ArticlePDF Available

Thermodynamic modeling for numerical simulations based on the generalized cubic equation of state


Abstract and Figures

We further elaborate on the generalized formulation for cubic equation of state proposed by Cismondi and Mollerup [Fluid Phase Equilib. 232, 74–89 (2005)]. With this formulation, all well-known cubic equations of state can be described with a certain pair of values, which allow for a generic implementation of different equations of state. Based on this generalized formulation, we derive a complete thermodynamic model for computational fluid dynamics simulations by providing the resulting correlations for all required thermodynamic properties. For the transport properties, we employ the Chung correlations. Our generic implementation includes the often used equations of state Soave–Redlich–Kwong and Peng–Robinson and the Redlich–Kwong–Peng–Robinson equation of state. The first two assume a universal critical compressibility factor and are, therefore, only suitable for fluids with a matching critical compressibility. The Redlich–Kwong–Peng–Robinson overcomes this limitation by considering the equation of state parameter as a function of the critical compressibility. We compare the resulting thermodynamic modeling for the three equations of state for selected fluids with each other and CoolProp reference data. Additionally, we provide a Python tool called real gas thermodynamic python library (realtpl). This tool can be used to evaluate and compare the results for a wide range of different fluids. We also provide an implementation of the generalized form in OpenFOAM.
Content may be subject to copyright.
Phys. Fluids 34, 116126 (2022); 34, 116126
© 2022 Author(s).
Thermodynamic modeling for numerical
simulations based on the generalized cubic
equation of state
Cite as: Phys. Fluids 34, 116126 (2022);
Submitted: 23 August 2022 • Accepted: 03 November 2022 • Published Online: 16 November 2022
T. Trummler, M. Glatzle, A. Doehring, et al.
This paper was selected as Featured
Application of the Maxwell–Stefan theory in modeling gas diffusion experiments into isolated
oil droplets by water
Physics of Fluids 34, 113327 (2022);
Simulation of liquid jet atomization and droplet breakup via a Volume-of-Fluid Lagrangian–
Eulerian strategy
Physics of Fluids 34, 113326 (2022);
A review on deep reinforcement learning for fluid mechanics: An update
Physics of Fluids 34, 111301 (2022);
Thermodynamic modeling for numerical
simulations based on the generalized cubic
equation of state
Cite as: Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277
Submitted: 23 August 2022 .Accepted: 3 November 2022 .
Published Online: 16 November 2022
T. Trummler,
M. Glatzle,
A. Doehring,
N. Urban,
and M. Klein
Institute of Applied Mathematics and Scientific Computing, Bundeswehr University Munich Werner-Heisenberg-Weg 39,
85577 Neubiberg, Germany
BooleWorks GmbH, Radlkoferstrasse 2, 81373 Munich, Germany
Author to whom correspondence should be addressed:
We further elaborate on the generalized formulation for cubic equation of state proposed by Cismondi and Mollerup [Fluid Phase Equilib. 232,
74–89 (2005)]. With this formulation, all well-known cubic equations of state can be described with a certain pair of values, which allow for a
generic implementation of different equations of state. Based on this generalized formulation, we derive a complete thermodynamic model for
computational fluid dynamics simulations by providing the resulting correlations for all required thermodynamic properties. For the transport
properties, we employ the Chung correlations. Our generic implementation includes the often used equations of state Soave–Redlich–Kwong and
Peng–Robinson and the Redlich–Kwong–Peng–Robinson equation of state. The first two assume a universal critical compressibility factor and
are, therefore, only suitable for fluids with a matching critical compressibility. The Redlich–Kwong–Peng–Robinson overcomes this limitation by
considering the equation of state parameter as a function of the critical compressibility. We compare the resulting thermodynamic modeling for
the three equations of state for selected fluids with each other and CoolProp reference data. Additionally, we provide a Python tool called real gas
thermodynamic python library (realtpl). This tool can be used to evaluate and compare the results for a wide range of different uids. We
also provide an implementation of the generalized form in OpenFOAM.
C2022 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://
Numerical flow simulations of super- and transcritical conditions
require appropriate thermodynamic models. A state-of-the-art ther-
modynamic model is based on a cubic equation of state (EoS) and
departure function formalism for the evaluation of enthalpy and
energy. This can be found in early
as well as recent computational
fluid dynamics (CFD) investigations.
For the cubic EoS, mostly the
(PR) or the Soave–Redlich–Kwong EoS
(SRK) is
employed. Both assume a universal critical compressibility and are,
therefore, only suitable for fluids with a matching critical compressibil-
ity. Volume translation methods
represent one possible solution
to improve the density prediction for fluids, which are not well
described by SRK or PR. Cismondi and Mollerup
suggested the
Redlich–Kwong–Peng–Robinson EoS (RKPR), introducing a third
EoS parameter and formulating all three EoS parameters as a function
of the critical compressibility. Building upon their work, Kim et al.
presented a thermodynamic modeling approach based on the RKPR
and demonstrated its advantages and suitability for different fluids.
Despite its advantages, only a few studies
have employed the RKPR
EoS recently for real gas CFD simulations of n-dodecane injections.
Within the suggestion of the RKPR, Cismondi and Mollerup
and ear-
lier Mollerup (see Michelsen and Mollerup
) also proposed a general
formulation of the cubic EoS by which all of the well-known cubic EoS
can be described with a particular set of values. Such a formulation
allows for a modularized implementation of all these cubic EoS, thus,
less code duplication and a better readability.
An alternative to a cubic EoS is the PC-SAFT EoS (perturbed-
chain statistical associating fluid theory),
which has successfully been
employed by Rodriguez et al.
and Rodriguez et al.
for two-
dimensional CFD simulations. Another approach is the usage of tabu-
lated reference data, yielding higher accuracy
and also a potentially
faster evaluation of the thermodynamic data.
Recently, also
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-1
CAuthor(s) 2022
Physics of Fluids ARTICLE
artificial neural networks, trained on tabulated data, have been
employed for thermodynamic modeling for real gas CFD simula-
However, cubic EoS are still mostly used due to their sim-
plicity and overall good accuracy. In addition to the EoS and the
relations for thermodynamic properties, CFD simulations also require
relations for the transport properties viscosity and thermal conductiv-
ity. Chung et al.
proposed correlations for these transport properties,
which are often employed for such real gas simulations.
Alternative methods are, for example, the Lucas method
for the vis-
cosity and the Stiel–Thodos method
for the thermal conductivity.
These methods have recently been employed by Sharan and Bellan
for nitrogen and showed a good agreement with NIST reference data.
Alternatively, the residual entropy scaling technique can be used for
the calculation of the thermal conductivity
and the viscosity
done by Koukouvinis et al.
In general, it is an important step in CFD
simulations to check the accuracy and suitability of a thermodynamic
model in advance, as included in several studies.
For different flu-
ids, pressure, and temperature ranges, such an evaluation can be com-
plicated and time-consuming. Apart from that, such a verification
requires an already successful implementation of the thermodynamic
model. This is usually not the case during the development or further
development of a CFD solver. The open source library CoolProp
provides implementations for the SRK and PR EoS. Bell et al.
a comprehensive thermodynamic library to evaluate chemical proper-
ties specifically targeted for chemical engineering. Our new tool
realtpl, on the other hand, has been specifically designed for appli-
cations in the context of CFD simulations and evaluates the entire
thermodynamic model required for these simulations.
In this paper, we aim to further promote the idea of the generalized
formulation of cubic EoS by Mollerup,
i.e., one formulation for all
three cubic EoS (PR, SRK, and RKPR). To this end, we describe in detail
how this formulation is solved and present the resulting relations for the
thermodynamic properties. We also outline the overall thermodynamic
model based on this generalized formulation. For the thermodynamic
model, we employ the Chung correlations for the evaluation of the
transport properties. We apply the thermodynamic model to selected
fluids and study its suitability. Therewith, we also demonstrate the good
applicability of the RKPR for all fluids with different critical compress-
ibility factors. In order to test the proposed thermodynamic model and
to apply it to different configurations, we provide an open source
Python tool called realtpl forarealgasthermodynamicpython
library. This tool can be used to evaluate and compare the results for a
wide range of different fluids. Additionally, we also provide the imple-
mentation of the generalized form in OpenFOAM.
The paper is structured as follows: Sec. II presents the thermody-
namic model based on the generalized cubic EoS. Then, in Sec. III,the
applicability and suitability of the thermodynamic model are assessed
for selected fluids comparing the model using SRK, PR, and RKPR.
Section IV contains information about the additionally provided
Python tool realtpl and the validation of the proposed
OpenFOAM implementation based on the generalized formulation.
Finally, the paper is summarized in Sec. V.
We present a thermodynamic model based on the generalized
cubic EoS. First, we present the EoS and describe in detail how it is
solved. Then, the correlations to evaluate the thermodynamic proper-
ties are presented and, finally, the Chung correlations for the transport
properties are briefly described.
A. Generalized cubic equation of state
We here consider the generalized formulation of a cubic EoS sug-
gested by Cismondi and Mollerup
and already earlier by Mollerup
pðv;TÞ¼ RT
The pressure pis a function of the molar volume vand the temperature
T. R denotes the universal gas constant with R¼8314:472 J=ðkmol KÞ.
aand brepresent the two traditional EoS parameters, considering
attractive forces with aand repulsive forces by the effective molecular
volume b. Both are determined by a proportionality factor and the criti-
cal properties p
and T
of the fluid (see Table I). Furthermore, ais mul-
tiplied by a correction factor athat is a function of reduced temperature
T=Tcand the acentric factor x. It is worth noting that for a¼0and
b¼0, the cubic EoS collapses to the ideal gas law. As a consequence,
mathematically, and also physically, the molar volume vhas to be larger
than the co-volume b(v>b). The common cubic EoS can be described
with special pairs of the values d
and d
is a supplementary
parameter defined as ð1d1Þ=ð1þd1Þ. Multiplying the denominator
out results in the well-known and often used formulation of
pðv;TÞ¼ RT
v2þubv þwb2;(2)
where u¼d1þd2and w¼d1d2. However, the first formulation [Eq.
(1)] yields simpler expressions of the derivations required for evaluat-
ing the thermodynamic properties (see Sec. II B)thanEq.(2). For the
widely used EoS SRK and PR, the proportionality factor in aand bis
constant and d
, or, respectively, uand w, are constants with d1¼1
(u¼1, w¼0) for SRK and d1¼1þffiffi
p(u¼2, w¼1) for PR.
Hence, for SRK and PR, a universal critical compressibility has been
assumed, which is about 0.285 for SRK and 0.263 for PR.
these two EoS are only well suited for a certain set of fluids with a cor-
responding similar critical compressibility. To overcome this limita-
tion, Cismondi and Mollerup
suggested to evaluate the EoS
parameters as a function of the critical compressibility resulting in the
RKPR EoS. For the detailed evaluation of the EoS parameters, see
Table I. For the RKPR, a different correlation than for SRK and PR is
used to evaluate a. Consequently, also the derivatives by temperature
@a=@Tand @2a=@ T2, required for the evaluation of the thermody-
namic properties, differ for the EoS. Concluding, all three cubic EoS
SRK, PR, and RKPR can be described by Eq. (1), where only the EoS
parameters a,b,andd
as well as the evaluation of achanges.
Figure 1 shows a reduced pressure–volume diagram for n-hexane
including cubic isotherms evaluated using the PR EoS. A subcritical
(T<Tc), a critical (T¼T
), and a supercritcal (T>Tc) isotherm are
plotted to illustrate the behavior in the different regimes. The subcriti-
cal isotherm crosses the two-phase region, which is bounded by the
bubble-point-line and the dew point-line. In thermodynamic equilib-
rium, the phase change follows the dashed line and the pressure
remains constant (see Fig. 1, dashed brown line). The cubic isotherm
shows a different path within the two-phase region describing meta-
stable states. Note that the part with the positive slope has physically
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-2
CAuthor(s) 2022
no meaning since pressure increases with volume (@p=@vjT).
Therefore, solving cubic EoS at subcritical conditions can be challeng-
ing, as described in more detail below.
In order to solve the cubic EoS [Eq. (1)], the equation is reformu-
lated using the dimensionless compressibility factor as follows:
RT ;(3)
as well as dimensionless expressions for aand bwith A¼paaðRTÞ2
and B¼pbðRTÞ1. Therewith, one obtains from Eq. (1)
Recasting results in the cubic form of all considered EoS in terms of Z,
which reads
Z3þa2Z2þa1Zþa0¼0 (5)
with the coefficients
The obtained cubic equation can be solved for real roots, which is well
described in the literature.
A cubic equation has either one real
and two imaginary roots or three real roots (see also Fig. 1). At super-
critical conditions usually only one real root is present. For this reason,
we recommend first checking for the existence of one real root when
evaluations are focused on supercritical conditions. Three real roots
are generally associated with the two-phase region present at subcriti-
cal conditions. As mentioned above, the volume has to be larger than
the co-volume b(v>b) or expressed in terms of the compressibility
factor Z>B. If the physical constraint Z>Bis full-filled, the smallest
root represents the liquid state and the largest root the vapor or gas
state (see also Fig. 1). From a thermodynamic perspective, the center
root is not stable and, therefore, physically meaningless. The correct
root of the two physically stable roots can be identified by comparing
the Gibbs energy (see Sec. II B). An alternative approach is to take
always the largest root, which usually corresponds to the vapor/gaseous
state (exception see below).
FIG. 1. Reduced pressure–volume (p–v) diagram for n-hexane evaluated using the
TABLE I. Parameters of the EoS adopted from Kim et al.
pd2þd1ðd3czZcÞd4þd5ðd3czZcÞd6with cz¼1:168;
d1¼0:428 363;d2¼18:496 215;
d3¼0:338 426;d4¼0:660 000;
d5¼789:723 105;d6¼2:512 392
A0:427 47 R2T2
! 0:457 24 R2T2
! 3y2þ3yd þd2þd1
B0:086 64 RTc
 0:0778 RTc
 1
with d¼1þd2
K0:485 08
þ1:551 71x
0:156 13x2
0:374 64
þ1:542 26x
0:269 92x2
þð5:273 45czZc0:258 26Þ
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-3
CAuthor(s) 2022
At very high and low pressures, the smallest root can be
smaller than the co-volume (ZB). In this case, there is only one
physical meaningful root, which is the largest one. At high pres-
sures, this root then corresponds to a liquid state, while at very low
pressures it corresponds to the gaseous/vapor state. It is important
to note that this can also occur in a clearly supercritical regime,
such as for n-dodecane at the ECN-Spray A condition with
p¼8 MPa in the temperature range T¼10971500 K us ing PR o r
RKPR. Consequently, a missing check for Z>Bresults in high
deviations for both EoS, while including the check yields moderate
deviations. Figure 4 illustrates the modeling of n-dodecane at the
mentioned conditions and shows that with the included check for
Z>Bonly small deviations occur.
The presented EoS can be extended to model a homogeneous
mixture for an arbitrary number of components. To this end, the EoS
parameter aaand bhave to be evaluated as a function of the mixture.
Details can be found in the literature.
B. Thermodynamic properties
In addition to the correlation of density, pressure, and tempera-
ture, also expressions for thermodynamic properties, such as the inter-
nal energy e, entropy s,enthalpyh, and specific heats cpand cv,are
needed for CFD simulations. The evaluation of these quantities
includes several thermodynamic derivatives, which can be solved using
the departure function formalism. For more detailed information, we
refer to Poling et al.
and Elliott and Lira,
and for details on the for-
mulations for the RKPR to Fathi et al.
and Kim et al.
The presented
formulations are obtained from Matheis
and recast to be valid for
the more generic formulation.
For the internal energy, this can be written as
where the subscript 0 refers to the ideal reference state. The solution of
the integral reads
where the term Kcontains the following expression:
bðd1d2Þln vþd1b
Consequently, the enthalpy his calculated with
hh0¼ee0þpv RT;(12)
resulting in the expression
Kþpv RT:(13)
The entropy sis obtained with
dvþRln ðZÞ;(14)
resulting in
ss0¼KT @aa
@TþRln 1 b
Finally, the Gibbs energy gis calculated using Eqs. (12) and (14),
gg0¼aaKþpv RT 1þln 1 b
As mentioned above, the Gibbs energy can be used to determine the
most stable root out of three real roots. If the smallest root is larger
than B[min(Z)>B], then the smallest root represents the liquid state
[Zl¼minðZÞ] and the largest one the vapor state [Zv¼maxðZÞ].
can be evaluated with
dg ¼gvgl
Bðd1d2Þln ðZlþd1BÞðZvþd2BÞ
ðZlZvÞþln ZlB
If dg <0(gv<gl), the vapor state is stable. Contrary, if dg >0
(gv>gl), the liquid state is stable.
Theheatcapacityatconstantvolumecvis calculated with
where cv0is evaluated using cv0¼cp0R.cp0, the heat capacity at
constant pressure at ideal reference state, is determined with the 7-
coefficient or the 9-coefficient NASA polynomials. For the corre-
sponding data for the 7-coefficient polynomials, we refer to Goos
et al.,
and for the 9-coefficient ones to McBride.
Special atten-
tion should be paid to the fact that the polynomials are adapted for
certain temperature ranges and that for a smooth calculation over
several temperature ranges an appropriate implementation has to
be done. Then, the heat capacity at constant pressure cpcan be
evaluated using
@vT¼ RT
with the denominator D
D¼ðvþd1bÞðvþd2bÞ¼v2þðd1þd2Þbv þd1d2b2:(23)
The speed of sound cis calculated using
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-4
CAuthor(s) 2022
C. Transport properties viscosity and heat conductivity
with the Chung correlations
For CFD simulations, also suitable relations for the transport
properties viscosity land heat conductivity kare necessary, where the
correlations by Chung et al.
are often employed.
Both quanti-
ties are composed of two terms referring to different pressure levels,
The first summand l
, respectively, dominates at low pressures
and is based on the Chapman–Enskog theory for diluted gases. The
second term, l
and k
, dominates at higher pressures and is based on
empirical correlations. The input for the model is composed of the
fluid properties, the temperature, the density, and the heat capacity cv,
where the latter only affects the evaluation of k. For a detailed descrip-
tion, see Chung et al.
and Poling et al.
Figure 2 shows the density q,theheatcapacityc
, the viscosity l,
and heat conductivity kfor the n-alkane methane (Zc¼0:2863), the
cycloalkane cyclopentane (Zc¼0:2813), the n-alkanes n-hexane
(Zc¼0:2664), and n-dodecane (Zc¼0:2497). All data have been
evaluated at a reduced pressure of p=pc¼1:5. Overall, the thermody-
namic modeling is able to reproduce the non-linear behavior for all
depicted quantities. As expected, the fluids with a critical compressibil-
ity close to 0.285 are well described by SRK, while for n-hexane
with Zc¼0:2664 PR (optimized for Zc¼0:263) gives good results.
FIG. 2. Comparison of the modeled density q, heat capacity c
, viscosity l, and heat conductivity kusing the thermodynamic model employing different EoS with reference val-
ues from CoolProp.
(a) Methane (Zc¼0:2863), (b) cyclopentane (Zc¼0:2813), (c) n-hexane (Zc¼0:2664), and (d) n-dodecane (Zc¼0:2497). All data have been evalu-
ated at a reduced pressure of p=pc¼1:5.
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-5
CAuthor(s) 2022
For n-dodecane, RKPR yields the best results. In all cases, the density
is modeled very well with the RKPR EoS. It is either comparable to
SRK and PR or much better, if the critical compressibility differs from
the values for which the two were designed.
Figure 3 visualizes the density variation qover the temperature
for these four fluids at different pressures p=pc¼f0:75;1:5;3:0g.
Again, the EoS can reproduce the non-linear density behavior at all
pressure levels. Furthermore, it can be seen that at all depicted pressure
levels, the trend of the suitability of a particular EoS remains about the
In Fig. 2, we have also included other thermodynamic quantities
and in the following we discuss the accuracy of the modeling of these
quantities. The specific heat capacity c
is evaluated with Eq. (20).The
peak at the pseudo-boiling is well captured by all EoS, but the maxi-
mum value is underestimated.
The overall behavior of the transport properties (l,k)iswell
described by the Chung correlations. In comparison with the density
evolution for the different cubic EoS, one can see that the error in the
modeling of the density qcorresponds to the error in the modeling of
these two quantities, see PR in Figs. 2(a) and 2(b). This is due to the
fact that the density qis an input parameter for the Chung model,
which directly affects the calculation of the high pressure empirical
terms l
and k
. Hence, the error of the Chung correlations increases
with an increasing modeling error of the density. Furthermore, for k,
the error of cvaffects the calculation of k
. As a consequence, the
Chung correlations yield better results the more accurate qand cvare
modeled. In addition to our findings, we refer to the recent work by
Longmire and Banuti,
who also evaluated and discussed the suitabil-
ity of the Chung correlations for modeling transport properties.
We provide an open source Python tool called realtpl (real
gas thermodynamic python library) for the presented thermodynamic
model. Additionally, we also provide the implementation of the gener-
alized form in OpenFOAM.
A. Python framework realtpl
Checking the accuracy of a thermodynamic model in advance is
a central step before conducting CFD simulations. For different fluids
as well as different pressure and temperature ranges, such an evalua-
tion can be complicated and especially time consuming. To this end,
we have written an open source Python tool to easily compare the
results obtained with the here described thermodynamic model based
on cubic EoS. The Python tool is called realtpl standing for our
real gas thermodynamic python library. It is directly coupled to the
open source library CoolProp
obtaining experimental reference data,
as well as uid properties, such as, for example, molar mass and critical
properties. In addition, a database was created for the NASA coeffi-
cients, which is also directly coupled to realtpl.Usingrealtpl,
thermodynamic modeling based on the cubic EoS PR, SRK, RKPR can
be compared and also contrasted with the reference data from
CoolProp. The current implementation is designed to evaluate results
over a temperature range (with defined number of temperature steps)
for a given pressure level. The data are displayed graphically and can
FIG. 3. Comparison of the modeled density qusing different EoS with reference values from CoolProp
at different pressures. Columns from left to right: (a) methane
(Zc¼0:2863), (b) cyclopentane (Zc¼0:2813), (c) n-hexane (Zc¼0:2664), and (d) n-dodecane (Zc¼0:2497). Rows from top to bottom: p=pc¼0:75;p=pc¼1:5, and
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-6
CAuthor(s) 2022
also be exported to a csv file for further processing. Moreover, also
evaluations over temperature and pressure ranges can be done, which
allows for table generation.
To this end, the ranges and also the
step width can be specified as configuration parameter. Apart from
that, this open source Python tool can serve as an inspiration for
implementing the present model into an internal ow solver.
Table II lists the process time for the evaluation of a representa-
tive configuration to provide estimates for the evaluation times using
realtpl. The data have been evaluated for 10, 10
perature steps using a standard laptop (Intel i5, seventh generation).
Startup refers to the reading of the config files and corresponding fluid
properties. Ref. data stands for extracting the reference data from
CoolProp. In the current version, the five quantities density, heat
capacity, speed of sound, viscosity, and heat conductivity are extracted.
This extraction from CoolProp cannot be vectorized and, thus, it has
to be looped over the temperature steps. Therefore, the evaluation
time required scales roughly linearly with the number of temperature
evaluations. This is the main time consumer when about 3 103tem-
perature evaluations are exceeded. For the thermodynamic model, we
have here listed the average of all three cubic EoS named Thermo.
model. To improve performance, the implementation of the thermo-
dynamic models has been recast as vectors, avoiding time-consuming
loops. For this reason, there is no linear scaling of the evaluation pro-
cess. Also at 10
temperature evaluations, the time per thermodynamic
model is still about 0.05 s. Here, it has to be noted that five quantities
depending on how often the check for B and the Gibbs evaluation has
to be done. The next contributions are then output and postprocessing
related. Figures refers to the visualization of all fivequantities including
the write out of the figures. For less than 3 103, this is the most time
consuming part. The last part Data-output is the write out of all evalu-
ated data to a csv file and does not consume significant time. For the
entire evaluation at, e.g., 10
temperature steps, approxim ately 6 s are
required with writing out of the figures and approximately 2.2 s with-
out. realtpl is available as a PIP Python package and also on
B. Generalized Formulation in OpenFOAM
OpenFOAM is a widely used open source software for simula-
tions, where currently the most recent versions are the foundation ver-
sion OpenFOAM-10
and the ESI version OpenFOAM2206.
both, the PR EoS is available as PengRobinsonGas.Insomein-house
extensions of OpenFOAM,
the SRK has additionally been imple-
mented. Despite the identical structure of SRK and PR, these two EoS
are often hard coded and, thus, lead to code duplicates. Following the
generalized formulation proposed above, we propose a more general
implementation of cubic EoS to avoid code duplication and to
improve readability. We provide this extension for OpenFOAM under In order to keep the traditional
OpenFOAM code structure and to not change the input les, we have
created three separate folders for the different EoS.
Figure 4 shows a validation of our OpenFOAM implementations
comparing the density distribution with that obtained using the Python
tool realtpl. As test configuration, we consider n-dodecane at a
pressure of p¼8 MPa and a temperature range of T¼500–1500 K,
matching the conditions of one operating point of the ECN Spray
As expected, both implementations lead to nearly identical
results. For RKPR, a very small deviation is visible, which is due to
rounding errors.
We have presented a thermodynamic model for real gas CFD
simulations based on the generalized formulation of cubic EoS. Using
this generalized formulation, all of the well-known cubic EoS can be
described with a particular value for d
[d2¼ð1d1Þ=ð1þd1Þ]. We
have provided a detailed presentation of the resulting generalized cubic
equation in Zand practical hints for solving it. To evaluate the ther-
modynamic properties, we presented formulations of the derivatives.
The transport properties are modeled with the Chung correlations.
The thermodynamic model allows for a modularized implementation
of several EoS.
For the cubic EoS, we have considered the well-known formula-
tions SRK and PR. These two are specifically designed for an assumed
critical compressibility factor and, therefore, their suitability is limited.
Additionally, we also considered the RKPR, where the EoS parameters
are functions of the critical compressibility factor. In this study, we
have assessed the applicability of the three EoS for selected fluids and
showed that the RKPR could be a good universally applicable choice
for the EoS. In addition to this, we have demonstrated that overall the
presented thermodynamic model can capture and reproduce the
TABLE II. Overall process time for different numbers of temperature evaluations
(t-ev), for details, see text.
10 t-ev (s) 10
t-ev (s) 10
t-ev (s) 10
t-ev (s)
Startup 0.020 0.020 0.020 0.020
Ref. data 0.037 0.179 1.489 15.894
Thermo. model 0.002 0.002 0.006 0.046
Figures 3.860 3.860 3.860 3.860
Data-output 0.018 0.034 0.146 0.365
Total 3.941 4.099 5.533 20.276
FIG. 4. Comparison of the modeled density for n-dodecane at a pressure of
p¼8 MPa. Comparison of python implementation in realtpl (colored solid
lines) and OpenFOAM implementation (colored dots) with the reference data (black
solid line) taken from CoolProp.
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-7
CAuthor(s) 2022
non-linear behavior of relevant thermodynamic quantities with an
acceptable error.
We provide our Python implementation of the generalized EoS
in the form of a ready-to-use open source tool, which can produce
results as those shown in this work for a wide range of fluids.
Additionally, we provide an implementation in OpenFOAM.
This project received funding by—Digitalization and
Technology Research Center of the Bundeswehr—under the project
MaST: Macro/Micro-simulation of Phase Decomposition in the
Transcritical Regime.
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Theresa Trummler: Conceptualization (lead); Data curation (equal);
Formal analysis (lead); Investigation (equal); Methodology (equal);
Software (equal); Validation (supporting); Writing original draft
(lead); Writing review & editing (lead). Martin Glatzle: Methodology
(equal); Software (equal); Validation (lead); Writing review & editing
(supporting). Alexander Doehring: Conceptualization (supporting);
Data curation (supporting); Investigation (supporting); Methodology
(supporting); Validation (supporting); Visualization (supporting);
Writing original draft (supporting); Writing review & editing
(supporting). Noah Urban: Data curation (supporting); Methodology
(supporting); Visualization (supporting); Writing original draft (sup-
porting). Markus Klein: Funding acquisition (lead); Writing original
draft (supporting); Writing review & editing (supporting).
The data that support the findings of this study are openly avail-
ableinpythontoolrealtpl as a PIP Python package https:// and github
realtpl. The data that support the findings of this study are openly
available in OpenFOAM on github
realFOAM with separate branches for the ESI and Foundation
M. Cismondi and J. Mollerup, “Development and application of a three-
parameter RK–PR equation of state,” Fluid Phase Equilib. 232, 74–89 (2005).
N. Zong, H. Meng, S.-Y. Hsieh, and V. Yang, “A numerical study of cryogenic
fluid injection and mixing under supercritical conditions,” Phys. Fluids 16,
4248–4261 (2004).
T. Kim, Y. Kim, and S.-K. Kim, “Numerical study of cryogenic liquid nitrogen
jets at supercritical pressures,” J. Supercrit. Fluids 56, 152–163 (2011).
T. S. Park, “LES and RANS simulations of cryogenic liquid nitrogen jets,”
J. Supercrit. Fluids 72, 232–247 (2012).
X. Petit, G. Ribert, G. Lartigue, and P. Domingo, “Large-eddy simulation of
supercritical fluid injection,” J. Supercrit. Fluids 84, 61–73 (2013).
M. Fathi, S. Hickel, and D. Roekaerts, “Large eddy simulations of reacting and
non-reacting transcritical fuel sprays using multiphase thermodynamics,”
Phys. Fluids 34, 085131 (2022).
C. Traxinger, H. M
C. Stemmer, N. Adams, and S. Hickel, “Experimental and numerical investigation
of phase separation due to multi-component mixing at high-pressure conditions,”
in 28th European Conference on Liquid Atomization and Spray Systems (Editorial
Universitat Polite`cnica de Vale`ncia, 2018), pp. 130–137.
J. Matheis and S. Hickel, “Multi-component vapor-liquid equilibrium model
for LES of high-pressure fuel injection and application to ECN Spray A,” Int. J.
Multiphase Flow 99, 294–311 (2018).
H. Muller, C. A. Niedermeier, J. Matheis, M. Pfitzner, and S. Hickel, “Large-
eddy simulation of nitrogen injection at trans- and supercritical conditions,”
Phys. Fluids 28, 015102 (2016).
N. Sharan and J. Bellan, “Investigation of high-pressure turbulent jets using
direct numerical simulation,” J. Fluid Mech. 922, A24 (2021).
C. Lagarza-Cort
es, J. Ram
ırez-Cruz, M. Salinas-V
azquez, W. Vicente-
ıguez, and J. M. Cubos-Ram
ırez, “Large-eddy simulation of transcritical
and supercritical jets immersed in a quiescent environment,” Phys. Fluids 31,
025104 (2019).
J. Poblador-Ibanez and W. A. Sirignano, “A volume-of-fluid method for
variable-density, two-phase flows at supercritical pressure,” Phys. Fluids 34,
053321 (2022).
R. Olmeda, A. Doehring, and C. Stemmer, “Study and application of wall-
roughness models in LES flows,” Int. J. Heat Fluid Flow 95, 108948 (2022).
N. Ly, A. Majumdar, and M. Ihme, “Regimes of evaporation and mixing behav-
iors of nanodroplets at transcritical conditions,” Fuel 331, 125870 (2023).
D.-Y. Peng and D. B. Robinson, “A new two-constant equation of state,” Ind.
Eng. Chem. Fundam. 15, 59–64 (1976).
M. S. Graboski and T. E. Daubert, “A modified soave equation of state for phase
equilibrium calculations. 1. Hydrocarbon systems,” Ind. Eng. Chem. Process
Des. Dev. 17, 443–448 (1978).
G. Soave, “Equilibrium constants from a modified Redlich-Kwong equation of
state,” Chem. Eng. Sci. 27, 1197–1203 (1972).
A. M. Abudour, S. A. Mohammad, R. L. Robinson, Jr., and K. A. Gasem,
“Volume-translated Peng–Robinson equation of state for saturated and single-
phase liquid densities,” Fluid Phase Equilib. 335, 74–87 (2012).
J. J. Martin, “Cubic equations of state-which?,” Ind. Eng. Chem. Fundam. 18,
81–97 (1979).
K. G. Harstad, R. S. Miller, and J. Bellan, “Efficient high-pressure state equa-
tions,” AIChE J. 43, 1605–1610 (1997).
J. Matheis, H. M
uller, C. Lenz, M. Pfitzner, and S. Hickel, “Volume translation
methods for real-gas computational fluid dynamics simulations,” J. Supercrit.
Fluids 107, 422–432 (2016).
S.-K. Kim, H.-S. Choi, and Y. Kim, “Thermodynamic modeling based on a gen-
eralized cubic equation of state for kerosene/LOx rocket combustion,”
Combust. Flame 159, 1351–1365 (2012).
K. Jung, Y. Kim, and N. Kim, “Real-fluid modeling for turbulent mixing pro-
cesses of n-dodecane spray jet under superciritical pressure,” Int. J. Automot.
Technol. 21, 397–406 (2020).
M. L. Michelsen and J. Mollerup, Thermodynamic Modelling: Fundamentals
and Computational Aspects (Tie-Line Publications, 2004).
J. Gross and G. Sadowski, “Perturbed-chain SAFT: An equation of state based
on a perturbation theory for chain molecules,” Ind. Eng. Chem. Res. 40,
1244–1260 (2001).
C. Rodriguez, P. Koukouvinis, and M. Gavaises, “Simulation of supercritical
diesel jets using the PC-SAFT EoS,” J. Supercrit. Fluids 145, 48–65 (2019).
C. Rodriguez, A. Vidal, P. Koukouvinis, M. Gavaises, and M. A. McHugh,
“Simulation of transcritical fluid jets using the PC-SAFT EoS,” J. Comput.
Phys. 374, 444–468 (2018).
P. Koukouvinis, A. Vidal-Roncero, C. Rodriguez, M. Gavaises, and L. Pickett,
“High pressure/high temperature multiphase simulations of dodecane injection
to nitrogen: Application on ECN Spray-A,” Fuel 275, 117871 (2020).
A. Doehring, T. Kaller, S. J. Schmidt, and N. A. Adams, “Large-eddy simulation
of turbulent channel flow at transcritical states,” Int. J. Heat Fluid Flow 89,
108781 (2021).
S. Jafari, H. Gaballa, C. Habchi, J.-C. De Hemptinne, and P. Mougin,
“Exploring the interaction between phase separation and turbulent fluid
dynamics in multi-species supercritical jets using a tabulated real-fluid model,”
J. Supercrit. Fluids 184, 105557 (2022).
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-8
CAuthor(s) 2022
P. Koukouvinis, C. Rodriguez, J. Hwang, I. Karathanassis, M. Gavaises, and L.
Pickett, “Machine learning and transcritical sprays: A demonstration study of
their potential in ECN Spray-A,” Int. J. Engine Res. 23, 1556–1572 (2022).
N. P. Longmire and D. T. Banuti, “Limits of fluid modeling for high pressure
flow simulations,” Aerospace 9, 643 (2022).
T. H. Chung, M. Ajlan, L. L. Lee, and K. E. Starling, “Generalized multiparame-
ter correlation for nonpolar and polar fluid transport properties,” Ind. Eng.
Chem. Res. 27, 671–679 (1988).
J. Matheis, “Numerical simulation of fuel injection and turbulent mixing under
high-pressure conditions,” Ph.D. thesis (Technische Universit
at Munchen,
B. E. Poling, J. M. Prausnitz, and J. P. O’Connell et al.,The Properties of Gases
and Liquids (McGraw-Hill, New York, 2001), Vol. 5.
M. Hopp and J. Gross, “Thermal conductivity of real substances from excess
entropy scaling using PCP-SAFT,” Ind. Eng. Chem. Res. 56, 4527–4538 (2017).
O. L
otgering-Lin and J. Gross, “Group contribution method for viscosities
based on entropy scaling using the perturbed-chain polar statistical associating
fluid theory,” Ind. Eng. Chem. Res. 54, 7942–7952 (2015).
I. H. Bell, J. Wronski, S. Quoilin, and V. Lemort, “Pure and pseudo-pure fluid
thermophysical property evaluation and the open-source thermophysical prop-
erty library CoolProp,” Ind. Eng. Chem. Res. 53, 2498–2508 (2014).
C. Bell and Contributors, “Thermo: Chemical properties component of chemi-
cal engineering design library (ChEDL),”
J. R. Elliott and C. T. Lira, Introductory Chemical Engineering Thermodynamics
(Prentice Hall, Upper Saddle River, NJ, 2012), Vol. 668.
See Penn-State-Colleage, for
“Cubic EOS and Their Behavior” accessed 10 July 2022.
E. Goos, A. Burcat, and B. Ruscic, “Third millennium ideal gas and condensed
phase thermochemical database for combustion,” (2009).
B. J. McBride, NASA Glenn Coefficients for Calculating Thermodynamic
Properties of Individual Species (National Aeronautics and Space
Administration, John H. Glenn Research Center, 2002).
See OpenFOAM-Foundation, for “OpenFOAM
Code (Foundation version) version 10” accessed 10 July 2022.
v2206 for “OpenFOAM Code (ESI version) version v2206” accessed 10 July
C. Traxinger, J. Zips, M. Banholzer, and M. Pfitzner, “A pressure-based solu-
tion framework for sub-and supersonic flows considering real-gas effects and
phase separation under engine-relevant conditions,” Comput. Fluids 202,
104452 (2020).
C. Traxinger, J. Zips, and M. Pfitzner, “Single-phase instability in non-
premixed flames under liquid rocket engine relevant conditions,” J. Propul.
Power 35, 675–689 (2019).
C. Traxinger, “Real-gas effects and single-phase instabilities during injection,
mixing and combustion under high-pressure conditions,” Ph.D. thesis
at der Bundeswehr M
unchen, 2021).
Physics of Fluids ARTICLE
Phys. Fluids 34, 116126 (2022); doi: 10.1063/5.0122277 34, 116126-9
CAuthor(s) 2022
... The transport properties viscosity µ and heat conductivity λ are calculated with the correlations by Chung et al. [38]. A detailed description of the employed thermodynamic model can be found in Trummler et al. [56]. ...
... To solve the cubic EoS (Equation (9)), the equation is reformulated using the dimensionless compressibility factor Z = pv/(RT). Details on solving cubic EoS can be found in the literature [13,15,39,56,58]. ...
... Especially for the higher temperatures, which correspond to the injection conditions, a good agreement can be observed. For more details on the modeling accuracy of hexane and pentane using the Peng-Robinson cubic EoS, we refer to Kim et al. [58] and Trummler et al. [56]. ...
Full-text available
Mixing under high pressure conditions plays a central role in several engineering applications, such as direct-injection engines and liquid rocket engines. Numerical flow simulations have become a complementary tool to study the mixing process under these conditions but require complex thermodynamic modeling as well as validation with accurate experimental data. For this reason, we use experiments of supercritical single-phase jet mixing from the literature, where the mixing is quantified by the mixture speed of sound, as a reference for our work. We here focus on the thermodynamic modeling of multi-component flows under high pressure conditions and the analytical calculation of the mixture speed of sound. Our thermodynamic model is based on cubic equations of state extended for multi-components. Using an extension of OpenFOAM, we perform large-eddy simulations of hexane and pentane injections and compare our results with the experimentally measured mixture speed of sound at specific positions. The simulation results show the same characteristic trends, indicating that the mixing effects are well reproduced in the simulations. Additionally, the effect of the sub-grid scale modeling is assessed by comparing results using different models (Smagorinsky, Vreman, and Wall-Adapting Local Eddy-viscosity). The comprehensive simulation data presented here, in combination with the experimental data, provide a benchmark for numerical simulations of jet mixing in high pressure conditions.
Thermodynamic effects of cryogenic medium have not been researched adequately for the accurate solution of the turbopump axial thrust, which is a key technique for the reusable rocket engine. In this paper, a liquid oxygen turbopump was chosen to reveal the influence of thermodynamic effects. Experimental tests using liquid nitrogen were carried out to verify the numerical model, and the numerical results under liquid oxygen were discussed to reveal the thermodynamic effects. The results show that the head coefficients and the efficiencies decrease under all operating conditions, due to the alterations of the physical properties caused by the thermodynamic effects of cryogenic medium. The total axial thrusts decrease in the range of 1.63% to 3.22%, the maximum variations of the axial thrust acting on the impeller shroud and hub are 2.96% and 2.69%, separately, owing to the divergences of the cavity structure. The entropy generation rate was chosen to analyze the power loss, and the minimum deviation caused by the thermodynamic effects is 5.01% at the normal condition; the distributions of the entropy generation rate in the rotor-stator cavities are obviously changed due to the addition of the thermodynamic effects. The new omega method was selected to compare the vortex distribution. The vortex strength changes slightly, owing to the reduction of the medium viscosity caused by the temperature rise. It is critical to consider the thermodynamic effects of cryogenic media for accurately calculating the axial thrust of a high power-density turbopump.
Full-text available
Flows in liquid propellant rocket engines (LRE) are characterized by high pressures and extreme temperature ranges, resulting in complex fluid behavior that requires elaborate thermo-physical models. In particular, cubic equations of state and dedicated models for transport properties are firmly established for LRE simulations as a way to account for the non-idealities of the high-pressure fluids. In this paper, we review some shortcomings of the current modeling paradigm. We build on the common study of property errors, as a direct measure of the density or heat capacity accuracy, to evaluate the quality of cubic equations of state with respect to pseudo boiling of rocket-relevant fluids. More importantly, we introduce the sampling error as a new category, measuring how likely a numerical scheme is to capture real fluid properties during a simulation, and show how even reference quality property models may lead to errors in simulations because of the failure of our numerical schemes to capture them. Ultimately, a further evolution of our non-ideal fluid models is needed, based on the gained insight over the last two decades.
Full-text available
We present a novel framework for high-fidelity simulations of inert and reacting sprays at transcritical conditions with highly accurate and computationally efficient models for complex real-gas effects in high-pressure environments, especially for the hybrid subcritical/supercritical mode of evaporation during the mixing of fuel and oxidizer. The high-pressure jet disintegration is modeled using a diffuse interface method with multiphase thermodynamics, which combines multi-component real-fluid volumetric and caloric state equations with vapor-liquid equilibrium calculations for the computation of thermodynamic properties of mixtures at transcritical pressures. Combustion source terms are evaluated using a finite-rate chemistry model, including real-gas effects based on the fugacity of the species in the mixture. The adaptive local deconvolution method (ALDM) is used as a physically consistent turbulence model for large-eddy simulation (LES). The proposed method represents multiphase turbulent fluid flows at transcritical pressures without relying on any semi-empirical break-up and evaporation models. All multiphase thermodynamic model equations are presented for general cubic state equations coupled with a rapid phase-equilibrium calculation method that is formulated in a reduced space based on the molar specific volume function. LES results show a very good agreement with available experimental data for the reacting and non-reacting Engine Combustion Network (ECN) benchmark Spray A at transcritical operating conditions.
Full-text available
Today, injection of liquid fuels at supercritical pressures is a frequently used technique to improve the efficiency of energy systems and address environmental constraints. This paper focuses on the analysis of the coupling between the hydrodynamics and thermodynamics of multi-species supercritical jets. Various phase transition phenomena, such as droplet formation process by condensation, which have been shown experimentally to significantly affect the flow and mixing dynamics of the jet, are studied. For this purpose, a tabulated multicomponent real fluid model assuming vapor-liquid equilibrium is proposed for the simulation of turbulent n-hexane jets injected with different inflow temperatures (480 K, 560 K, 600 K) into supercritical nitrogen at 5 MPa and 293 K. Numerical results are compared with available experimental data but also with published numerical studies, showing a good agreement. In addition, comparisons between different turbulence models, including the LES Sigma, Smagorinsky and RANS K − ϵ models have been performed, showing the relevance of the LES Sigma model for these very complex two-phase flows.
Full-text available
Injection, mixing and combustion under high-pressure conditions are key processes in modern energy conversion machines. Driven by the demand for higher efficiency and reduction of pollutants, intensive investments are made in recent years in the further development of especially two types of fuel-fired engines: liquid-propellant rocket engines (LREs) and gas engines (GEs). This arises from the fact, that LREs will remain an essential component for payload launchers in the foreseeable future and that GEs fired with hydrogen or natural gas are a possible solution to gradually diversify towards cleaner energy conversion machines. Computational fluid dynamics (CFD) can contribute to a better understanding of the injection, mixing and combustion processes within these types of engines. Here, especially one thermodynamic topic is of paramount interest within recent years: phase separation processes under initially supercritical conditions. This work presents a CFD tool that enables the thorough investigation of these processes. Both a pressure- and a density-based solver framework are introduced. The first comprises different formulations of the pressure equation to cover a wide range of Mach numbers. A double-flux scheme specifically tailored for real-gas flows is the core of the density-based solver. The thermodynamic framework relies on a rigorous and fully conservative description of the thermodynamic state. Cubic equations of state and the departure function concept form the basis of the thermal and caloric closure. Consequently, real-gas effects are included inherently. Multicomponent phase separation processes are considered by means of a minimization of the Gibbs energy. For the investigation of the non-premixed combustion process, a tabulated combustion model based on the flamelet concept is employed. Overall, measurement data from five different experimental test campaigns are used to validate the numerical framework. Both Large-Eddy Simulations and Reynolds-Averaged Navier-Stokes simulations are performed. Most of the simulations are conducted with the pressure-based framework. In the first step, real-gas effects in underexpanded jets are investigated. Very good agreement with experimental speed of sound measurements is found. Further investigations demonstrate the importance of the consideration of real-gas effects to correctly capture the jet mixing process. Next, the phase separation process in an underexpanded argon jet is studied. In the fully developed jet, the single-phase instabilities are found downstream of the nozzle exit and upstream of the Mach disk. This is in excellent agreement with experimental Mie scattering measurements. Next, the possibility of phase separation under GE-like operating conditions is investigated. Two different fuels - hydrogen- and methane-based - are considered. For the latter, pronounced phase separation processes are found which are triggered by a strong expansion and a mixing with the ambient gas. No two-phase effects occur in the hydrogen-based fuel as the critical temperature of the less volatile component is dramatically lower as in the methane-based fuel. For the investigation of phase separation processes under LRE-like operating conditions a combined experimental and numerical study together with the University of Stuttgart is conducted. Three different test cases are defined. The characteristics of the phase formation process agree well between experiments and simulations. The single-phase instability is caused solely by a mixing process of the injected fuel with the ambient gas. Next, the prediction capabilities of the pressure- and the density-based solver are assessed in detail. For the pressure-based approach a very good agreement with three experimental test cases is found. The density-based method, in contrast, yields possibly nonphysical states indicated by a strong entrainment into the two-phase region. Finally, phase separation effects in a hydrogen and a methane flame under LRE-typical operating conditions are studied. Single-phase instabilities are found on both sides of the flamelet caused by the low temperatures and the presence of water. For the methane flame, a Large-Eddy Simulation for a reference experiment is conducted. The results show that the region of phase separation is mostly restricted to the oxygen core. The OH* emission images indicate that both flame length and shape are in good agreement with the experimental results.
Full-text available
The present work investigates the application of Machine Learning and Artificial Neural Networks for tackling the complex issue of transcritical sprays, which are relevant to modern compression-ignition engines. Such conditions imply the departure of the classical thermodynamic perspective of ideal gas or incompressible liquid, necessitating the use of costly and elaborate thermodynamic closures to describe property variation and simulation methods. Machine Learning can assist in several ways in speeding up such calculations, either as a compact, trained thermodynamic model that can be coupled to the flow solver, or as a surrogate predictive tool of spray characteristics. In this work, such applications are demonstrated and their performance is assessed against more traditional approaches. Such applications involve the prediction of macroscopic spray characteristics, for example, the spray penetration over time, or the spray distribution in space and time, and predictions of fluid properties for the thermodynamic states encountered in such applications. Macroscopic characteristics can be adequately predicted by relatively simple network structures, involving just a hidden layer of 3–4 neurons, whereas prediction of thermodynamic states requires several layers of 5–20 neurons each. The results of integrating Artificial Neural Networks in transcritical sprays are rather promising; prediction of thermodynamic properties at pressures greater than 1bar has effectively zero error, yielding simulations indistinguishable from standard tabulated approaches with minimal overhead. When used as a regression method for time-histories either of spray characteristics or spray distributions, the results are within experimental uncertainty of similar experiments, not included in the training dataset.
Full-text available
We present well-resolved large-eddy simulations (LES) of a channel flow solving the fully compressible Navier–Stokes equations in conservative form. An adaptive look-up table method is used for thermodynamic and transport properties. A physically consistent subgrid-scale turbulence model is incorporated, that is based on the Adaptive Local Deconvolution Method (ALDM) for implicit LES. The wall temperatures are set to enclose the pseudo-boiling temperature at a supercritical pressure, leading to strong property variations within the channel geometry. The hot wall at the top and the cold wall at the bottom produce asymmetric mean velocity and temperature profiles which result in different momentum and thermal boundary layer thicknesses. Different turbulent Prandtl number formulations and their components are discussed in context of strong property variations.
The objective of this paper is to examine the fundamental mechanisms responsible for the transition between subcritical evaporation and supercritical dense-fluid-mixing in the absence of convection effects, specifically focusing on the liquid–vapor interfacial dynamics. To isolate the dynamics of this transition process, we characterize the different physical behaviors exhibited by an n-dodecane nanoscale droplet placed in different nitrogen ambient conditions across the fuel’s critical point. We employ a continuum-based interface-resolving diffuse-interface method to explore the underlying phase-exchange mechanisms that bring about such distinct dynamics. Following the comparison against molecular dynamics simulations and experiments of evaporating droplets and experimental data for vapor–liquid equilibria, a parametric study at various ambient conditions and droplet sizing is performed to identify four regimes of evaporation/mixing behaviors: sub- and supercritical droplet evaporation, and sub- and supercritical dense-fluid-mixing. It is shown that the distinction in the phase-exchange mechanisms in these four regimes are brought about by the different thermodynamic phases the droplet center can exhibit during the evaporation/mixing process: subcritical liquid, supercritical liquid-like, subcritical gaseous, and supercritical gas-like, respectively. It is shown that the subcritical dense-fluid-mixing behavior is a direct result of nanoconfinement of the liquid–vapor interfacial structure and thus is not present for large droplet sizes. The present study also shows that the supercritical phase-exchange dynamics can follow two different pathways: supercritical droplet-like evaporation and supercritical dense-fluid-mixing. Furthermore, promoting the early transition to supercritical dense-fluid-mixing can significantly expedite the phase-exchange process through the disintegration of the liquid-like droplet core.
A two-phase, low-Mach-number flow solver is created and verified for variable-density liquid and gas with phase change. The interface is sharply captured using a split volume-of-fluid method generalized for a non-divergence-free liquid velocity and with mass exchange across the interface. Mass conservation to machine-error precision is achieved in the limit of incompressible liquid. This model is implemented for two-phase mixtures at supercritical pressure but subcritical temperature conditions for the liquid, as it is common in the early times of liquid hydrocarbon injection under real-engine conditions. The dissolution of the gas species into the liquid phase is enhanced, and vaporization or condensation can occur simultaneously at different interface locations. Greater numerical challenges appear compared to incompressible two-phase solvers that are successfully addressed for the first time: (a) local thermodynamic phase equilibrium and jump conditions determine the interface solution (e.g., temperature, composition, surface-tension coefficient); (b) a real-fluid thermodynamic model is considered; and (c) phase-wise values for certain variables (e.g., velocity) are obtained via extrapolation techniques. The increased numerical cost is alleviated with a split pressure-gradient technique to solve the pressure Poisson equation for the low-Mach-number flow. Thus, a fast Fourier transform method is implemented, directly solving the continuity constraint without an iterative process. Various verification tests show the accuracy and viability of the current approach. Then, the growth of surface instabilities in a binary system composed of liquid n-decane and gaseous oxygen at supercritical pressures for n-decane is analyzed. Other features of supercritical liquid injection are also shown.
This paper develops a new treatment of rough walls in turbulent high-enthalpy combustion chamber environments where the effect of the roughness on the (averaged) temperature profile can no longer be neglected. As the velocity and temperature profiles are affected by the roughness, a new framework is presented to take into account the effect of normalized sand grain roughnesses on the so called downward shift of the profiles in the logarithmic region. The application of the two presented methods exploits the roughness functions which can be found in the literature. The modeling presented in this paper is among the first attempts to combine a roughness model with a chemically reacting flow. Specifically, LES simulations are performed to verify the implementation of the models proposed for two setups, a periodic channel running on air and a rocket combustion chamber running on methane and oxygen. A Flamelet combustion model is used to model the chemical reactions. At the chamber walls, a roughness model is implemented in order to observe the effects on the temperature fields. The results are compared with the available experimental data.
Direct numerical simulations of free round jets at a Reynolds number (Re_D) of 5000, based on jet diameter (D) and jet-exit bulk velocity (U_e), are performed to study jet turbulence characteristics at supercritical pressures. The jet consists of Nitrogen (N₂) that is injected into N₂ at same temperature. To understand turbulent mixing, a passive scalar is transported with the flow at unity Schmidt number. Two sets of inflow conditions that model jets issuing from either a smooth contraction nozzle (laminar inflow) or a long pipe nozzle (turbulent inflow) are considered. By changing one parameter at a time, the simulations examine the jet-flow sensitivity to the thermodynamic compressibility factor (Z), inflow condition, and pressure (p) spanning perfect- to real-gas conditions. The inflow affects flow statistics in the near-field (containing the potential core closure and the transition region) as well as further downstream (containing fully-developed flow with self-similar statistics) at both atmospheric and supercritical p. The sensitivity to inflow is larger in the transition region, where the laminar-inflow jets exhibit dominant coherent structures that produce higher mean strain rates and higher turbulent kinetic energy than in turbulent-inflow jets. Decreasing Z at a fixed supercritical ambient pressure (p∞) enhances pressure and density fluctuations (non-dimensionalized by local mean pressure and density, respectively), but the effect on velocity fluctuations depends also on local flow dynamics. When Z is reduced, large mean strain rates in the transition region of laminar-inflow jets significantly enhance velocity fluctuations (non-dimensionalized by local mean velocity) and scalar mixing, whereas the effects of decreasing Z are minimal in jets from turbulent inflow.