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In this study, we evaluate the thermodynamic structure of laminar hydrogen/oxygen flames at supercrit- ical pressures using 1D flame calculations and large-eddy simulation (LES) results. We find that the real fluid mixing behavior differs between inert (cold flow) and reactive (hot flow) conditions. Specifically, we show that combustion under transcritical conditions is not dominated by large-scale homogeneous real-fluid mixing: similar to subcritical atomization, the supercritical pure oxygen stream undergoes a distinct transition from liquid-like to gas-like conditions; significant mixing and combustion occurs pri- marily after this transition under ideal gas conditions. The joint study of 1D flame computations and LES demonstrates that real-fluid behavior is chiefly confined to the bulk LOX stream; real fluid mixing oc- curs but in a thin layer surrounding the LOX core, characterized by water mass fractions limited to 3%. A parameter study of 1D flame solutions shows that this structure holds for a wide range of relevant injec- tion temperatures and chamber pressures. To analyze the mixing-induced shift of the local fluid critical point, we introduce a state-space representation of the flame trajectories in the reduced temperature and reduced pressure plane which allows for a direct assessment of the local thermodynamic state. In the flame, water increases the local mixture critical pressures, so that subcritical conditions are reached. This view of limited mixing under supercritical conditions may yield more efficient models and an improved understanding of the disintegration modes of supercritical flows.
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Combustion and Flame 196 (2018) 364–376
Contents lists available at ScienceDirect
Combustion and Flame
journal homepage: www.elsevier.com/locate/combustame
Thermodynamic structure of supercritical LOX–GH2 diffusion flames
Daniel T. Banuti
a
,
, Peter C. Ma
b
, Jean-Pierre Hickey
c
, Matthias Ihme
a
,
b
a
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
b
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
c
Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada
a r t i c l e i n f o
Article history:
Received 26 September 2017
Revised 12 November 2017
Accepted 11 June 2018
Keywo rds:
Counterflow diffusion flame
Flamelet
Transcritical
Supercritical
Injection
Rocket
a b s t r a c t
In this study, we evaluate the thermodynamic structure of laminar hydrogen/oxygen flames at supercrit-
ical pressures using 1D flame calculations and large-eddy simulation (LES) results. We find that the real
fluid mixing behavior differs between inert (cold flow) and reactive (hot flow) conditions. Specifically,
we show that combustion under transcritical conditions is not dominated by large-scale homogeneous
real-fluid mixing: similar to subcritical atomization, the supercritical pure oxygen stream undergoes a
distinct transition from liquid-like to gas-like conditions; significant mixing and combustion occurs pri-
marily after this transition under ideal gas conditions. The joint study of 1D flame computations and LES
demonstrates that real-fluid behavior is chiefly confined to the bulk LOX stream; real fluid mixing oc-
curs but in a thin layer surrounding the LOX core, characterized by water mass fractions limited to 3%. A
parameter study of 1D flame solutions shows that this structure holds for a wide range of relevant injec-
tion temperatures and chamber pressures. To analyze the mixing-induced shift of the local fluid critical
point, we introduce a state-space representation of the flame trajectories in the reduced temperature and
reduced pressure plane which allows for a direct assessment of the local thermodynamic state. In the
flame, water increases the local mixture critical pressures, so that subcritical conditions are reached. This
view of limited mixing under supercritical conditions may yield more efficient models and an improved
understanding of the disintegration modes of supercritical flows.
©2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction
Injection at supercritical pressures is a technology ubiquitous
in transportation; it is used in Diesel engines, gas turbines, and
rocket engines [1,2] . At these conditions, subcritical break-up into
ligaments and droplets, Fig. 1 a, is replaced by a turbulent mixing
process, Fig. 1 b, when the surface tension vanishes under super-
critical conditions [2–5] .
However, a categorization based on the pure propellant critical
pressures has been shown to be too simplistic: the critical pressure
of a mixture may significantly exceed the critical pressures of the
components [6] , thus an injection process that is supercritical with
respect to the injected fluid and the background gas, henceforth
referred to as ‘nominally supercritical’, may still exhibit subcritical
break-up characteristics. This has been demonstrated experimen-
tally [7,8] and numerically [9–11] .
The view of Fig. 1 has also informed the development of the
respective numerical models: the sharp interfaces in subcritical
Corresponding author.
E-mail addresses: daniel@banuti.de (D.T. Banuti), mihme@stanford.edu (M.
Ihme).
injection typically lead to an approach in which interfaces are
tracked, whereas the diffuse mixing in supercritical injection is
treated as a problem of accurately modeling the thermodynamics
of the real fluid-state behavior and mixing. A ‘real fluid’ in this
context is characterized by significant intermolecular forces that
render the ideal gas equation of state inapplicable, consistent with
the view in physical chemistry [12] . The canonical model was in-
troduced by Oefelein and Yang [13] .
This approach has been used extensively to study nominally
supercritical combustion of liquid oxygen (LOX) with gaseous hy-
drogen (GH2), representative of liquid rocket engines. In such en-
gines, the flame is anchored in a recirculation zone behind the LOX
post of a coaxial injector [3,4,13] . Juniper et al. [14] noted that
the quenching strain rate of a GH2/LOX flame exceeds the max-
imum values found in rocket engines by an order of magnitude.
Indeed, it can be assumed in numerical modeling of rocket en-
gines that the reactions reach a chemical equilibrium [15,16] . Ex-
periments [3,4] and simulations [13,17,18] show that the flame ef-
fectively separates oxygen from hydrogen; the flame encloses the
oxygen stream due to hydrogen-rich operating conditions. For spe-
cific flow conditions, combustion and mixing may take place in
the ideal gas limit [19,20] . Thus, liquid propellant rocket engines
https://doi.org/10.1016/j.combustflame.2018.06.016
0010-2180/© 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
D.T. Banuti et al. / Combustion and Flame 196 (2018) 364–376 365
Fig. 1. Characterization of subcritical and supercritical injection.
exhibit combustion conditions that closely resemble idealized dif-
fusion flames. Ribert et al. [21] and Pons et al. [22] used this in-
sight to address counterflow diffusion flames under trans- and su-
percritical conditions, an approach that has since been adopted by
other groups [23–25] . Since, large-eddy simulations (LES) of super-
critical injection have become state-of-the-art [26,27] , with funda-
mentals still being investigated [28,29] .
Furthermore, ample evidence for a structure of the super-
critical state space has been collected. Specifically, experimental
[30,31] and theoretical [32,33] results indicate that distinct liquid-
like and gas-like supercritical states can be distinguished. These
states are separated by the Widom line, an extension to the co-
existence line characterized by peaks in the isobaric specific heat
capacity. The Widom line is a general fluid property [34] and can
even be identified in mixtures [35] . The relevance of the Widom
line for injection lies in pseudoboiling [2] , the transition from
liquid-like to gas-like supercritical states across the Widom line.
Pseudoboiling resembles subcritical boiling, where a peak in the
thermal expansivity causes a sudden drop in density, and a peak
in isobaric specific heat capacity acts as an energy sink akin to the
latent heat of vaporization [36] . Pseudoboiling has been identified
as an important process in injection and combustion [23,37] .
So far, a systematic investigation of the thermodynamic struc-
ture of supercritical non-premixed flames with respect to mix-
tures and transitions has not been carried out. Therefore, we ad-
dress three questions in the present work: (1) In which part of
the flame are real-fluid effects (pure fluid and mixing) relevant?
(2) How does the local thermodynamic state and the critical point
change throughout the flame? 3) How sensitive are these results
with respect to changes in the injection conditions? To address
these questions, we compute LES and 1D flame structures of the
cryogenic LOX/GH2 diffusion-flame at nominally supercritical pres-
sure, covering inert and reactive cases. Variation of the 1D condi-
tions over a range of strain rates from near-equilibrium to quench-
ing conditions, pressures, injection temperatures, and dilution, are
evaluated to assess the generality of the initial results. The spa-
tial distribution of real fluid thermodynamics is demonstrated us-
ing LES results.
2. Numerical methods
This section discusses the numerical methods used for the
present study. First, the thermodynamic model used for real fluid
mixtures is introduced. The discussions of the 1D flame solver and
the LES solver, which use this thermodynamic model, follow. The
section concludes with relations used in the analysis of the results.
2.1. Thermodynamic relations
The Peng–Robinson equation of state (PR EoS) [38] is used in
this study for the evaluation of thermodynamic quantities. A cubic
EoS has been chosen for its computational efficiency and readily
available mixing rules; PR has been chosen for its accuracy espe-
cially for supercritical fluids [13] . Real-fluid effects are accounted
for by departure functions that are derived from the state equation
to ensure thermodynamic consistency of the governing equations.
The PR state equation is expressed as
p =
RT
v b
a
v
2
+ 2 bv b
2
, (1)
where R is the gas constant, v is the specific volume, and the coef-
ficients a and b are functions of temperature and composition, ac-
counting for effects of intermolecular attractive forces and volume
displacement. For mixtures of N
S
species, the coefficients a and b
in Eq. (1) are evaluated as [6]
a =
N
S
α=1
N
S
β=1
X
αX
βa
αβ , (2a)
b =
N
S
α=1
X
αb
α, (2b)
where X
αis the mole fraction of species α. The coefficients a
αβ
and b
αare calculated using the mixing rules recommended by
Harstad et al. [39] , with
a
αβ = 0 . 457236
(RT
cr,αβ )
2
p
cr,αβ 1 + c
αβ1
T /T
cr,αβ2
, (3a)
b
α= 0 . 077796
RT
cr,α
p
cr,α
, (3b)
c
αβ = 0 . 37464 + 1 . 54226 ω
αβ 0 . 26992 ω
2
αβ . (3c)
The critical properties of the major species are taken from
the NIST database [40] , the critical properties of the intermedi-
ate species are determined based on their Lennard–Jones poten-
tials and their acentric factor is set to zero, following Giovangigli
et al. [41] . The critical properties for all species considered in this
study are compiled in Table 1 .
In mixtures, the local critical pressure and temperature vary
with composition. While the mixture critical temperature lies be-
tween the component pure fluid values, the mixture critical pres-
sure may significantly exceed the pure fluid values. Both can be de-
termined using the pseudocritical method described by Reid et al.
[6] , where
T
cr , mix
=
N
S
α=1
X
αT
cr ,α, (4a)
p
cr , mix
= RT
cr , mix
N
S
α=1
X
αZ
cr ,α
N
S
α=1
X
αv
cr ,α
. (4b)
The compressibility factor Z[6] is an important nondimen-
sional parameter in real fluid thermodynamics, quantifying the de-
viation from ideal gas behavior. It is defined as
Z =
pv
RT
. (5)
The critical compressibility factor Z
cr [6] corresponds to the com-
pressibility factor evaluated for T
cr
, p
cr
, and v
cr
. Previous results
366 D.T. Banuti et al. / Combustion and Flame 196 (2018) 364–376
Tabl e 1
Critical properties of species in H
2
/O
2
combustion, where T
cr
, p
cr
, v
cr
, and
ω represent, respectively, the critical temperature, critical pres-
sure, critical molar volume and acentric factor of the species.
Parameters H
2 O
2 H
2
O O H OH H
2
O
2 HO
2 N
2
T
cr
[K] 33.0 154.58 647.10 105.28 190 .8 2 105.28 141.34 141.3 4 126.19
p
cr
[MPa] 1.284 5.043 22.064 7.0 88 31.013 7. 08 8 4.786 4.786 3.395
v
cr
[cm
3
/mol] 64.28 73.37 55.95 41.21 17.0 7 41.21 81.93 81.93 89.41
ω 0 . 216 0.022 0.344 0.0 0.0 0.0 0.0 0.0 0.0372
[20] showed that the exact ideal gas condition Z = 1 is only met
along a line in p
r T
r space, implying that the ideal gas law is
strictly not fulfilled for most practical CFD conditions. Allowing for
a 5% deviation from ideal gas behavior, i.e. 0 . 95 < Z < 1 . 05 , ex-
tends the quasi-ideal gas regime to practical values of p
r and T
r
.
Then, quasi-ideal behavior can be expected for T
r
> 2 and p
r
< 3.
For simplicity, we will refer to this as ‘ideal’ in the remainder of
the paper.
2.2. CFD solver
The finite-volume solver CharLES
x
is used in this study. The nu-
merical solver and the corresponding numerical methods are dis-
cussed in detail elsewhere [42–44] , only a brief overview focusing
on the models required for real fluids will be given here. The gov-
erning equations solved are the conservation of mass, momentum,
energy, and species. The PR EoS, Eq. (1) , is used to close the sys-
tem. Details on how to evaluate thermodynamic quantities can be
found in Ma et al. [43] . The dynamic viscosity and thermal conduc-
tivity are evaluated using Chung’s method with high-pressure cor-
rection [45,46] . Takahashi’s high-pressure correction [47] is used to
evaluate binary diffusion coefficients. A diffuse interface method is
used and no surface tension effects are considered.
The convective flux is discretized using a sensor-based hybri d
scheme in which a high-order, non-dissipative scheme is combined
with a low-order, dissipative scheme to minimize numerical dissi-
pation [48] . A central scheme, which is fourth-order accurate on
uniform meshes, is used along with a second-order ENO scheme
for the hybrid scheme and a density sensor [42,43] is adopted in
this study. An entropy-stable flux correction technique [43] ensures
the physical realizability of the numerical solution including the
positivity of scalars and the damping of non-linear instabilities in
the numerical solution.
Transcritical flow is characterized by strong non-linearities in-
herent in the real-fluid EoS imposing severe numerical difficul-
ties. To remedy spurious pressure oscillations generated by a fully
conservative scheme [43,49] , a double-flux method is extended
to the transcritical regime [43] . A strong stability preserving 3rd-
order Runge–Kutta (SSP-RK3) scheme [50] is used for time ad-
vancement, a Strang-splitting scheme [51] is applied in this study
to separate the convection operator from the remaining operators
of the system. For reacting cases, a transcritical extension of the
flamelet/progress variable approach [52,53] is adopted [43,44] .
2.3. Counterflow diffusion flame
The flamelet model [52,54] was found to be applicable to coax-
ial rocket injectors [23] . With this, a profile through a turbulent
non-premixed flame can be represented by a 1D-counterflow diffu-
sion flame. The axisymmetric, laminar counterflow diffusion flame
admits a self-similar solution and can be simplified to a one-
dimensional steady state problem [25,54] . The governing equations
for continuity, radial momentum, species, and temperature can be
written as
d
d x
(ρu ) + 2 ρV = 0 , (6a)
ρu
d V
d x
+ ρV
2
=
d
d x
μd V
d x
, (6b)
ρu
d Y
α
d x
= d J
α
d x
+ ˙ ω
α, (6c)
c
p
ρu
d T
d x
=
d
d x
λd T
d x
N
s
α=1
J
α
d h
α
d x
N
s
α=1
˙ ω
αh
α, (6d)
where conventional notation is used [25] , V = w/r, = ( p/ r) /r,
h
αis the sensible enthalpy of species α, and J
α= ρY
αV
αis the dif-
fusion flux for species α, accounting for multi-component diffusion
[55] . The governing equations are closed with the PR EOS, dis-
cussed in Section 2.1 . The mathematical boundary conditions are
determined by the operating conditions of the engine, namely the
propellant injection temperatures and the chamber pressure.
The governing equations and the equation of state are imple-
mented in the Cantera package [56] . Special care has been taken
to fully resolve the thermodynamic nonlinearities of pseudoboiling
near the Widom line [36] by modifying the grid adaptation strat-
egy to account for the large variation in fluid properties.
A validation of the method with the detailed numerical sim-
ulation of a two-dimensional counterflow diffusion flame in the
transcritical regime by Lacaze and Oefelein [23] was undertaken.
The oxygen and hydrogen streams are injected at T
LOX
= 120 K and
T
GH2
= 295 K, respectively, at a combustion pressure of 7.0 MPa.
The strain rate, calculated as the velocity difference between the
injectors divided by the separation distance [23] , is 10
5 s
1
.
We perform the comparison in composition space, using the
definition of a mixture fraction based on atomic hydrogen mass
fraction [23] ,
Z
H
= W
H
2
Y
H
2
W
H
2
+
Y
H
W
H
+ 2
Y
H
2
O
W
H
2
O
+
Y
OH
W
OH
+
Y
HO
2
W
HO
2
+ 2
Y
H
2
O
2
W
H
2
O
2
,
(7)
where Y
αand W
αare mass fraction and molecular weight for
species α, respectively. In the pure LOX/GH2 case, the flame is sit-
uated at the stoichiometric mixture fraction Z
H
= Z
st
= 0 . 11 .
Figure 2 shows temperature, composition, density, and specific
heat capacity of the current simulation and the reference results.
Mesh-convergence studies were performed to ensure that the so-
lution is mesh independent. Results with and without Soret effect
are presented. It can be seen that the Soret effect has a marginal
but noticeable influence on all major flame properties, improving
agreement with the reference solution, except for the lean tem-
perature distribution. All 1D results presented in the following are
simulated with the Soret effect taken into account. The real-fluid
effects near the oxidizer injector are evidenced in Fig. 2 c and d,
showing profiles of density and heat capacity. The drop in den-
sity and the local peak in the specific heat capacity upon crossing
the Widom line are well captured by the current model, as are the
other flame properties.
D.T. Banuti et al. / Combustion and Flame 196 (2018) 36 4–376 367
Fig. 2. Validation of the near-equilibrium flame (lines) with the detailed numerical simulation by Lacaze and Oefelein [23] (symbols). Baseline conditions are T
LOX
= 120 K,
T
GH2
= 295 K, p = 7.0 MPa.
3. Results
In this section, we discuss results of the 1D flame calculations
with respect to the thermodynamic mixing behavior, and intro-
duce a new diagram in terms of the local reduced pressure and
reduced temperature, which allows an intuitive evaluation. To as-
sess the generality, we then perform a parameter study in terms
of strain rate, pressure, injection temperatures, and dilution of the
fuel and the oxidizer stream. Finally, we analyze multidimensional
CFD results and show that the conclusions from the 1D study are
applicable to more complex approaches.
3.1. Strain rate limits
We will first investigate how the limits of chemical equilibrium
and quenching affect the overall flame structure for the baseline
case introduced in Fig. 2 ( T
LOX
= 120 K, T
GH2
= 295 K, p = 7.0 MPa).
Figure 3 compares the structure of a near-equilibrium flame on
the burning branch to the structure of a flame close to quenching,
corresponding to the points on the top branch of the classical S-
curve [54] . In the near-equilibrium case, hydrogen and oxygen mix
only in a small region around Z
H
= 0 . 11 = Z
st
. The flame close to
quenching conditions behaves very differently: oxygen and hydro-
Fig. 3. Influence of strain rate on the 1D flame structure. Composition and tem-
perature of near-equilibrium flame (solid line), a = 1 . 0 ×10
5
1/s and of flame near
quenchi ng (dashed line), a = 1 . 78 ×10
7
1/s.
gen are present throughout the composition space, the maximum
temperature is reduced by more than 10 0 0 K due to competition
between heat release and diffusion.
368 D.T. Banuti et al. / Combustion and Flame 196 (2018) 364–376
3.2. Mixing and transition from a real to an ideal fluid
It is clear from Fig. 2 c that the flame solution includes regions
with liquid-like density in the oxidizer limit, and transitions to
gaseous densities for higher mixture fractions. However, density at
high pressures only indirectly yields information about the ther-
modynamic state. Furthermore, the actual composition at the tran-
sition is unclear. We will now investigate this in more detail.
Figure 4 shows profiles of the compressibility Z, as well as
mass fractions of hydrogen, oxygen, and water, for low strain rates
and the near-quenching flame. At baseline conditions, real fluid be-
havior is observed only on the oxidizer side. For the condition near
equilibrium, Fig. 4 a shows how the compressibility factor reaches
unity for Z
H
> 2 . 0 ×10
3
, while Y
O2
has only marginally reduced
from unity. Near quenching, ideal gas conditions are reached at the
slightly higher mixture fraction of Z
H
> 3 . 0 ×10
3
. Figure 4 shows
the species mass fraction at which transition to an ideal gas occurs
for H
2
, H
2
O, and OH. In order to allow for a quantitative evaluation
of the purity of oxygen at the transition, (1 Y
O2
) is shown instead
of Y
O2
. The curves for Y
H2O
and (1 Y
O2
) are indistinguishable, in-
dicating that Y
H2O
+ Y
O2
1 during conversion to ideal gas condi-
tions; the radicals remain confined to the reaction zone. We obtain
a water mass fraction of 2% at the transition to an ideal gas. Only
traces of H
2
and OH are present as real fluids, with mass fractions
four orders of magnitude less than that of water. We can conclude
that the transition from dense transcritical oxygen to an ideal gas
occurs under almost pure conditions in equilibrium flames. This is
consistent with earlier results [20] .
We will now evaluate the supercritical transition of liquid-like
oxygen to gaseous supercritical oxygen in more detail. Figure 2 c
shows a very steep change in density at Z
H
1 . 0 ×10
3
, accom-
panied by a distinct maximum in isobaric specific heat capacity, as
shown in Fig. 2 d. This peak has been identified as a main effect
hindering the heating of the LOX core by acting as a thermal bar-
rier [23] and causing an increased thermal sensitivity in pure fluid
injection [37] . It is thus important to identify the origin of the c
p
peak in the context of diffusion flames. Earlier studies associated
the peak in heat capacity to the mixture reaching the mixture crit-
ical temperature [18] or the transcritical nature of the fluid [23] .
We have seen that the transition of dense oxygen to an ideal
gas at baseline conditions occurs almost as a pure fluid, specifi-
cally, in a binary mixture of oxygen and water with a water mass
fraction of 2%. Significant fractions of both components have to
be present at the mixture critical temperature for high pressure
phase separation to occur (in the H
2
/O
2
system investigated by
Yan g [18] , the critical mixture temperature at p = 8 MPa is reached
for an oxygen mole fraction of 0.8). These conditions are not met
in the present case. Instead, we recognize the c
p peak as an in-
stance of pseudoboiling [2,36] , the transition from a supercritical
liquid-like to a gas-like state involving a rapid expansion and re-
duced temperature increase when heat is added, albeit over a fi-
nite temperature interval.
3.3. The reduced state plot as system representation
After studying the transition from a real to an ideal fluid in the
cryogenic limit, we now proceed to examine the thermodynamic
trajectory of the whole flame solution. The goal is to extend the
approach of Lacaze and Oefelein [23] by introducing a more in-
tuitive representation, and then to evaluate the generality of the
results via a parametric study.
The critical point of a mixture may deviate substantially from
the critical point of its constituents and can be calculated us-
ing the mixing rules of Eqs. (4a) and (4b) . In order to evaluate
the profile of the thermodynamic state, Dahms et al. [57] intro-
duced plots of reduced temperature T
r
= T /T
cr and reduced pres-
Fig. 4. Species mass fractions and compressibility factor for equilibrium (solid) and
near-quenching (dashed) flame. In the equilibrium case, Z 1 for Z
H
> 2 . 0 ×10
3
,
for near quenching at Z
H
> 3 . 0 ×10
3
. Evaluation in terms of deviation from oxygen
mass fraction reveals that only water mixes with oxygen under real-fluid conditions
in the equilibrium case.
D.T. Banuti et al. / Combustion and Flame 196 (2018) 36 4–376 369
Fig. 5. Projections of the flamelet trajectories in Z
H
p
r
T
r
- space, evaluated for baseline conditions ( Fig. 2 ).
sure p
r
= p/p
cr
versus mixture fraction to analyze mixture profiles.
Only when reduced pressure and reduced temperature are simul-
taneously smaller than unity, subcritical phase separation is possi-
ble. Lacaze and Oefelein [23] applied this approach to reactive mix-
tures. Figure 5 a and b shows results for the investigated conditions
near equilibrium and for inert mixing, neither of which indicates
phase separation.
Interpretation of Fig. 5 a and b is not straightforward, even
though each graph represents but a single operating condition
in terms of injection temperature and chamber pressure. Further-
more, it is not obvious how changes in operating conditions will
affect the thermodynamic structure. There is a more intuitive way
of representing the data by collapsing the mixture fraction in both
plots and combining them in a reduced pressure/reduced tempera-
ture diagram, as shown in Fig. 5 d. In this graph, every flamelet cor-
responds to a trajectory in the T
r p
r state space obtained from
Eqs. (4a) and (4b) , connecting the oxidizer and fuel injection states
‘O2’ and ‘H2’ for a specific set of boundary conditions. Then, the
criterion for identifying a phase change is straightforward: it will
occur when the thermodynamic trajectory intersects the coexis-
tence line. Consequently, when a trajectory passes the critical point
at supercritical pressure, the mixing rules predict that no phase
separation occurs, as the trajectory does not intersect the coexis-
tence line.
Any variation of the reduced pressure is a function of only the
local composition when the combustion pressure is kept constant.
Variations of the reduced temperature are caused by both changes
in temperature and in composition. Thus, horizontal parts of the
trajectory would mark regions where the fluids heat without a
change in composition.
Figure 5 d combines a trajectory near chemical equilibrium
(blue), and an inert mixing line (green), for the baseline operating
condition ( T
LOX
= 120 K, T
GH2
= 295 K, p = 7 . 0 MPa). No phase sep-
aration is predicted in either case. For the flame solution, oxygen is
seen to heat with limited mixing, until a loop is formed at subcrit-
ical pressures. This loop is caused by the presence of water in the
reaction zone, which significantly changes the local critical param-
eters due to its comparatively high critical pressure and temper-
ature, see Table 1 . Towar ds the pure hydrogen limit, the reduced
temperature reaches a maximum. The inert mixing case constitutes
a monotonic trajectory, with an inflexion point close to the critical
temperature. It is interesting to note that the inert cryogenic mix-
ing case passes the critical temperature at a higher reduced pres-
sure than the reactive flamelet and thus appears less susceptible to
phase separation.
Figure 5 d is useful for a number of reasons. First, it allows for
the direct assessment as to whether the coexistence line is crossed
or not instead of inferring this from two graphs as in Fig. 5 a and
b. Second, it can combine several flamelet trajectories, without af-
fecting readability. This means that it not only highlights a sin-
gular condition, but instead may be used to assess the physical
envelope of an engine in the p
r T
r phase space, as determined
370 D.T. Banuti et al. / Combustion and Flame 196 (2018) 364–376
Fig. 6. Flame structure trajectory between pure oxygen and pure hydrogen in terms
of the local reduced pressure and temperature. Gray contours mark the compress-
ibility Zfor oxygen (solid) and hydrogen (dashed). The blue line is the coexistence
line. Data are evaluated from NIST [40] .
by the chamber pressure and the injection conditions for LOX and
GH2. Third, the plot allows to develop intuition as to what effect a
change in operating conditions has on the trajectories.
3.3.1. Analysis of the thermodynamic structure
The previous section introduced the T
r p
r reduced state plot
as a way to assess local flame conditions, but left a number of de-
tails unclear, such as the nature of the nonlinearity of the inert
mixing line upon crossing T
cr
, or the actual position of the flame
in the plot. We will proceed by performing a more detailed analy-
sis to examine these aspects further.
Figure 6 shows the trajectory of a flame at baseline conditions
with the fluid compressibility superimposed. Oxygen is injected
in a liquid-like state with Z 0 . 2 . When the temperature of the
flame increases by heat diffusion from the flame, ideal gas behav-
ior is reached for T
r
> 2, consistent with earlier results [20] . The
dashed contours show that hydrogen reaches ideal gas conditions
at higher temperatures and sustains ideal gas behavior for higher
pressures. We can see that the reduced conditions of the hydrogen
injection correspond to ideal gas behavior for hydrogen, but would
show emerging real fluid behavior for oxygen.
Figure 7 is an extension of Fig. 5 d, adding the point of maxi-
mum temperature ( ×), the point where the oxygen mass fraction
drops to 0.98 (
), the coexistence line (solid line), and the Widom
line (dashed line). Different fluids may exhibit different Widom
lines [34] , however, due to the predominant presence of oxygen
when the Widom line is crossed by the state trajectory, we show
the Widom line of oxygen. The arrow points in the direction of
an increasing value of the parameter. In Fig. 7 , this parameter is
the strain rate, we add the flamelet trajectory corresponding to the
near-quenching point.
Counterintuitively, hot reacting cases are more prone to phase
separation than cryogenic inert mixing, judged by the reduced
pressure when passing the critical temperature. This can be at-
tributed to the formation of water in the flame, which signifi-
cantly influences the mixture critical properties due to its compa-
rably high T
cr and p
cr
, see Table 1 . In turn, the resulting reduced
pressures decrease, pushing the trajectories closer to the critical
point. This effect of water is strongest where the highest temper-
atures are reached, pushing the associated loop in the trajectory
to reach even subcritical pressures: the highest temperature is not
found at the highest reduced temperature, but instead near the
lowest reduced pressure. The highest reduced temperature is in-
Fig. 7. Flame structure trajectories between pure oxygen and pure hydroge n in
terms of the local reduced pressure and temperature. The asterisk (
) denotes an
oxygen mass fraction of 0.98, the cross ( ×) marks the state of highest tempera-
ture; black solid and dashed lines are the coexistence line and pseudoboiling line,
respectively, separated by the critical point (CP).
Fig. 8. Effect of transport model on flame trajectory.
stead found on the rich side of the flame, and caused by the very
low critical temperature of hydrogen. For the reactive cases, oxy-
gen is heated by thermal diffusion from the reaction zone to tem-
peratures exceeding 2 T
cr
, thus reaching ideal gas conditions be-
fore significant mixing occurs. In the cold mixing case, however,
hydrogen is found to diffuse into the dense liquid oxygen in a
non-ideal mixing process. Indeed, the horizontal part of the tra-
jectory near pure oxygen conditions indicates an interesting phys-
ical phenomenon: for constant combustion pressures, the reduced
pressure only changes with the composition according to Eq. (4b) .
Then, horizontal parts of the trajectory signify regions in which
heat transfer occurs with very limited mixing. This corresponds to
high Lewis numbers in near-critical fluids, consistent with results
of Harstad & Bellan [58] and Lacaze & Oefelein [23] .
Figure 8 demonstrates the influence of the transport model on
the thermodynamic flame structure in more detail. In contrast to
the noticeable effect of including the Soret term on the tempera-
ture and density distributions, shown in Fig. 2 , the fuel-rich side of
the flame trajectory in Fig. 8 is significantly modified. Assuming a
D.T. Banuti et al. / Combustion and Flame 196 (2018) 36 4–376 371
unity Lewis number is furthermore found to overestimate diffusive
mixing towards the cryogenic oxygen stream.
3.3.2. Effect of parameter variations
We extend our analysis by considering effects of operating con-
ditions and propellant dilution on the state-space representation.
Figure 9 a shows the impact of combustion pressure on the trajec-
tories. Increasing the pressure mainly leads to a translation of the
curves towards higher reduced pressures, as the fluid critical pres-
sures and the composition remain approximately constant. While
the maximum temperature increases with pressure, this effect is
hardly discernible. A slight drop in reduced pressure moving away
from the pure oxygen condition can be seen prior to reaching the
Widom line; both move to higher temperatures as the pressure is
increased. The dilution of the oxygen stream (
) is not affected by
a variation in pressure.
The impact of the oxidizer temperature on the trajectories is
shown in Fig. 9 b. The fuel-rich side from pure hydrogen to the
flame loop at the highest temperature remains essentially un-
changed, the oxygen side of the flame is translated to higher re-
duced temperatures upon raising the oxygen temperature. The bot-
tom of the flame loop is hardly affected: while a higher oxygen
temperature provides a higher enthalpy which in turn raises the
combustion temperature, this effect is practically negligible com-
pared to the heat released in the flame.
Variation of the fuel temperature is shown in Fig. 9 c. The oxi-
dizer side of the flame remains unchanged, while the pure hydro-
gen boundary condition moves to lower reduced temperatures as
the hydrogen temperature is decreased. The lower the hydrogen
temperature, the longer is the horizontal initial part of the flamelet
trajectory emanating from the hydrogen injection condition, signi-
fying heating without mixing. The impact of the hydrogen temper-
ature on the flame position is visible, and can be attributed to the
higher heat capacity of hydrogen and the larger temperature inter-
val compared to the oxidizer variation shown in Fig. 9 b.
Finally, we study the effect of diluting the propellant streams,
using nitrogen in the oxidizer flow (representative for air-breathing
combustion), and water in the fuel stream (representative for pre-
burner combustion). Figure 10 shows how an increasing nitrogen
fraction in the oxidizer stream affects the thermodynamic structure
of the flame. Due to the lower critical temperature and pressure
of nitrogen compared to oxygen, we see in Fig. 10 a how the oxi-
dizer injection condition moves towards higher reduced pressures
and temperatures as the mass fraction is increased. The flame loop
narrows and shifts to higher reduced pressures, before vanishing
altogether as the composition of air is approached. The maximum
flame temperature is reduced, the temperature peak moves to
lower mixture fractions as the stoichiometric mixture ratio shifts,
see Fig. 10 b.
Addition of water to the hydrogen stream can be evaluated
from Fig. 11 . Figure 11 a shows that the reduced state of the injec-
tion condition shifts with diluent mass fraction. Due to the large
critical properties of water, the effect is more pronounced than
for the oxygen dilution case –the fuel injection point eventu-
ally reaches subcritical pressures. The flame loop opens and moves
to lower reduced temperatures. The maximum temperature of the
flame reduces with diluent mass fraction, and shifts to higher mix-
ture fractions, see Fig. 11 b.
We can conclude that changes in the boundary conditions do
indeed lead to well behaved variations in the flame trajectories
that allow to develop intuition.
3.4. Application to multidimensional simulation results
In order to allow for an assessment of the spatial thermody-
namic structure, we analyze the LOX/GH2 shear layer configura-
Fig. 9. Flame structure trajectories between pure oxygen and pure hydrogen in
terms of the local reduce d pressure and temperature. Shown are variati ons of pres-
sure, and the impact of varyin g oxidizer and fuel temperature. The asterisk (
) de-
notes an oxygen mass fraction of 0.98, the cross ( ×) marks the state of highest
temperature; black solid and dashed lines are the coexistence line and the oxygen
Widom line, respectively.
372 D.T. Banuti et al. / Combustion and Flame 196 (2018) 364–376
Fig. 10. Flame trajectories between oxidizer and pure hydro gen in terms of the local reduced pressure and temperature. Shown is the effect of added nitrogen in the
oxidizer stream. The ( ×) marks the state of highest temperature; black solid and dashed lines are the coexistence line and oxygen Widom
line, respectively. Pressure and
inflow temperatures are identical to the baseline case.
Fig. 11. Flame structure trajectories between pure oxygen and fuel in terms of the local reduce d pressure and temperature. Shown is the effect of added water in the fuel
stream. The ( ×) marks the state of highest temperature; black solid and dashed lines are the coexistence line and oxygen
Widom line, respectively.
tion that has been proposed as a benchmark case by Ruiz et al.
[59] . The computational domain is 2D, thus a realistic structure of
turbulent flow features cannot be expected, however, we are inter-
ested here in the local mixing behavior, which we expect to not be
affected by the 2D domain. The case is set up as follows: The LOX
stream is injected at a temperature of 100 K, and GH2 is injected at
a temperature of 150 K, at a chamber pressure of 10 MPa. The op-
erating conditions are purely supercritical for the hydrogen stream
and transcritical for oxygen. The GH2 and LOX jets have velocities
of 125 m/s, and 30 m/s, respectively. Simulations are performed
using the method described in Section 2.2 . The two streams are
separated by an injector lip of height h = 0 . 5 mm, which is also
included in the computational domain. The region of interest ex-
tends from 0 to 10 h in the axial direction with the origin set at the
center of the lip face. A sponge layer of length 5 h at the end of the
domain is included to absorb acoustic waves. The computational
mesh has 100 grid points across the injector lip. A uniform mesh
is used in both directions for the region from 0 to 10 h in axial di-
rection and from 1 . 5 h to 1. 5 h in transverse direction; stretching
is applied with a ratio of 1.02 only in the transverse direction out-
side this region. Adiabatic no-slip wall conditions are applied at
the injector lip and adiabatic slip wall conditions are applied for
the top and bottom boundaries of the domain. A 1/7th power law
for the near-wall velocity profiles is used for both the LOX and GH2
streams. The CFL number is set to 0.8 and no sub-grid scale model
is used.
3.4.1. Flow field
Oefelein [19] carried out a thorough analysis of spatial profiles
of this case. In particular, profiles of the compressibility factor and
the density demonstrate the thin region in which the transition
from a liquid-like to a gas-like supercritical fluid occurs. We extend
this work by analyzing the complete 2D region of interest with
regard to its thermodynamic characteristics. Specifically, we iden-
tify regions of real fluid behavior under the criterion that the local
compressibility factor, as introduced in Eq. (5) , is unity ±5%. We
identify real fluid mixing when the local fluid behaves like a real
fluid and more than one species is present in a significant fraction.
D.T. Banuti et al. / Combustion and Flame 196 (2018) 36 4–376 373
Fig. 12 . Domains of inert and reactive LOX (bottom stream) GH2 (top stream) shear
layers at nominally supercritical pressure. Light and dark blue denote ideal gas
( 0 . 95 < Z < 1 . 05 ) and real fluid behavior ( Z < 0 . 95 Z > 1 . 05 ), respectively. Real
mixing ( Z < 0 . 95 Y
O
2
< 0 . 999 ) is marked red, the flame ( T > 20 0 0 K) is yellow. The
black line corresponds to the stoichiometric mixture fraction Z
H , st
= 0 . 11 . (For inter-
pretation of the references to color in this figure legend, the reader is referred to
the web version of this article).
Figure 12 shows the flow fields of the inert and the reactive
computation. In both cases, the GH2 stream (on top) can be con-
sidered an ideal gas, whereas LOX (on the bottom) behaves like a
real fluid. In the inert case in Fig. 12 a, real mixing occurs mainly
in a layer between the streams, some cells with real mixing can
be seen in the recirculation zone behind the injector lip. Figure 12 b
shows that the real mixing layer in the reacting case is thinner and
clearly separated from the stoichiometric line. In some positions,
localized regions of real mixing can be observed on the fuel-rich
side of the flame.
3.4.2. Mixing analysis
Figure 13 is the reduced state plot for the CFD shear layer,
capturing the characteristics of the 1D flame solutions. The re-
sult matches the unity Lewis number 1D flame solution shown in
Fig. 8 , most notably in the absence of the flame loop. More impor-
tantly, no significant reduction in the reduced pressure is observed
during the bulk transition from real fluid to ideal gas conditions,
consistent with the 1D results.
A more quantitative view of real fluid mixing is provided by
Fig. 14 . It shows the count of cells, representative of computational
cost, for a given compressibility - oxygen mass fraction combina-
tion. Figure 14 a indicates that in the inert case, more than 20% di-
lutant has to be mixed with oxygen before an ideal gas state is
Fig. 13. Flame structure trajectories between pure oxygen and pure hyd rogen in
terms of the local re duced pressure and temperature from LES results of a LOX/GH2
shear layer calculation.
Fig. 14. Scatter plots of ideal to real fluid transition in terms of oxygen mass frac-
tion and compressibility factor. LOX and GH2 denote the respective inflow condi-
tions, which exceed a count of 40 0,0 0 0 each and are capped here to improve read-
ability. The trace of orange fields thus represents the main mixing trajectory. The
horizontal line represents the Z > 0 . 95 threshold to ideal gas conditions. The bin