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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

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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 [4–6].

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 ZH≈1×10−3, 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)

10−410−310−210−1100

Mixture Fraction

0

5

10

15

20

cp(kJ/kgK)

(b)

10−410−310−210−1100

Mixture Fraction

0

2

4

6

8

Prandtl number

6 MPa

7 MPa

8 MPa

9 MPa

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

10−1

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

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