 scheme is condi 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. The iscous and molecula diffusion flux
unc ions a e de e mined hanks o an eigh h-
o de cen e ed diffe ence scheme. The empo al
in eg a ion is pe o med by using an explici hi d-
o de TVD Runge-Ku a algo i hm. Fu he de-
ails abou he nume ical me hods as well as an
exhaus i e e ifica ion o he sol e a e p o ided
by Ma inez Fe e e al (2014).
3. Nume ical se up
The s udied simula ion consis s in ai eleased
om a high p essu e essel in o he quiescen a -
mosphe e. Ai is conside ed as a wo-species mix-
u e (O2and N2) desc ibed wi h a iable hea
capaci ies and anspo p ope ies hus a oid-
ing he eso o simpli ying hypo heses such as
cons an hea capaci y a io alue. The co e-
sponding elease eloci y is 630 m/s a he exi
(Ma = 1). The flow field is ini ialized wi h he
mix u e cha ac e is ic o ai a a p essu e o 1 a m
and a empe a u e o 300 K. The diame e o he
injec o exi is se o D= 0.001 m, which co -
esponds o a Reynolds numbe o 77500. The
inflow pa ame e s e ained o pe o m he p esen
simula ion a e lis ed in able 1 and co espond o
a sonic unde -expanded je wi h a nozzle p essu e
a io (NPR) based on s a ic p essu e o fi een.
Table 1. Unde -expanded je flow pa ame e s.
Injec o F ee-s eam
P(a m) 15.0 1.0
T(K) 1000.0 300.0
M a 1.0 0.05
u(m/s) 630.0 20.0
YO20.233 0.233
YN20.767 0.767
The compu a ional domain dimension, non-
dimensionalized by he diame e o injec ion D,
a e L∗
x1
= 14 and L∗
x2
=L∗
x3
= 6. This domain is
disc e ized wi h Nx1×Nx2×Nx3= 880×449×449
nodes Ca esian g id, which co esponds o ap-
p oxima ely 180 millions nodes. Sponge egions
combining bo h g id coa sening and explici fil e -
ing a e used in o de o a oid spu ious nume ical
wa e eflec ions and o make easie he p ocessing
o open bounda y condi ions. The esolu ion in
he highly esol ed egion is Δx= Δy= Δz=
D/60. Pe ec ly non- eflec ing bounda ies con-
di ions a e applied a he op, bo om, backside
and on side bounda ies. Pa ially non- eflec ing
bounda y condi ion is imposed a he ou flow.
The alue o he CFL numbe is se o 0.75 and
he Fou ie numbe Fo is se o 0.1.
y/D
Figu e 1. Nume ical schlie en image o he nea -field
o he je .
21s In l. Shock In e ac . Symp.
189
3 - 8 Aug. 2014, Riga, La ia
4. Desc ip ion o he eloci y field
The s uc u e o he axisymme ic ee je expand-
ing h ough he small o ifice in o quiescen a mo-
sphe e is displayed in Fig. 1. The whole comp ess-
ible s uc u e is clea ly iden ified.
As he flow lea es he nozzle he high p es-
su e misma ch causes i o expand and accele -
a e (Fig. 2). Expansion wa es o igina e nea he
expansion poin , p opaga e and mee he ou e
bounda y o he je , whe e hey a e eflec ed as
comp ession wa es. Coalescence o hese p essu e
wa es esul s in a cu ed ba el shock su ounding
he immedia e supe sonic egion. The eflec ion o
he inciden shock is no egula and a Mach disk
pa e n appea s in he nea -field o he je . The
flow is subsonic jus behind he Mach disk, while
i emains supe sonic downs eam o he ba el
shock. The iple poin connec s a ious discon i-
nui ies and becomes he o igin o a new slip line,
which gi es ise o a supe sonic shea laye .
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 2 4 6 8 10 12 14
P*
Ma
x/D
Figu e 2. Axial p ofiles o Mach numbe and no mal-
ized p essu e P∗=P/Pa.
0
1
2
3
0 2 4 6 8 10 12 14
P esen s udy
Lehnasch e al. (2005)
Veliko odny e al. (2012)
Yuceil e al. (2003)
Chau eau e al. (2006)
u∗
x/D
Figu e 3. No malized axial eloci y p ofiles u∗=u/ue
along he cen e line o he je .
Figu e 2 shows cen e line alues o he no -
malised p essu e P∗=P/Paand Mach numbe .
Due o he expansion o he gas, he p essu e and
he empe a u e dec ease significan ly while he
eloci y and he Mach numbe inc ease (Fig. 2
and Fig. 3). The Mach numbe d op indeed allows
o delinea e he Mach disk loca ion a app oxi-
ma ely x/D = 3.55. The posi ion o he Mach
disk is checked agains he empi ical co ela ion
o Ashkenas e al (1966) xDM /D = 0.67�P0/Pa,
which p o ides a simila es ima e: xDM /D =
3.60. Downs eam o he Mach disk, he p essu e
s abilizes a ound he a mosphe ic p essu e while
he flow eaccele a es p og essi ely and becomes
supe sonic a a dis ance x/D ≈13 om he in-
jec o . Figu e 3 epo s compa isons o s eam-
wise eloci y cen e line alues wi h p e ious ex-
pe imen al and nume ical da a. Conside ing he
difficul ies associa ed wi h high eloci y measu e-
men s abo e x/D = 2, he p esen eloci y p ofile
displays a sa is ac o y le el o ag eemen wi h ex-
pe imen al da a.
5. Scala field: u bulen mixing
We now in es iga e scala mixing in such highly
comp essible flow. To his pu pose, we sol e an
addi ional anspo equa ion o a passi e scala
ξwich is ad ec ed wi h an uni y Lewis numbe .
The molecula diffusi i y o ξis aken equal o he
he mal diffusi i y and Lewis numbe effec s asso-
cia ed o diffe en ial diffusion a e hus d opped off
om he p esen analysis. This ace ξis defined
o be uni y in he je and ze o elsewhe e. Figu e 4
p esen s he mean Mach numbe field supe im-
posed wi h ou iso-con ou s o ξ(0.1,0.4,0.7,0.95).
0 5.3
x/D
Figu e 4. Mean Mach numbe field supe imposed wi h
ξiso-con ou s.
0 1
x/D
Figu e 5. Mean ξfield supe imposed wi h ξiso-
con ou s.
21s In l. Shock In e ac . Symp.
190
3 - 8 Aug. 2014, Riga, La ia
Figu e 5 epo s he mean field o he ace ξ.
The bounda y o he mean je beha es like an im-
pe meable memb ane. In his configu a ion, he
u bulence de elops a his bounda y and g ows
in he supe sonic shea laye s. Figu e 6 and 7
display he adial p ofiles o he mean alue and
a iance o he passi e scala a diffe en c oss-
sec ions o he je . The quan i y �
�2cha ac e izes
he dispe sion o he scala om i s a e age alue.
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
�
ξ
/D
Figu e 6. Radial p ofiles o �
ξ.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.5 1 1.5 2 2.5 3
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
�
�2
/D
Figu e 7. Radial p ofiles o �
�2.
P oduc ion o a iance eflec s he inhomogene-
i y o he local mix u e. Con e sely, i s des uc-
ion cha ac e izes he ac ion o molecula p o-
cesses ough he mean alue o he SDR (scala
dissipa ion a e) χξ=D(∂ξ/∂xk)(∂ξ/∂xk) o he
passi e ace . The scala mixing akes place in
he supe sonic shea s laye s as shown in Fig. 6.
One can no ice ha a x/D = 12, he alue o
�
ξon he symme y axis ( /D = 0) is lowe han
uni y. The shea laye s indeed in e sec he cen-
e line o he je a a dis ance o x/D ≈12 which
co esponds app oxima ely o he leng h o he
subsonic h oa . Figu e 7 illus a es he mix u e
homogeniza ion and he des uc ion o a iance
aking place be ween he plane x/D = 6 and he
plane x/D = 14. The a iance �
�2is also plo -
ed agains he mean alue �
ξin Fig. 8. Thus, he
esul ing p ofiles can be compa ed o he max-
imum ealizable alue o he scala a iance, as
gi en by �
ξ(1 −�
ξ). Figu e 9 and 10 ep esen e-
spec i ely ξ�2and χξalong ξiso-con ou s defined
p e iously. Fo x/D ≤3, p ofiles a e less ep e-
sen a i e due o he lack o poin in his egion
o cap u e he dynamics which may explain he
pe sis ence o esidual oscilla ions on he p ofiles.
The peak o bo h ξ�2and χξa x/D ≈4 pe -
cep ible on he iso-con ou s ξ= 0.4, 0.7 and 0.95
a e explained by he impac o he eflec ed shock
wa e while he iso-con ou ξ=0.1 does no c oss
he eflec ed shock wa e and does no exhibi he
p esence o such a peak. This is consis en wi h
he p e ious in es iga ion o Huh e al (1996) who
poin ed ou he shock wa e enhancemen o mix-
ing by deflec ing s eamlines in mixing egion and
gene a ing shock-gene a ed o ici y.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.2 0.4 0.6 0.8 1
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
�
�2
�
ξ
Figu e 8. Scala a iance plo ed e sus �
ξ.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 2 4 6 8 10 12 14
ξ= 0.1
ξ= 0.4
ξ= 0.7
ξ= 0.95
-
-
-
-
ξ�2
x/D
Figu e 9. P ofiles o ξ�2along diffe en s ξiso-con ou s.
P obabili y densi y unc ions (PDFs) o ξa e
21s In l. Shock In e ac . Symp.
191
3 - 8 Aug. 2014, Riga, La ia
epo ed on figu e 11. These PDFs a e e alua ed
on iso-con ou s o ξ o diffe en alues o x/D.
The shape o he PDF, acco ding o hei posi-
ion in he je , a e consis en wi h hei heo e ical
coun e pa s (Bilge (1980)). On he bounda y o
he je and in e nal side o he je , i.e. ξ= 0.1
and ξ= 0.95, he influence o molecula mixing
effec s on ξis less app eciable. In con as , in-
side he shea laye , i.e. ξ= 0.4 and ξ= 0.7,
he PDF shapes igh en a ound he mean alue.
Scala dissipa ion a e is known o play a c ucial
ole o non-p emixed condi ions since mixing is
a p e equisi e be o e combus ion occu s. In he
field o u bulen combus ion modeling app oach,
i emains a common p ac ice o close he a e age
SDR ha appea s in he RHS o he scala a i-
ance anspo equa ion (1) (see appendix) by in-
oking a simila i y hypo hesis be ween scala and
eloci y u bulence spec a which esul s in he
classical app oxima ion τξ�Cξτ wi h Cξa mod-
eling cons an . This leads o he Linea elaxa ion
model (LRM) which consis s in �εξ=�
ξ��2/τξ�
�
ξ��2/Cξτ .
0
1000
2000
3000
4000
5000
6000
7000
0 2 4 6 8 10 12 14
ξ= 0.1
ξ= 0.4
ξ= 0.7
ξ= 0.95
-
-
-
-
χξ
x/D
Figu e 10. P ofiles o χξalong diffe en s ξiso-
con ou s.
The alidi y o he simplified LRM closu e is
analysed by in es iga ing he p opo ionali y con-
s an be ween τξand τ , i.e. he scala o u -
bulence ime scale a io Cξ=τξ/τ in he p esen
configu a ion. Figu e 12 epo s he scala mixing
ime scale defined as τξ=�
ξ��2/�εξalong he iso-
con ou s o he mean alue ξ. Figu e 13 epo s
he u bulence ime scale defined as τ =k/ε along
ξiso-con ou s. The quan i y kis he u bulen ki-
ne ic ene gy while εdeno es i s dissipa ion a e.
Finally, Fig. 14 epo s he p ofile o he scala
o u bulence ime scale a io. In his figu e,
i is no ewo hy ha he shock-wa e impac on
he mixing laye does no significan ly impac he
scala o u bulence ime scale a io. Mo eo e ,
i is also ema kable ha , in highly comp essible
si ua ions such as hose conside ed he ein, he
hypo hesis o a cons an alue, as implied by he
LRM closu e, is a he well e ified and no sig-
nifican ly al e ed by comp essibili y effec s.
0
1
2
3
4
5
6
7
0 0.2 0.4 0.6 0.8 1
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
P(ξ)
ξ
(a) ξ= 0.1
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
P(ξ)
ξ
(b) ξ= 0.4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
P(ξ)
ξ
(c) ξ= 0.7
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1
x/D = 6
x/D = 8
x/D = 10
x/D = 12
x/D = 14
P(ξ)
ξ
(d) ξ= 0.95
Figu e 11. Pd o ξon ξ= 0.1 (a), ξ= 0.4 (b), ξ= 0.7
(c) and ξ= 0.95 (d) iso-con ou s.
21s In l. Shock In e ac . Symp.
192
3 - 8 Aug. 2014, Riga, La ia
0
5e-06
1e-05
1.5e-05
2e-05
2.5e-05
3e-05
3.5e-05
0 2 4 6 8 10 12 14
ξ= 0.1
ξ= 0.4
ξ= 0.7
ξ= 0.95
-
-
-
-
τξ
x/D
Figu e 12. P ofiles o τξalong diffe en s ξiso-
con ou s.
0
5e-06
1e-05
1.5e-05
2e-05
2.5e-05
3e-05
3.5e-05
0 2 4 6 8 10 12 14
ξ= 0.1
ξ= 0.4
ξ= 0.7
ξ= 0.95
-
-
-
-
τ
x/D
Figu e 13. P ofiles o τ along diffe en s ξiso-con ou s.
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12 14
ξ= 0.4
ξ= 0.7
-
-
Cξ
x/D
Figu e 14. P ofiles o Cξ=τξ/τ along diffe en s ξ
iso-con ou s.
6. Conclusions
Highly esol ed nume ical simula ions o highly
unde -expanded u bulen gas je s ha e been
conduc ed. The compa isons pe o med be ween
he p esen esul s and expe imen al da a se s o
empi ical co ela ions gi e ise o a sa is ac o y
le el o ag eemen . Special emphasis has been
placed on he desc ip ion o u bulen mixing
downs eam o he Mach disk s uc u e. The
s udy has been ocused on he applicabili y o
he linea elaxa ion model (LRM) as a possible
closu e o he mean scala dissipa ion a e and
especially on he mapping o he scala o u bu-
lence ime scale a io Cξ. The ob ained esul s
show ha he hypo hesis o a cons an alue, as
implied by he LRM closu e, is a he well e ified
and no significan ly al e ed by comp essibili y
effec s. Fu u e wo ks will consis in compu ing
highly unde -expanded hyd ogen/ai je so as o
de e mine he flammabili y index (FI) map in
such condi ions. This will p o ide e y aluable
insigh s in o secu i y issues ele an o hyd ogen
explosion haza ds
Acknowledgemen s
The p esen wo k is pa o he PhD hesis o Ro-
main Bu ay, financially suppo ed by CNRS and
Region Poi ou-Cha en es. This wo k was g an ed
access o he HPC esou ces o IDRIS unde he
alloca ions x20142a0912 and x20142b7251 made
by GENCI (G and Equipemen Na ional de Cal-
cul In ensi ).
Appendix
The anspo equa ion o he scala a iance
w i es :
∂
∂ (ρξ��2) + ∂F ξ��2
k
∂xk
=−2ρD ∂ξ��
∂xk
∂ξ��
∂xk
−2ρu��
k�
∂�
ξ
∂xk
(1)
wi h he scala flux F�2
k= (ρukξ��2−ρD ∂ξ��2
∂xk
).
In his anspo equa ion, he fi s e m in he
le hand side is he accumula ion e m and he
second is he (conse a i e) flux e m (con ec ion
and diffusion). On he igh hand side o eq. (1),
he fi s e m co esponds o mean SDR while he
second is he p oduc ion associa ed o mean con-
cen a ion g adien s.
Re e ences
Ashkenas H., She man F.S. (1966), S uc u e and
u iliza ion o supe sonic ee je s in low densi y
wind unnels, Ra efied Gas Dynamics, 2 84–
105.
Beguie C., Deskeyse L., Launde B.E. (1986),
Ra io o scala and eloci y dissipa ion ime
scales in shea flow u bulence, Physics o Flu-
ids A 21, 307310.
Bilge R. (1980), ”Tu bulen flows wi h non-
p emixed eac an s”, in Tu bulen Reac ing
Flows, Topics in Applied physics.
21s In l. Shock In e ac . Symp.
193
3 - 8 Aug. 2014, Riga, La ia
Chau eau C., Da idenko D.M., Sa h B.,
Gokalp I., A ashko V., Fab e C. (2006), PIV
measu emen s in an unde -expanded ho ee
je , 10 h In e na ional Symposium on Applica-
ion o Lase Techniques o Fluid Mechanics,
Lisbon, Po ugal, 26–29 june 2006.
Ewan B.C.R., Moodie K. (1986), S uc u e and
eloci y measu emen s in unde -expanded je s,
Combus ion Science and Technology, 45 275.
Gome L., Robin V., Mu a A. (2012), Influence
o esidence and scala mixing ime scales in
non-p emixed combus ion in supe sonic u -
bulen flows, Combus ion Science and Tech-
nology, 184:10-11, 1471–1501.
Huh H., D iscoll J.F. (1996), Shock wa e enhance-
men o he mixing and he s abili y limi s o
supe sonic hyd ogen-ai je flames, P oceed-
ings o he 26 h In e na ional Symposium on
Combus ion, 1996, 2933–2939.
Lehnasch G. (2005), Con ibu ion a l’e ude nu-
me ique des je s supe soniques sous-de endus,
PhD Thesis, Uni e si y o Poi ie s, 2005.
Ma inez Fe e P.J., Bu ay R., Lehnasch G.,
Mu a A. (2014), A de ailed e ifica ion p oce-
du e o comp essible eac i e mul icomponen
Na ie -S okes sol e , Compu e s & Fluids, 89
88–110.
Veliko odny A., Kud iako S. (2012), Nume i-
cal s udy o he nea -field o highly unde -
expanded u bulen gas je , In e na ional
Jou nal o Hyd ogen Ene gy, 37 17390–17399.
Yuceil K.B., O ugen M.V., A ik E. (2003), In e -
e ome ic Rayleigh sca e ing and PIV mea-
su men s in he nea -field o unde -expanded
sonic je s, 41s Ae ospace Science Mee ing and
Exhibi , Reno, Ne ada, 6–9 Janua y 2003.
21s In l. Shock In e ac . Symp.
194
3 - 8 Aug. 2014, Riga, La ia
