ArticlePDF Available

Abstract and Figures

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.
Content may be subject to copyright.
Citation: Longmire, N.P.; Banuti, D.T.
Limits of Fluid Modeling for High
Pressure Flow Simulations. Aerospace
2022,9, 643. https://doi.org/
10.3390/aerospace9110643
Academic Editor: Kevin Lyons
Received: 31 August 2022
Accepted: 11 October 2022
Published: 24 October 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
aerospace
Article
Limits of Fluid Modeling for High Pressure Flow Simulations
Nelson P. Longmire and Daniel T. Banuti *
Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA
*Correspondence: dbanuti@unm.edu
Abstract:
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.
Keywords: supercritical; transcritical; pseudo boiling; computational fluid dynamics; simulation
1. Introduction
Few areas in engineering have undergone a transformation as sudden as modeling
supercritical injection, when new experimental evidence contradicted the prevalent un-
derstanding of the key physical phenomena. Coming from an intuitive view of injection
and break-up [
1
], informed by our experience with jets of water in air, a large part of
injection for high-pressure rocket engines was focused on the break-up of propellants into
droplets, the further secondary break-up of droplets, and their ultimate vaporization [
2
,
3
];
numerical studies of spherical oxygen droplets at supercritical pressures naturally assumed
the existence of such droplets [46].
A series of experiments, conducted in Germany [
7
10
], France [
11
,
12
], and the United
States [
13
16
] in the mid 1990s and early 2000s completely changed our view. Rather than
a multi-phase break-up, transcritical injection, i.e., injection of a propellant at subcritical
temperature into an environment at supercritical pressure and temperature, could be better
described as a turbulent mixing process between two fluids of different density. Instead of
a droplet cloud, a compact stream of transcritical oxygen leaves the injector.
Summarized [
17
], “The acceptance of this change in understanding is perhaps best
reflected in the shift of boundary conditions posed by the Rocket Combustion Modeling
(RCM) workshops 2001 [
18
] and 2006 [
19
]. For the same configuration of a single injec-
tor combustion chamber (but at a different oxidizer/fuel ratio) at supercritical pressures,
the 2001 workshop specified a spectrum of oxygen droplets to be prescribed in the CFD
calculation. Five years later, injection velocity and density were deemed appropriate injec-
tion boundary conditions.” (It is curious to see that the Diesel and jet engine communities,
despite dealing with similar supercritical injection conditions [
20
,
21
], have not completely
adopted this Eulerian continuum view).
The model that has been adopted almost exclusively [
22
] was introduced by Oefelein
and Yang [
23
]. In it, the propellants are considered a chemically reacting single phase
mixture, in which caloric properties are evaluated using a real fluid equation of state (EOS)
Aerospace 2022,9, 643. https://doi.org/10.3390/aerospace9110643 https://www.mdpi.com/journal/aerospace
Aerospace 2022,9, 643 2 of 21
and transport coefficients are used based on high-pressure correlations [
24
,
25
]. The local
thermodynamic state is evaluated using van der Waals mixing rules in a single fluid mixing
approach [
22
], while different equations of state have been explored (e.g., the Benedict–
Webb–Rubin EOS [
26
] in [
23
]), mostly cubic equation of states are used today [
22
], namely
the Peng–Robinson [27] and Soave–Redlich–Kwong [28] EOS.
Unlike liquids or gases, transcritical fluid properties can exhibit steep gradients, sharp
peaks, and be non-monotonous, as illustrated in Figure 1. The peak of the isobaric specific
heat capacity was shown to correspond to the latent heat of a transcritical liquid–gas
transition, pseudo boiling [
29
31
] and thus of particular importance. It was linked to a
thermal break-up of transcritical jets [
32
] and the supercritical analog to the subcritical
boiling crisis–heat transfer deterioration [33].
100 150 200 250 300
T(K)
0
500
1000
1500
ρ(kg/m3)
6 MPa
7 MPa
8 MPa
9 MPa
100 150 200 250 300
T(K)
0
5
10
15
cp(kJ/kgK)
6 MPa
7 MPa
8 MPa
9 MPa
100 150 200 250 300
T(K)
0.00
0.05
0.10
0.15
k(W/mK)
6 MPa
7 MPa
8 MPa
9 MPa
100 150 200 250 300
T(K)
0
50
100
150
µ(µPas)
6 MPa
7 MPa
8 MPa
9 MPa
100 150 200 250 300
T(K)
0
2
4
6
8
Prandtl number
6 MPa
7 MPa
8 MPa
9 MPa
100 150 200 250 300
T(K)
0.0
0.1
0.2
0.3
0.4
α(W/mK)
6 MPa
7 MPa
8 MPa
9 MPa
Figure 1. Near-critical fluid properties, oxygen data from CoolProp [34].
Due to the complexity and nonlinearity of the Navier–Stokes equations numerical
errors exist in all LES simulations [
35
]. Ghosal in 1995 did a study to quantify and analyze
three different types of numerical errors that occur in LES simulations [
35
]. The three types
of errors highlighted in the study are “discretization errors”, “aliasing errors”, and “model-
ing errors”. Chow et al. [
36
] then built off of this study by comparing the numerical errors in
LES simulations to DNS data to better quantify the numerical errors that are present. At the
end of the study Chow et al. states “Results from our DNS dataset are similar to Ghosal’s
statistical analysis, confirming the need for careful selection of numerical parameters in
LES” [
36
]. This stresses the importance to quantify and understand these numerical errors
in order to limit them to create accurate numerical simulations of complex flow problems.
Consider the fundamental counterflow diffusion flame investigated thoroughly by
Lacaze and Oefelein [
37
]. Figure 2shows how properties change in the mesh along the
symmetry plane, specifically the isobaric heat capacity
cp
and the Prandtl number Pr as the
ratio of momentum diffusivity to thermal diffusivity. In particular, both
cp
and Pr exhibit
distinct peaks at a mixture fraction ZH1×103, as expected from Figure 1.
Despite the extremely high grid resolution of 230,000 cells for a laminar 2D case
of a smooth and straight flame, a close look suggests that these peaks are not resolved,
but appear truncated. At
ZH
1
×
10
3
the fluid is essentially pure oxygen with an
impurity of 0.1%; it is thus justifiable to compare the CFD data to tabulated pure oxygen
data from the NIST reference fluid database [
38
]. At the lowest pressure, the
cp
peak values
represented in the used numerical model (Peng–Robinson EOS) may exceed the values
captured with CFD by almost a factor of two; in addition, the actual physical reference data
exceeds the Peng–Robinson model by another 40%. A quantitative comparison between
the property value captured in CFD and reference data [
34
,
38
] is compiled in Table 1.
The simulation captures the fluid properties at sufficiently high pressures (8 MPa and
9 MPa), where the
cp
peak is less pronounced and extends over a wider temperature range.
Aerospace 2022,9, 643 3 of 21
The simulation does not capture the
cp
peak at the lower pressures 6 MPa and 7 MPa, where
it only extends over a few Kelvin width.
(a)
104103102101100
Mixture Fraction
0
5
10
15
20
cp(kJ/kgK)
(b)
Figure 2.
Isobaric specific heat capacity
cp
(a) and Prandtl number Pr (b) sampled in computational
mesh; data from Lacaze and Oefelein [
37
]. Actual
cp
peak value as horizontal lines from reference
data (NIST): dashed; Peng-Robinson: dotted.
Table 1. Fluid property sampling error in [37]. Reference data from NIST [38].
Pressure cref
p,max cCFD
p,max Ratio Rel. Error Prref
max PrCFD
max Ratio Rel. Error
in MPa J/kg/K J/kg/K
60 15.6 6.46 2.41 58.59% 9.38 4.24 2.21 54.83%
70 7.78 5.40 1.44 30.59% 5.13 3.47 1.48 32.29%
80 5.42 4.84 1.12 10.73% 3.76 3.14 1.20 16.52%
90 4.31 4.00 1.08 7.19 % 3.10 2.58 1.20 16.69%
Thus, in addition to the ‘classical’ errors, transcritical fluid properties introduce new
types of errors. The example of the counterflow diffusion flame [
37
] shows that the
thermodynamically important
cp
peak mostly is not represented by the prevalent model
of cubic EOS. In addition, the example shows that the representation of a flow field in a
discretized domain poses a fundamental challenge to the accurate sampling of physical
properties for near-critical fluids, while this issue is present in all discretized representations,
it is much less pronounced in liquid or gaseous property models, which are smooth,
monotonous, and do not feature sudden changes in their slopes. In that sense, the character
of near-critical fluid properties, particularly across the pseudo boiling transition, turns this
into a qualitatively different problem. Two main cases can be identified from the properties
shown in Figure 1: (i) sudden changes in the slope of properties that form a ‘knee’, visible in
the thermal conductivity, viscosity, and speed of sound; (ii) distinct local extrema, present
in the isobaric and isochoric heat capacities, or the Prandtl number.
Thus, in this paper, we will evaluate two main types of error sources for real fluid
simulations: (i) property errors, as a direct measure of values for density or heat capacity
and newly identified (iv) sampling errors, as a measure of how likely a numerical scheme
is to capture the strongly non-linear real fluid properties during a simulation.
2. Materials and Methods
In this section, we discuss fluid property models and computational fluid dynam-
ics (CFD) solvers. More specific methodologies will be introduced in the respective
results sections.
2.1. Fluid Properties
Real fluid properties, such as shown in Figure 1, are characterized by striking non-
linear deviations from ideal fluid models. Generally, we can distinguish thermal, caloric,
and transport properties. The thermal equation of state relates pressure
p
, density
ρ
,
and temperature
T
of a fluid. The caloric EOS relates fluid energy (e.g., enthalpy) to
pvT
.
Aerospace 2022,9, 643 4 of 21
Finally, transport properties, such as viscosity or thermal conductivity are not directly
linked to any EOS and mostly stand-alone correlations.
Here, we use the cubic equations of state developed by Peng and Robinson [
27
]
(PR) and Soave, Redlich, and Kwong [
28
] (SRK). Cubic EOS are comparatively cheap
computationally, but not very accurate. Today, high-accuracy reference EOS are typically
based on expressions of the Helmholtz free energy [
39
]. These equations have been reported
countless times and will thus not be repeated here.
It is important to note that the functional form of either approach has no physical
relevance [
39
]. Cubic equations of state are arbitrary extensions of the van der Waal
EOS [
40
]; Helmholtz EOS were developed because relevant fluid properties can be obtained
through differentiation rather than integration from the Helmholtz free energy, increasing
the flexibility and hence accuracy significantly. Thus, there is no fundamental advantage of
these EOS over other fluid property representations, such as neural networks [33,41].
Here, we will use the CoolProp library [
34
] and NIST database [
38
] to determine
baseline fluid properties using the aforementioned EOS.
The mixture critical point will be evaluated using the pseudo-critical method [
42
,
43
],
where the effective mixture critical properties are computed based on the composition and
the pure fluid properties.
2.2. Computational Fluid Dynamics
We used the open source CFD library SU2 [
44
] with its low-Mach number approxi-
mation solver [
45
] for simulations in this paper. We extended the solver to use tiny neural
networks (TNN) for efficient and accurate fluid property modeling [
33
,
41
,
46
]. A thorough
discussion is given in Longmire and Banuti [33] and will be omitted here.
Artificial neural networks are used to model the fluid’s density
ρ
, isobaric specific
heat capacity
cp
, isochoric specific heat capacity
cv
, thermal conductivity
k
, and viscosity
µ
shown in Figure 3. The ANN models match the NIST data well; the Peng–Robinson
equation of state overestimates the low-temperature density and underestimates the maxi-
mum isobaric specific heat capacity by about 20%. We chose CO
2
because, from the species
compared in Figure 4, carbon dioxide shows the smallest bias for either SRK or PR accuracy,
while still being relevant for rocket engines.
200 400 600 800
T(K)
0
500
1000
1500
ρ(kg/m3)
CO2 12MPa - 1x 3x tanh
ANN
Tpb
NIST Data
200 400 600 800
T(K)
0
2500
5000
7500
cp(J/kgK)
CO2 12MPa - 1x 10x tanh
ANN
Tpb
NIST Data
200 400 600 800
T(K)
0
100
200
µ(µPas)
CO2 12MPa - 1x 4x tanh
ANN
Tpb
NIST Data
200 400 600 800
T(K)
0.00
0.05
0.10
0.15
0.20
k(W/mK)
CO2 12MPa - 1x 4x tanh
ANN
Tpb
NIST Data
200 400 600 800
T(K)
0
500
1000
1500
cv(J/kgK)
CO2 12MPa - 1x 7x tanh
ANN
Tpb
NIST Data
Figure 3.
Comparison of NIST data and ANN fit of CO
2
properties. From top to bottom: density
ρ
, isobaric specific heat capacity
cp
, viscosity
µ
, thermal conductivity
k
, and isochoric specific heat
capacity
cv
. From left to right:
p={
10, 12, 20
}
MPa. The ANN was fit for
T={
250, 700
}
K, data
are shown to 200 K to demonstrate ANN behavior outside of the fitted interval. The pseudo boiling
temperature Tpb at which cpreaches a maximum increases with rising pressure. From [47].
Aerospace 2022,9, 643 5 of 21
100 150 200 250 300
T(K)
0
200
400
600
ρ(kg/m3)
CH4 6 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
200
400
600
ρ(kg/m3)
CH4 10 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
200
400
600
ρ(kg/m3)
CH4 20 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
500
1000
1500
ρ(kg/m3)
O2 6 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
500
1000
1500
ρ(kg/m3)
O2 10 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
500
1000
1500
ρ(kg/m3)
O2 20 MPa
SRK
PR
HEOS
200 300 400 500 600
T(K)
0
500
1000
1500
ρ(kg/m3)
CO2 6 MPa
SRK
PR
HEOS
200 300 400 500 600
T(K)
0
500
1000
1500
ρ(kg/m3)
CO2 10 MPa
SRK
PR
HEOS
200 300 400 500 600
T(K)
0
500
1000
1500
ρ(kg/m3)
CO2 20 MPa
SRK
PR
HEOS
400 500 600 700 800
T(K)
0
200
400
600
800
1000
ρ(kg/m3)
H2O 6 MPa
SRK
PR
HEOS
400 500 600 700 800
T(K)
0
200
400
600
800
1000
ρ(kg/m3)
H2O 10 MPa
SRK
PR
HEOS
400 500 600 700 800
T(K)
0
200
400
600
800
1000
ρ(kg/m3)
H2O 20 MPa
SRK
PR
HEOS
Figure 4.
Comparison of Peng–Robinson and Soave–Redlich–Kwong EOS to reference data for
density
ρ
of, from top to bottom, methane, oxygen, carbon dioxide, water; at three different pressures
from left to right p= [6, 10, 20]MPa.
3. Results
In this section, we discuss (1) property representation errors and show how they
impact real fluid simulations; (2) sampling errors as a numerical error type not encountered
in gas or liquid flow simulations.
In particular, we focus on the common propellants oxygen and methane, and the
main combustion products carbon dioxide and water. The chosen pressures represent
technically relevant cases of near-critical experiments (6/7 MPa), gas generator cycles
(10 MPa), and staged combustion cycles (20 MPa).
3.1. Property Representation Error
The most striking effect of real fluid property models is, naturally, their difference to
idealized models for ideal gases (
p=ρRT
), or incompressible fluids (
ρ=const
). In this
section we will discuss the magnitude and impact of inaccurate fluid property models,
e.g., by using cubic EOS. Naturally, this aspect has been investigated since the advent of
the very first real fluid CFD models [
23
]. It is long known that cubic EOS may introduce,
e.g., errors in density of 10% to 20%.
Here, we will discuss the relevance of the density and the isobaric specific heat capacity.
3.1.1. Density Errors
Why is the density relevant? The jet break-up length and thus position of heat release
in coaxial injectors, where an inner jet is surrounded by an annular flow and their respective
injection velocities u, scales mostly with the momentum flux ratio between both streams,
Aerospace 2022,9, 643 6 of 21
J=(ρu2)outer
(ρu2)inner
. (1)
On the other hand, the global chamber conditions are mostly determined by the overall
oxidizer-fuel ratio OFR,
OFR =˙
mOxidizer
˙
mFuel
=(ρuA)Oxidizer
(ρuA)Fuel
, (2)
where Ais the injector area.
If we do not get the density right in our simulations, we cannot capture both the overall
chamber conditions such as temperature or speed of sound and the correct propellant
jet break-up and heat release. However, both are essential parameters for combustion
instabilities [
2
]. Figure 4compares reference quality data for density
ρ
at rocket-relevant
conditions 6, 10, and 20 MPa, for the rocket-relevant species methane, oxygen, carbon
dioxide, and water.
It becomes clear that cubic EOS always involve a trade-off: SRK data is more accurate
towards lower temperatures for the propellants oxygen and methane, but PR captures the
liquid–gas transition much better. For the reaction products carbon dioxide and water PR
data seems slightly better, but we have to conclude that neither cubic EOS can represent
the fluid properties accurately.
3.1.2. Heat Capacity Errors
The isobaric specific heat capacity
cp
plays a more indirect, yet equally important
role for propellants. Figure 1shows the distinct peak in heat capacity, which represents
the pseudo boiling condition [
29
], marking the position of maximum change during the
supercritical liquid–gas transition.
Heat Transfer
On the one hand,
cp
is a dominating factor in the calculation of the Prandtl number Pr,
c.f. Figure 1, which in turn governs transient heat transfer. Figure 1furthermore shows the
dramatic local peak in Pr which is manifest as a layer of locally high Pr in a fluid exhibiting
a temperature distribution across pseudo boiling, and acting as an insulator that inhibits
heat transfer across that layer.
Figure 5compares reference quality data for
cp
at rocket-relevant conditions 6, 10,
and 20 MPa, for the rocket-relevant species methane, oxygen, carbon dioxide, and water.
Particularly at low supercritical pressures, reference data exceed cubic EOS data by as much
as 30%. The difference between PR and SRK EOS is less pronounced for the heat capacity
compared to density data.
Generally, an underestimated Pr leads to an overestimated heat transfer rate. Thus,
the insulating properties of the pseudo boiling layer will not be reflected in our simulation
if our fluid property model misses that value by 30%.
Pseudo Boiling
Another important aspect of the heat capacity peak is its physical meaning as a super-
critical latent heat [
29
,
30
]. Ernst Schmidt was the first to realize [
48
,
49
] that the widening
heat capacity distribution around saturation conditions at high subcritical pressures plays
the role of a distributed latent heat. This can be seen in Figure 5for water, where a more
and more pronounced peak and tail of
cp
are developed for increasing subcritical pressures.
We now understand that this view can be extended to supercritical pressures [
29
,
30
], where
the instantaneous latent heat has been completely replaced by the wide
cp
peak. Somewhat
surprisingly, the integral of the
cp
across the complete transition is invariant for a wide
range of pressures, establishing a generalized latent heat that can be identified for both
subcritical and supercritical transitions.
Aerospace 2022,9, 643 7 of 21
100 150 200 250 300
T(K)
0
5
10
15
20
cp(kJ/kgK)
CH4 6 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
5
10
15
cp(kJ/kgK)
CH4 10 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
1
2
3
4
5
cp(kJ/kgK)
CH4 20 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
5
10
15
cp(kJ/kgK)
O2 6 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
1
2
3
4
5
cp(kJ/kgK)
O2 10 MPa
SRK
PR
HEOS
100 150 200 250 300
T(K)
0
1
2
3
4
5
cp(kJ/kgK)
O2 20 MPa
SRK
PR
HEOS
200 300 400 500 600
T(K)
0
2
4
6
8
10
cp(kJ/kgK)
CO2 6 MPa
SRK
PR
HEOS
200 300 400 500 600
T(K)
0
2
4
6
8
10
cp(kJ/kgK)
CO2 10 MPa
SRK
PR
HEOS
200 300 400 500 600
T(K)
0
1
2
3
4
5
cp(kJ/kgK)
CO2 20 MPa
SRK
PR
HEOS
400 500 600 700 800
T(K)
0
2
4
6
8
10
cp(kJ/kgK)
H2O 6 MPa
SRK
PR
HEOS
400 500 600 700 800
T(K)
0
2
4
6
8
10
cp(kJ/kgK)
H2O 10 MPa
SRK
PR
HEOS
400 500 600 700 800
T(K)
0
10
20
30
cp(kJ/kgK)
H2O 20 MPa
SRK
PR
HEOS
Figure 5.
Comparison of Peng–Robinson and Soave–Redlich–Kwong EOS to reference data for
isobaric specific heat capacity cpof, from top to bottom, methane, oxygen, carbon dioxide, water; at
three different pressures from left to right p= [6, 10, 20]MPa.
This naturally leads to the questions of whether cubic EOS capture the location and
the strength of the pseudo boiling transition. We understand the transition location as the
temperature at which the heat capacity and the isobaric thermal density gradient reach
maxima [
29
]; we understand the strength in terms of the boiling number
B2
[
29
], which
measures the height of the
cp
peak in relation to the limit value towards liquid conditions
at lower temperatures. This becomes intuitively clear when regarding oxygen data in
Figures 4and 5: at low pressures, we see a distinct drop in density to occur simultaneously
with the heat capacity peak. At higher pressures, this transition has moved to higher
temperatures while becoming much less pronounced.
Figure 6shows that cubic EOS are not good at capturing the pseudo boiling transition.
For the regarded species, they generally predict the transition to occur at temperatures
that are too low (for the case of water at a reduced pressure of 5 by 80 K). The magnitude
measured by
B2
is systematically underpredicted at lower reduced pressures. At reduced
pressures exceeding 3, it depends strongly on the species: PR data matches reference data
well for carbon dioxide; however, identical distributions are predicted for oxygen, methane,
and water, even though the reference data shows distinct curves.
We have to conclude that cubic EOS simultaneously overestimate heat transfer to a
transcritical fluid, underestimate the amount of energy needed to transform it to a gaseous
state during the transition (i.e., heat capacity peak), and predict the wrong temperature at
which this transition should occur. This suggests that cubic EOS will tend to underestimate
the timescale of pseudo boiling processes, such as the thermal break-up of transcritical
Aerospace 2022,9, 643 8 of 21
jets [
32
], the vaporization of transcritical droplets, or the formation of vapor layers during
film boiling [33].
1.0 1.1 1.2 1.3
Reduced temperature
1
2
3
4
5
Reduced pressure
O2
CH4
H2O
CO2
12345
Reduced pressure
101
100
101
102
B2
O2
CH4
H2O
CO2
Figure 6.
Pseudo boiling properties compared between PR EOS (dashed) and reference data (solid).
Left: The pseudo boiling line as the locus of
cp
peaks shows pronounced differences between both
data sets. (Right) the pseudo boiling strength
B2
[
29
] is underpredicted for reduced pressures up to 3;
beyond is well matched for carbon dioxide, but not for the other species.
3.1.3. 2D Heat Transfer
To show the importance of the EOS and how it impacts the results of simulations,
compared the results of the highly accurate TNN shown in Figure 3to the Peng–Robinson
EOS in a parameter study of supercritical CO
2
of a laminar heated flat plate. We chose
CO
2
because, from the species compared in Figure 4, carbon dioxide shows the smallest
bias for either SRK or PR accuracy, while still being relevant for rocket engines. An in
depth explanation of an original test case can be found in [
33
]. At a Reynolds number of
Re
=
112 and at a bulk temperature of
T=
270 K, we set the wall temperature to different
temperatures, at and above pseudo boiling which heated the bulk fluid. These simulations
were performed for two different pressures of
p=12, 20
MPa, and for two different EOS
using the ANN and a second time using the PR EOS. To compare the results boundary
layer profiles were taken and the heat flux along wall for each simulation was extracted to
create a boiling curve.
The plot in Figure 7is a density contour plot for the simulation with a wall temperature
of
Tw=
390 K. The contour plot shows a steep density change for the fluid close to wall as
a thin vapor film forms close to the wall.
Figure 7.
Density contour plots at 12 MPa for wall temperature
Tw=
390 K. The figure shows the
density distribution for a dense/cool flow coming from the left, and passing over a heated plate at
the bottom. At the heated wall, a low-density vapor-like layer forms.
Figure 8compares the density
ρ
in the boundary layer and the isobaric heat capacity
cp
in the boundary layer for the results using the two different EOS. The top are the results
for the 12 MPa case with
Tw=
390 and the bottom are results for the 20 MPa case with
Tw=
390. The density boundary layer plots show that the PR EOS over predicts the liquid
density in the bulk fluid while also showing a larger thickness of the vapor film near the
wall. The heat capacity plots show the PR EOS under predicts the maximum heat capacity
while also being offset from the TNN results.
Aerospace 2022,9, 643 9 of 21
0.00 0.01 0.02 0.03 0.04 0.05
y(m)
0
500
1000
1500
ρ(kg/m3)
ANN
Peng-Robinson
0.00 0.01 0.02 0.03 0.04 0.05
y(m)
0
2000
4000
6000
cp(J/kgK)
ANN
PR
HEOS max
PR max
250 300 350 400
T(K)
0.0
0.5
1.0
1.5
q(kW/m2)
12 MPa
ANN
Peng-Robinson
constant
Figure 8.
Fluid properties in the boundary layer for a wall temperature of
Tw=
390 at a pressure
of
p=
12 MPa comparing the ANN (blue) simulation to the Peng–Robinson (red) simulation. Left:
density profile; Right: heat capacity profile; Right: average heat flux along the flat plate for heated
flat plate of different wall temperatures.
Figure 8further compares the heat transfer results of the different EOS for the two
different pressures. The plots show the average heat along the wall for each wall temper-
ature which is a boiling curve. There is little to no variation in the heat transfer results
for wall temperatures that are close to the bulk fluid temperature because there is little
to no variation in the fluid properties but as the wall temperature increases there is large
variation between the simulations using the two different EOS. For the 12 MPa case the
ANN show a larger local maximum compared to the PR EOS and as the wall temperature
increases more the PR EOS under predicts the rate of heat transfer compared to the highly
accurate ANN models.
The simulation reveals that the introductory remark of an underestimated Prandtl
number and supercritical latent heat do not necessarily lead to direct decrease in heat
transfer. Here, Figure 8shows that this is indeed the case for low wall temperatures.
However, as a result, an insulating wall vapor layer can more easily form and inhibit
further heat transfer. Case in point, PR underestimates the maximum heat transfer, overes-
timates the vapor layer thickness, and overall predicts a more smooth transition to heat
transfer deterioration.
3.2. Sampling Error
We have addressed the extent and impact of inaccurate fluid properties as derived
from using cubic equations of state. However, our opening example in Figure 2highlights
another potential source or error, one that, to the best of our knowledge, has not received
any attention so far.
Consider the cartoon of the flow field behind a coaxial injector in Figure 9. Behind a
LOX post of thickness
δ
, streams of gaseous methane and liquid oxygen meet, mix, and react.
We can identify the adiabatic flame temperature
Tad
as the maximum temperature reached
in the field. Following the path of oxygen [
50
], it is injected with a temperature
TLOX,in
,
heats through pseudo boiling at
Tpb
, before mixing and reacting. With flames mostly
anchored behind the LOX post [
7
,
12
], this means that the flow behind the LOX post sees a
temperature increase from
100 K in the LOX stream to
3500 K in the flame, or to
300 K
in an ambient temperature gaseous methane stream.
Figure 9.
Shear layer with anchored flame behind coaxial injector. LOX post thickness
δ
, mesh resolu-
tion d, LOX temperature TLOX,in, adiabatic flame temperature Tad, pseudo boiling temperature Tpb.
If we want our solvers to capture the extremely narrow
cp
peak maximum, we need a
mesh of sufficiently small cells of size
d
to ensure a
1 K temperature resolution, over a
δ
width space. As a naive order of magnitude evaluation, to resolve a 1 K interval over the
Aerospace 2022,9, 643 10 of 21
200 K temperature difference during inert mixing, we will need
O(
100
)
cells. However,
as this assumes a perfectly regular distribution of both mesh resolution and temperature,
the actual number to ensure capturing even under adverse conditions will likely be higher.
If the mesh is insufficiently resolved, our method will not be able to capture the relevant
fluid property. This is what happens in Figure 2, where even the extremely fine mesh skips
over the cppeaks.
We will refer to this as sampling error. The simplest configuration to potentially exhibit this
sampling error is a 1D temperature difference. In the following, we discuss two fundamentally
different approaches to quantify the impact of a sampling error: First, a Monte Carlo approach
that is solver agnostic; second, a 1D transport problem evaluated in the open source SU2
solver. The overall strategy in both cases is to locally extract the heat capacity and transport
properties directly on the mesh, in order to quantify how often extrema in transport properties
are skipped by the solver, and what the corresponding errors are.
3.2.1. Monte Carlo
The idea behind our Monte Carlo transport analysis is that on a computational mesh
capturing some state and property, the precise location of a particular state on a grid point
is essentially random. For a sufficiently resolved simulation, the form of the underlying
mesh should be irrelevant; the flow may additionally move across the mesh so that locally
resolved states change temporally and spatially.
This is not an issue for fluids with moderate or smooth property variations, such as
ideal gases or liquids. Figure 10 shows how linear interpolation from certain discrete points
can be used as very good approximations for the actual physical property between the
sampled points, using oxygen data from the NIST database [38] at p=0.01 MPa.
(a)
100 150 200 250 300
Temperature in K
0.0
0.2
0.4
0.6
0.8
1.0
Heat capacity in J/kgK
NIST
interpolation
(b)
100 150 200 250 300
Temperature in K
0.00
0.01
0.02
0.03
Therm. Cond. in W/mkg
NIST
interpolation
Figure 10.
Comparison of interpolation to physical data for ideal gas. Data from NIST database [
38
]
for oxygen at 0.01 MPa, i.e., at ideal gas conditions. (
a
) Comparison of isobaric heat capacity.
(b) Comparison of thermal conductivity.
However, for transcritical flows, two complications need to be accounted for when
regarding fluid properties in adjacent cells: First, fluid property curves may exhibit a local
curvature that introduces a significant error if a fluid property is estimated from linear
interpolation or distinct local extrema are present that cannot be recovered from averaging
and require for a mesh point to reach the exact state of the extremum. Second, it is these
extrema rather than local or averaged property values that may constrain transport fluxes,
and thus act as a bottle neck.
Figure 11 shows how the isobaric specific heat capacity and thermal conductivity for a
near-critical fluid (oxygen at 6 MPa) exhibit this structure that is extremely challenging to
sample accurately. In both cases, the value obtained from interpolation deviates by more
than a factor of two from the actual value at the point of interest; for
cp
, the error compared
to the maximum value in the respective temperature interval is, however, much larger.
We have two core concepts to this Monte Carlo transport analysis:
1.
Within bounds defined by the boundary conditions, states can be randomly sam-
pled from the physical fluid properties to obtain information about how numerical
properties are reconstructed on a discrete representation.
Aerospace 2022,9, 643 11 of 21
2.
Extremal values of fluid properties that minimize transport need to be taken into
account as they act as bottle necks.
As an example, the evaluation shown in Figure 11 is performed 10,000 times, i.e., within
the boundary conditions 100 K and 300 K, two random temperatures are determined fol-
lowing a uniform distribution, then the relevant physical (
φ
), interpolated (
¯
φ
), and extremal
values
φext
are identified. Figure 12 shows the results, where the direct error is the ratio of
physical and interpolated values, and max/min are the ratios of the interpolated and the
extremal values.
(a)
100 150 200 250 300
Temperature in K
0
5
10
15
20
Heat capacity in J/kgK
NIST
interpolation
max error
interp. error
(b)
100 150 200 250 300
Temperature in K
0.00
0.05
0.10
0.15
Therm. Cond. in W/mkg
NIST
interpolation
max error
interp. error
Figure 11.
Illustration of error from underresolved sampling of non-monotonous fluid properties.
(
a
) isobaric specific heat capacity
cp
; (
b
) thermal conductivity
k
. The black solid line is NIST reference
data; the orange circles in ‘interpolation’ mark sample positions in a mesh, where the middle value
¯
φ
is
the interpolation of the available mesh values
¯
φ=1
2(φ(T) + φ(T+))
; the purple square ‘NIST’ marks
the evaluation of the property
ˆ
φ
from NIST data at the average temperature, i.e.,
ˆ
φ=φ(1
2(T+T+))
;
the red circle marks the local maximum (
cp
) or minimum (
k
), the interpolation error (dashed red) is
the difference between
¯
φ
and
ˆ
φ
, the max error (dashed blue) marks the difference between
¯
φ
and the
maximum (cp) or minimum (k) value φext.
For the specific heat capacity in Figure 11a, the majority of cases do find a ratio close
to unity when random temperatures are sampled that are close together or far away from
the extremum, such that a linear interpolation is a close approximation. Because the
second derivative of
cp
is mostly positive except for the narrow region around the peak,
the interpolation overestimates the value except when the peak is enclosed, such that the
peak in the histogram for the direct error in Figure 11a is moved to a value slightly larger
than unity. Over all, we see that the simple interpolation may be wrong by more than
an order of magnitude, with a higher error when compared to the extremum. The same
pattern can be seen in Figure 11b for the thermal conductivity.
(a)
100101
Ratio
100
101
102
103
104
Count
direct
max
(b)
100101
Ratio
100
101
102
103
104
Count
direct
min
Figure 12.
Results of Monte Carlo analysis over single random interval
[Tmin
,
Tmax]
for 10,000 samples.
(
a
) isobaric specific heat capacity
cp
; (
b
) thermal conductivity
k
. ‘Direct’ is ratio
φ/¯
φ
; ‘max/min’ are
ratios φext/¯
φ. A ratio of 100means that the physical value is exactly sampled.
We find that for two randomly sampled mesh temperatures, there is indeed a sub-
stantial error to the physical property to be sampled based on the unique structure of
supercritical fluid properties—something not observed for ideal gases or liquids. The pre-
vious analysis, however, has the shortcoming of not being mesh-size sensitive. In the
Aerospace 2022,9, 643 12 of 21
following, we thus extend the study to account for a provided mesh resolution. In order to
quantify mesh resolution effects, an extended study is performed using the following steps:
1. Prescribe a mesh resolution nand the boundary conditions Tmin and Tmax.
2.
Determine
n
1 random temperatures (uniform distribution) in the interval
[Tmin
,
Tmax]
.
Together with the boundary conditions
Tmin
and
Tmax
,
n
temperature intervals are thus
identified.
3. For each interval, perform the analysis illustrated in Figure 11.
4. Save the maximum error obtained across all intervals [Tmin,Tmax].
5.
Repeat the above steps
N
times to analyze a distribution of the error for a given
resolution n.
6. Repeat the above steps for different nto study the impact of the resolution.
Figures 13 and 14 show the results of an evaluation of the heat capacity
cp
for tem-
perature boundary conditions of 100 K and 300 K, for mesh resolutions
n
of 4 to 800 cells,
with different numbers of sample runs
N
from 100 to 5000. It can be seen that
N=
1000
samples already give a good estimate of the final result.
For the coarse meshes in Figure 13 the ratios between the maximum value and the
numerical value may by as high as a factor of ten. Even for 50 cells, a significant number of
samples will estimate numerical values that are more than a factor of 2 from the physical
values. Towards higher resolution, this effect gets smaller and smaller. Figure 14 shows
that errors are mostly below 10 % for a resolution of 800 cells.
5 10
Ratio
0
5
10
Count
5 10
Ratio
0
50
100
Count
5 10
Ratio
0
200
400
600
Count
5 10
Ratio
0
5
10
15
Count
5 10
Ratio
0
20
40
60
80
Count
5 10
Ratio
0
100
200
300
400
Count
2.5 5.0 7.5
Ratio
0
5
10
Count
5 10
Ratio
0
50
100
Count
2.5 5.0 7.5 10.0
Ratio
0
200
400
600
Count
2 4
Ratio
0
5
10
15
Count
246
Ratio
0
50
100
150
Count
2.5 5.0 7.5
Ratio
0
500
1000
Count
Figure 13.
Histograms of maximum error ratio between interpolated and maximum value of
cp
.
From top to bottom: 4, 10, 20, and 50 cells resolution. From left to right columns: 100, 1000, and
5000 samples. Note the change in x scale.
Aerospace 2022,9, 643 13 of 21
123
Ratio
0
5
10
15
20
Count
2 4
Ratio
0
100
200
300
Count
2 4
Ratio
0
500
1000
1500
Count
1.0 1.5 2.0
Ratio
0
5
10
15
20
Count
123
Ratio
0
100
200
300
400
Count
123
Ratio
0
500
1000
1500
Count
1.0 1.2 1.4 1.6
Ratio
0
10
20
30
Count
1.00 1.25 1.50 1.75
Ratio
0
100
200
300
Count
1.0 1.5 2.0
Ratio
0
1000
2000
Count
1.05 1.10
Ratio
0
5
10
15
20
Count
1.0 1.2 1.4
Ratio
0
200
400
600
Count
1.0 1.2 1.4 1.6
Ratio
0
1000
2000
3000
Count
Figure 14.
Histograms of maximum error ratio between interpolated and maximum value of
cp
.
From top to bottom: 100, 200, 400, and 800 cells resolution. From left to right columns: 100, 1000,
and 5000 samples. Note the change in x scale.
3.2.2. 1D Heat Transfer
The other test case to demonstrate the discretization error is a 1D transport problem.
Specifically, the chosen test case is split between a cool liquid-like side and a warm gas-like
side. The temperature difference of the the two sides will induce a heat flux. The fluid of
interest here is oxygen at 7 MPa. Artificial neural networks were used to model the fluid
properties [
41
,
47
]. The computational domain is 1 cm long; six different mesh resolutions
were used
n={
10, 20, 40, 80, 100, 800
}
. The goal of the 1D mixing case is to assess whether
there is a clear difference in transport between the coarse and the fine meshes.
The case is set up so that there is a cold liquid-like side on one half of the domain, and a
warm gas-like state on the other side, where the temperature difference induces a heat flux.
The fluid for the 1D heat transfer case is oxygen at 7 MPa (
pr=
1.389). The cold liquid-like
oxygen is initially set to a temperature of 100 K as this temperature is before pseudo boiling,
and the warm gas-like oxygen has an initial temperature of 300 K which is a temperature
above pseudo boiling. The pseudo boiling temperature for oxygen is at 163 K, so the fluid
goes through the pseudo boiling process during the heating process. The computational
domain is 10 mm long. The elements are equidistant, and six different mesh resolutions
were used where
n
is the number of elements
n={
10, 20, 40, 80, 100, 800
}
. The number
of elements and grid spacing is shown in Table 2. The top and bottom boundaries of
the domain are Euler walls, and the left and right boundaries of the domain are pressure
Aerospace 2022,9, 643 14 of 21
boundaries that are set so there is no back pressure. The time simulated is 10 s; the spreading
interfacial layer reaches the boundaries after about 3 s.
Table 2. Table of the grid spacing for the computational domains used.
# Elements Spacing din mm
10 1.0
20 0.5
40 0.25
80 0.125
100 0.1
800 0.0125
Physically, as the density interface moves through the fluid, the
cp
peak associated
with pseudo boiling is present at each point in time.
For the different meshes,
cp
in each cell is calculated using the temperature, and the
maximum
cp
throughout the computational domain for each time step is taken and plotted
as seen in Figure 15. The plot shows that only the most refined mesh of 800 elements
fully recovers the maximum of the isobaric heat capacity. In contrast, the coarser meshes
only capture the maximum periodically and randomly throughout the simulation. Since
the coarser meshes underestimate the maximum
cp
, it is expected that the heat transfer
rate would be faster in the coarser meshes. None of the meshes can recover the NIST
maximum of the isobaric heat capacity fully for the plot on the right, but this is due to
the Peng–Robinson fluid property model as it underestimates the maximum isobaric heat
capacity by about 20%.
(a)
0.0 2.5 5.0 7.5 10.0
t(s)
0
2
4
6
8
cp(kJ/kg*K)
10
20
40
80
100
800
NIST Max cp
PR Max cp
(b)
0.0 2.5 5.0 7.5 10.0
t(s)
0
2
4
6
8
cp(kJ/kg*K)
10
20
40
80
100
800
NIST Max cp
Figure 15.
Plots of the maximum heat capacity versus time for O
2
at 7.0 MPa in 1D mixing simulation
using Peng–Robinson fluid properties (a) and ANN fluid properties (b).
Further, for each mesh at each time step, the mean thermal conductivity is calculated
between each cell and compared to what the NIST thermal conductivity would be based
on the mean temperature between each cell. A ratio between the simulation value and
the NIST value was calculated, and the maximum ratio was taken for each time step and
plotted, shown in Figure 16. For the coarse meshes, the ratios are large, meaning there is
a significant error between the simulation value and the NIST value. The error decreases
as the number of elements in the mesh increases, and the curves converge because more
points are available at later timesteps. The error is negligible in the most refined mesh
of 800 elements as the ratio between the simulation value, and the NIST value is 1. This
shows that for the coarse meshes, the solver overestimates the mean thermal conductivity
compared to the fine meshes, again indicating that the heat transfer in the coarse meshes
should occur at a faster rate than in a fine mesh.
Aerospace 2022,9, 643 15 of 21
0.0 2.5 5.0 7.5 10.0
t(s)
1
2
3
4
Ratio
10
20
40
80
100
800
Figure 16.
Plot of the maximum error ratio between simulation value and NIST value for thermal
conductivity versus time for O2at 7.0 MPa
The spreading of the interface is shown in Figure 17. At
t=
0, the left half of the
domain is 100 K, and the right half of the domain is 300 K. As time goes on, the liquid and
gaseous oxygen will mix, and the temperature curve will begin to flatten until eventually a
linear distribution of the temperature is reached. The plot shows that the heat moves faster
in the coarse mesh than in the fine mesh.
(
a
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
n= 10
n= 800
(
b
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
c
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
d
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
e
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
f
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
Figure 17.
Plot of the temperature along the domain of the mesh at different time steps, (
a
f
) corre-
sponding to
t={
0, 0.25, 0.5, 1, 2, 3
}
(s). The red line is a simulation using ANN fluid property models
on a coarse mesh of
n=
10 elements, and the black line is a case on a fine mesh of
n=
800 elements
using ANN fluid property models.
To better quantify and analyze the movement of the heat, the position of a high and
low-temperature threshold is tracked. As time increases, this high temperature threshold
will move, and its
x
position will increase, and the low temperature threshold
x
position
will decrease. How fast these positions change is a measure how quickly the heat moves in
each simulation. Figure 18 shows the results of this analysis and compares the difference
of the results between a coarse mesh of 10 elements and a fine mesh of 800 elements.
The change of the position of the low temperature and high temperature thresholds show
that the thresholds move faster in the coarse mesh than in the fine mesh.
Sampling error on practical meshes cause the thermal conductivity to be overestimated,
and the isobaric heat capacity to be underestimated. To counter act the overestimation and
underestimation, fixes to the thermal conductivity and isobaric heat capacity models are
applied. The overall strategy is to ensure that the extrema values of
cp
and
k
are preserved.
For the thermal conductivity three fixes are demonstrated. For the first, two adjacent cells’
temperature are checked; if the left cell is above 154 K and the right cell is below 300 K,
Aerospace 2022,9, 643 16 of 21
the minimum thermal conductivity between the two cells is used instead of calculating the
mean thermal conductivity between the two cells.
k=(min(ki,kj)if Ti>154 K Tj<300 K;
ki+kj
2otherwise. (3)
For the second fix, two adjacent cells’ temperature are checked; if the left cell is above
154 K and the right cell is below 300 K, the half of the minimum thermal conductivity
between the two cells is used instead of calculating the mean thermal conductivity between
the two cells.
k=(min(ki,kj)
2if Ti>154 K Tj<300 K;
ki+kj
2otherwise. (4)
For the third fix, two adjacent cells’ temperature are checked; if the left cell is above
154 K and the right cell is below 300 K, the thermal conductivity is set to zero.
k=(0 if Ti>154 K Tj<300 K;
ki+kj
2otherwise. (5)
For the isobaric heat capacity fix, the temperature in each cell is checked, and if the
cell’s temperature is in the range of 154 K to 170 K, the heat capacity is set to the maximum
value of 7641 J/kgK.
cp=cmax
pif Ti>154 K Tj<170 K;
ANN value otherwise. (6)
0123
t(s)
0.004
0.003
0.002
0.001
0.000
x(m)
n= 10
n= 800
0123
t(s)
0.000
0.001
0.002
0.003
0.004
x(m)
0123
t(s)
0.000
0.002
0.004
0.006
x(m)
0123
t(s)
0.004
0.003
0.002
0.001
0.000
x(m)
n= 10
n= 800
0123
t(s)
0.000
0.001
0.002
0.003
0.004
x(m)
0123
t(s)
0.000
0.002
0.004
0.006
x(m)
0123
t(s)
0.004
0.003
0.002
0.001
0.000
x(m)
n= 10
n= 800
0123
t(s)
0.000
0.001
0.002
0.003
0.004
x(m)
0123
t(s)
0.000
0.002
0.004
0.006
x(m)
Figure 18.
Plot of the movement of the temperature curves. The left column of plots tracks the
x
position of a certain lower temperature threshold, from top to bottom the temperature thresholds
were
T={
100.1, 101, 120
}
. The middle column of plots tracks the
x
position where a certain high
temperature threshold, from top to bottom the temperature thresholds were
T={
299.9, 299, 280
}
.
The far right column is then the difference between the lower and higher temperature thresholds.
The plots were scaled using the initial value so that the plots start at the 0.
Aerospace 2022,9, 643 17 of 21
Figure 19 are plots of the temperature along the domain at different time steps. These
plots show that the cases on the coarse mesh with the
cp
fix and the case with the
k
in
Equation (3), the heat moves at a slower rate than the coarse mesh with no property fix,
but the heat is moving faster than in the case using the fine mesh. The cases with the other
kfixes move slower than both the coarse mesh and the fine mesh.
(
a
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
n= 10
n= 800
n= 10 (k fix)
n= 10 (cp fix)
n= 10 (k/2 fix)
n= 10 (k=0 fix)
(
b
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
c
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
d
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
e
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
(
f
)
0.003 0.004 0.005 0.006 0.007 0.008
x(m)
100
150
200
250
300
T(K)
Figure 19.
Plot of the temperature along the domain of the mesh at different time steps, (
a
f
) corre-
sponding to
t={
0, 0.25, 0.5, 1, 2, 3
}
s. The red line is a simulation using ANN fluid property models
on a coarse mesh of
n=
10 elements, the light blue line is a case on a coarse mesh of
n=
10 elements
using ANN fluid property models but the isobaric heat capacity fix was applied, the purple line is a
case on a coarse mesh of
n=
10 elements but the thermal conductivity fix shown in Equation (3),
the blue line is a case on a coarse mesh of
n=
10 elements but the thermal conductivity fix shown
in Equation (4), the dark blue line is a case on a coarse mesh of
n=
10 elements but the thermal
conductivity fix shown in Equation (5) and the black line is a case on a fine mesh of
n=
800 elements
using ANN fluid property models.
Again to better quantify and analyze the movement of the heat in the flow, the position
of a high and low temperature threshold is tracked. As time increases, this high temperature
threshold will move, and its
x
position will increase, and the low temperature threshold
x
position will decrease. Figure 20 shows the results of this analysis and compares the
difference of the results between a coarse mesh of 10 elements, a fine mesh of 800 elements,
a coarse mesh of 10 elements with three different fixes to the thermal conductivity
k
, and a
coarse mesh of 10 elements with a fix to the isobaric heat capacity. For the movement of
the low temperature threshold, the case on the coarse mesh with no property correction
and the case with the
cp
fix were very similar. Furthermore, the movement for the low
temperature threshold shows that the case with the kfix using Equation (3) moves slower
than the coarse mesh case with no correction but faster than the fine mesh, but the cases
using the other two fixes to the thermal conductivity, the heat moves faster than the coarse
and fine mesh. The movement of the high temperature threshold shows the same thing
as the low temperature threshold. The difference between the low and high temperature
threshold movement shows that the heat transfer is slower for the case with the
cp
fix and
the case with the
k
fix using Equation (3) than the coarse mesh, but the heat transfer is
faster for these cases than the case using the fine mesh. Furthermore, the plots show that
the cases with the other two
k
fixes, the heat moves slower than both the coarse mesh and
the fine mesh. The difference between the low and high temperature threshold movement
also shows that the heat transfer for the cases with the
k
fixes are slower than the case with
the cpfix.
Aerospace 2022,9, 643 18 of 21
0123
t(s)
0.004
0.003
0.002
0.001
0.000
x(m)
n= 10
n= 800
n= 10 (cp fix)
n= 10 (k fix)
n= 10 (k/2 fix)
n= 10 (k=0 fix)
0123
t(s)
0.000
0.001
0.002
0.003
0.004
x(m)
0123
t(s)
0.000
0.002
0.004
0.006
x(m)
0123
t(s)
0.004
0.003
0.002
0.001
0.000
x(m)
n= 10
n= 800
n= 10 (cp fix)
n= 10 (k fix)
n= 10 (k/2 fix)
n= 10 (k=0 fix)
0123
t(s)
0.000
0.001
0.002
0.003
0.004
x(m)
0123
t(s)
0.000
0.002
0.004
0.006
x(m)
0123
t(s)
0.004
0.003
0.002
0.001
0.000
x(m)
n= 10
n= 800
n= 10 (cp fix)
n= 10 (k fix)
n= 10 (k/2 fix)
n= 10 (k=0 fix)
0123
t(s)
0.000
0.001
0.002
0.003
0.004
x(m)
0123
t(s)
0.000
0.002
0.004
0.006
x(m)
Figure 20.
Plot of the movement of the temperature curves now including the results where fixes for
the thermal conductivity and isobaric heat capacity were used. The left column of plots tracks the x
position of a certain lower temperature threshold, from top to bottom the temperature thresholds
were
T={
100.1, 101, 120
}
. The middle column of plots tracks the
x
position where a certain high
temperature threshold, from top to bottom the temperature thresholds were
T={
299.9, 299, 280
}
.
The far right column is then the difference between the lower and higher temperature thresholds.
The plots were scaled using the initial value so that the plots start at the 0.
4. Conclusions
The present analysis showed that property representation of near-critical fluids (e.g.,
heat capacity, thermal conductivity, viscosity) in a solver is strongly affected and degraded
by the highly nonlinear and non-monotonous fluid behavior. In particular, the estimation
of inter-cell properties, such as a mean thermal conductivity, may yield values that exceed
physical values by multiples.
For heat conduction, the relevant properties (heat capacity, thermal conductivity) yield
errors that always lead to an overestimation of heat conduction predictions. This may have
an impact on predictions of regenerative cooling, wall heat flux for film cooling, or liquid
core length determinations for injection simulations.
Effectively, this behavior introduces a new—thermodynamic—mesh resolution re-
quirement in addition to classical constraints such as
y+
or Kolmogorov length scales.
Two different methods of estimation (Monte Carlo, 1D CFD) show that a
O(
800
)
point
resolution is required to accurately capture the extremal fluid properties in supercritical
oxygen for the temperature interval 100-300 K. Consider the LOX post shown in Figure 9:
for cold flow mixing of gaseous CH4 with LOX, 800 cells over a
O(
0.5 mm) width is
prohibitive; for reactive flow, the temperature range behind the LOX post is much higher
(from cryogenic temperatures, to the adiabatic flame temperature exceeding 3500 K, back to
cryogenic temperatures), imposing even higher resolution requirements. Thus, capturing
these properties is prohibitive and a model is required.
Finally, our results suggest that previously shown marginal impact of the equation
of state may just be a numerical artifact when the solver does not resolve the difference
between EOS.
Aerospace 2022,9, 643 19 of 21
Author Contributions:
Conceptualization, D.T.B. and N.P.L.; methodology, D.T.B.; software, N.P.L.
and D.T.B.; validation, N.P.L.; resources, D.T.B.; data curation, N.P.L.; writing—original draft prepara-
tion, N.P.L. and D.T.B.; writing—review and editing, D.T.B.; visualization, N.P.L.; supervision, D.T.B.;
project administration, D.T.B.; funding acquisition, D.T.B. All authors have read and agreed to the
published version of the manuscript.
Funding:
We gratefully acknowledge the support of NASA MSFC through the Cooperative Agree-
ment 80NSSC20M0256, and of AFRL/RBK through DEPSCoR FA9550-22-1-0306, which funded
this research.
Data Availability Statement:
The reference data used is publicly available through NIST [
38
] at
https://webbook.nist.gov/chemistry/fluid/ (accessed on 30 August 2022) and CoolProps [
34
] at http:
//www.coolprop.org/coolprop/HighLevelAPI.html#propssi-function (accessed on 30
August 2022
).
Acknowledgments:
We gratefully acknowledge the support of NASA MSFC through the Cooperative
Agreement 80NSSC20M0256 which funded this research.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design
of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or
in the decision to publish the results.
Abbreviations
The following abbreviations are used in this manuscript:
CFD Computational Fluid Dynamics
EOS Equation Of State
HEOS Helmholtz EOS
SRK Soave Redlich Kwong EOS
PR Peng Robinson EOS
References
1. Villermaux, E. Mixing and spray formation in coaxial jets. J. Propuls. Power 1998,14, 807–817.
2. Yang, V.; Anderson, W. (Eds.) Liquid Rocket Engine Combustion Instability; AIAA: Washington, DC, 1995.
3.
Candel, S.; Herding, G.; Synder, R.; Scouflaire, P.; Rolon, C.; Vingert, L.; Habiballah, M.; Grisch, F.; Péalat, M.; Bouchardy, P.; et al.
Experimental Investigation of Shear Coaxial Cryogenic Jet Flames. J. Propuls. Power 1998,14, 826–834.
4.
Delplanque, J.P.; Sirignano, W. Numerical study of the transient vaporization of an oxygen droplet at sub- and super-critical
conditions. Int. J. Heat Mass Transf. 1993,36, 303–314.
5.
Yang, V.; Lin, N.; Shuen, J. Vaporization of Liquid Oxygen (LOX) Droplets in Supercritical Hydrogen Environments. Combust. Sci.
Technol. 1994,97, 247–270.
6. Sirignano, W.; Delplanque, J.P. Transcritical vaporization of liquid fuels and propellants. J. Propuls. Power 1999,15, 896–902.
7.
Mayer, W.; Tamura, H. Propellant Injection in a Liquid Oxygen/Gaseous Hydrogen Rocket Engine. J. Propul. Power
1996
,
12, 1137–1147.
8.
Mayer, W.; Ivancic, B.; Schik, A.; Hornung, U. Propellant Atomization and Ignition Phenomena in Liquid Oxygen/Gaseous
Hydrogen Rocket Combustors. J. Propuls. Power 2001,17, 794–799.
9.
Oschwald, M.; Schik, A. Supercritical nitrogen free jet investigated by spontaneous Raman scattering. Exp. Fluids
1999
,
27, 497–506.
10.
Oschwald, M.; Smith, J.J.; Branam, R.; Hussong, J.; Schik, A.; Chehroudi, B.; Talley, D. Injection of Fluids into Supercritical
Environments. Combust. Sci. Technol. 2006,178, 49–100.
11.
Habiballah, M.; Orain, M.; Grisch, F.; Vingert, L.; Gicquel, P. Experimental studies of high-pressure cryogenic flames on the
mascotte facility. Combust. Sci. Technol. 2006,178, 101–128.
12.
Candel, S.; Juniper, M.; Singla, G.; Scouflaire, P.; Rolon, C. Structures and dynamics of cryogenic flames at supercritical pressure.
Combust. Sci. Technol. 2006,178, 161–192.
13.
Chehroudi, B.; Talley, D.; Coy, E. Initial growth rate and visual characteristics of a round jet into sub- and supercritical environment
of relevance to rocket, gas turbine, and Diesel engines. In Proceedings of the 37th Aerospace Sciences Meeting and Exhibit, Reno,
NV, USA, 11 January–14 January 1999.
14.
Chehroudi, B.; Talley, D.; Coy, E. Visual characteristics and initial growth rates of round cryogenic jets at subcritical and
supercritical pressures. Phys. Fluids 2002,14, 850–861.
15.
Davis, D.; Chehroudi, B. The effects of pressure and acoustic field on a cryogenic coaxial jet. In Proceedings of the 42nd Aerospace
Sciences Meeting and Exhibit, Reno, NV, USA, 5 January–8 January 2004.
Aerospace 2022,9, 643 20 of 21
16.
Davis, D.; Chehroudi, B. Measurements in an acoustically driven coaxial jet under sub-, near-, and supercritical conditions. J.
Propuls. Power 2007,23, 364–374.
17.
Banuti, D.T. Thermodynamic Analysis and Numerical Modeling of Supercritical Injection. Ph.D. Thesis, University of Stuttgart,
Stuttgart, Germany, 2015.
18.
Habiballah, M.; Zurbach, S. Test case RCM-3—Mascotte single injector 60 bar. In Proceedings of the 2nd International
Workshop Rocket Combustion Modeling—Atomization, Combustion and Heat Transfer, DLR, Lampoldshausen, Germany, 25–27
March 2001.
19.
Vingert, L.; Nicole, A.; Habiballah, M. Test Case RCM-2, Mascotte single injector. In Proceedings of the 3rd International
Workshop on Rocket Combustion Modeling, Vernon, France, 13–15 March 2006.
20.
Oefelein, J.C.; Dahms, R.N.; Lacaze, G.; Manin, J.L.; Pickett, L.M. Effects of pressure on fundamental physics of fuel injection in
Diesel engines. In Proceedings of the ICLASS, Heidelberg, Germany, 2–6 September 2012.
21.
Banuti, D.T. A thermodynamic look at injection in aerospace propulsion systems. In Proceedings of the AIAA Scitech 2020
Forum, Orlando, FL, USA, 6–10 January 2020.
22.
Banuti, D.T.; Hannemann, V.; Hannemann, K.; Weigand, B. An efficient multi-fluid-mixing model for real gas reacting flows in
liquid propellant rocket engines. Combust. Flame 2016,168, 98–112.
23.
Oefelein, J.C.; Yang, V. Modeling High-Pressure Mixing and Combustion Processes in Liquid Rocket Engines. J. Propul. Power
1998,14, 843–857.
24.
Ely, J.F.; Hanley, H. Prediction of transport properties. 1. Viscosity of fluids and mixtures. Ind. Eng. Chem. Res.
1981
,20, 323–332.
25.
Ely, J.F.; Hanley, H. Prediction of transport properties. 2. Thermal conductivity of pure fluids and mixtures. Ind. Eng. Chem. Res.
1983,22, 90–97.
26.
Benedict, M.; Webb, G.; Rubin, L. An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their
Mixtures: I. Methane, Ethane, Propane, and n-Butane. J. Chem. Phys. 1940,8, 334–345.
27. Peng, D.Y.; Robinson, D.B. A new two-constant equation of state. Ind. Eng. Chem. Res. 1976,15, 59–64.
28. Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972,27, 1197–1203.
29. Banuti, D.T. Crossing the Widom-line—Supercritical pseudo-boiling. J. Supercrit. Fluids 2015,98, 12–16.
30. Banuti, D.T. The latent heat of supercritical fluids. Period. Polytech. Chem. Eng. 2019,63, 270–275.
31.
Banuti, D.; Raju, M.; Ihme, M. Between supercritical liquids and gases–reconciling dynamic and thermodynamic state transitions.
J. Supercrit. Fluids 2020,165, 104895. https://doi.org/10.1016/j.supflu.2020.104895.
32.
Banuti, D.T.; Hannemann, K. The absence of a dense potential core in supercritical injection: A thermal break-up mechanism.
Phys. Fluids. 2016,28, 035103.
33.
Longmire, N.; Banuti, D. Onset of heat transfer deterioration caused by pseudo-boiling in CO
2
laminar boundary layers. Int. J.
Heat Mass Transf. 2022,193, 122957. https://doi.org/10.1016/j.ijheatmasstransfer.2022.122957.
34.
Bell, I.H.; Wronski, J.; Quoilin, S.; Lemort, V. Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-
Source Thermophysical Property Library CoolProp. Ind. Eng. Chem. Res.
2014
,53, 2498–2508. https://doi.org/10.1021/ie4033999.
35. Ghosal, S. Analysis of Discretization Errors in LES; Center for Turbulence Research Annual Research Briefs, California, 1995.
36.
Chow, F.K.; Moin, P. A further study of numerical errors in large-eddy simulations. J. Comput. Phys.
2003
,184, 366–380.
https://doi.org/10.1016/S0021-9991(02)00020-7.
37.
Lacaze, G.; Oefelein, J.C. A non-premixed combustion model based on flame structure analysis at supercritical pressures. Combust.
Flame 2012,159, 2087–2103.
38.
Linstrom, P.J.; Mallard, W.G. NIST Chemistry Webbook, NIST Standard Reference Database Number 69; National Institute of
Standards and Technology: Gaithersburg, MD, USA, 2001. Available online: http://webbook.nist.gov/chemistry (accessed on 1
March 2021).
39.
Prausnitz, J.M.; Lichtenthaler, R.N.; de Azevedo, E.G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall:
Hoboken, NJ, USA, 1985.
40.
van der Waals, J. Over de Continuiteit van den Gas- en Vloeistoftoestand. Ph.D. Thesis, University of Leiden, Leiden,
The Netherlands, 1873.
41.
Banuti, D.T. A critical assessment of adaptive tabulation for fluid properties using neural networks. In Proceedings of the AIAA
Aerospace Sciences Meeting, Virtual Event, 11–15, 19–21 January 2021.
42. Harstad, K.G.; Miller, R.S.; Bellan, J. Efficient high-pressure state equations. AIChE J. 1997,43, 1605–1610.
43. Reid, R.C.; Prausnitz, J.M.; Poling, B.E. The Properties of Gases and Liquids, 4th ed.; McGraw Hill: New York, NY, USA, 1987.
44.
Pini, M.; Vitale, S.; Colonna, P.; Gori, G.; Guardone, A.; Economon, T.; Alonso, J.; Palacios, F. SU2: The Open-Source Software for
Non-ideal Compressible Flows. J. Phys. Conf. Ser. 2017,821, 012013. https://doi.org/10.1088/1742-6596/821/1/012013.
45.
Economon, T.D. Simulation and Adjoint-Based Design for Variable Density Incompressible Flows with Heat Transfer. AIAA J.
2020,58, 757–769. https://doi.org/10.2514/1.J058222.
46.
Longmire, N.P.; Banuti, D. Extension of SU2 using neural networks for thermo-fluids modeling. In Proceedings of the AIAA
Propulsion and Energy 2021 Forum, Virtual Event, 9–11 August 2021. https://doi.org/10.2514/6.2021-3593.
47.
Longmire, N.; Banuti, D.T. Modeling of the supercritical boiling curve by forced convection for supercritical fluids in relation to
regenerative cooling. In Proceedings of the AIAA Aerospace Sciences Meeting, Virtual Event, 11–15, 19–21 January 2021.
Aerospace 2022,9, 643 21 of 21
48.
Schmidt, E.; Eckert, E.; Grigull, E. Jahrbuch der Deutschen Luftfahrtforschung Bd. II, AAF Translation Nr. 527, Air Material Command;
Wright Field: Dayton, OH, USA, 1939; p. 53158.
49.
Schmidt, E. Wärmetransport durch natürliche Konvektion in Stoffen bei kritischem Zustand. Int. J. Heat Mass Transf.
1960
,
1, 92–101.
50.
Banuti, D.T.; Ma, P.C.; Hickey, J.P.; Ihme, M. Thermodynamic structure of supercritical LOX–GH2 diffusion flames. Combust.
Flame 2018,196, 364–376. https://doi.org/10.1016/j.combustflame.2018.06.016.
... While the volume translation methods [74,75] have been shown to increase the accuracy of PR estimations significantly, the associated complexity in obtaining analytical solutions of departure functions for thermodynamic properties has limited its applicability in CFD [22]. Both PR and SRK have also been found to significantly underestimate the specific heat capacity peaks at supercritical pressures [76]. Other non cubic, empirical EoS such as Younglove's [28,29] Modified Benedict Webb Rubin (MBWR) EoS have also been used but on a more limited basis (see for example [77]) due to the associated complexity. ...
Article
Full-text available
Computational Fluid Dynamics (CFD) frameworks of supercritical cryogenic fluids need to employ Real Fluid models such as cubic Equations of State (EoS) to account for thermal and inertial driven mechanisms of fluid evolution and disintegration. Accurate estimation of the non-linear variation in density, thermodynamic and transport properties is required to computationally replicate the relevant thermo and fluid dynamics involved. This article reviews the availability, performance and the implementation of common Real Fluid EoS and data-based models in CFD studies of supercritical cryogenic fluids. A systematic analysis of supercritical cryogenic fluid (N2, O2 and CH4) thermophysical property predictions by cubic (PR and SRK) and non-cubic (SBWR) Real Fluid EoS, along with Chung’s model, reveal that: (a) SRK EoS is much more accurate than PR at low temperatures of liquid phase, whereas PR is more accurate at the pseudoboiling region and (b) SBWR EoS is more accurate than PR and SRK despite requiring the same input parameters; however, it is limited by the complexity in thermodynamic property estimation. Alternative data-based models, such as tabulation and polynomial methods, have also been shown to be reliably employed in CFD. At the end, a brief discussion on the thermophysical modelling of cryogenic fluids affected by quantum effects is included, in which the unsuitability of the common real fluid EoS models for the liquid phase of such fluids is presented.
... 29,30 Recently, also artificial neural networks, trained on tabulated data, have been employed for thermodynamic modeling for real gas CFD simulations. 31, 32 However, cubic EoS are still mostly used due to their simplicity 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 conductivity. ...
Article
Full-text available
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.