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Numerical framework for transcritical real-ﬂuid

reacting ﬂow simulations using the ﬂamelet progress

variable approach

Peter C. Ma∗

, Daniel T. Banuti†

, and Matthias Ihme‡

Stanford University, Stanford, CA 94305, USA

Jean-Pierre Hickey§

University of Waterloo, Waterloo, ON N2L 3G1, Canada

An extension to the classical FPV model is developed for transcritical real-ﬂuid com-

bustion simulations in the context of ﬁnite volume, fully compressible, explicit solvers. A

double-ﬂux model is developed for transcritical ﬂows to eliminate the spurious pressure

oscillations. A hybrid scheme with entropy-stable ﬂux correction is formulated to robustly

represent large density ratios. The thermodynamics for ideal-gas values is modeled by a

linearized speciﬁc heat ratio model. Parameters needed for the cubic EoS are pre-tabulated

for the evaluation of departure functions and a quadratic expression is used to recover the

attraction parameter. The novelty of the proposed approach lies in the ability to account

for pressure and temperature variations from the baseline table. Cryogenic LOX/GH2

mixing and reacting cases are performed to demonstrate the capability of the proposed

approach in multidimensional simulations. The proposed combustion model and numerical

schemes are directly applicable for LES simulations of real applications under transcritical

conditions.

I. Introduction

Liquid rocket engines (LRE) are one of the many practical applications which operate near or above

the critical point of the working ﬂuid. The large expansion ratio needed to generate the required thrust

means that the combustion chamber operates at extremely high pressures, typically between 30 and 200

bar. In liquid rocket engines, the high mass ﬂow rate is achieved by injecting high energy density cryogenic

fuel and oxidizer into the combustion chamber through an extensive array of coaxial or impinging jets,

see Cheroudi1for a review of high-pressure injection strategies. In the most common coaxial setup, the

oxidizer generally exits the lip of the injectors below the critical temperature while being above the critical

pressure of the ﬂuid. In order to initiate chemical reactions for combustion, the cryogenic ﬂuid must undergo a

transcritical phase change to a supercritical state. In the transcritical regime, the thermo-physical properties

of the ﬂuid undergoes drastic changes for minute perturbations of the baseline thermodynamic state. This

highly non-linear ﬂow behavior requires the use of a generalized state equation to account for the complex

thermodynamic properties.

A physics-based understanding of real ﬂuid eﬀects is needed to fully address the modeling challenges

for trans- and supercritical combustion ﬂow. Early experiments sought to address the physics of a sub-

to supercritical jet phase change. The transcritical eﬀects are most clearly observed in the visualization of

transcritical pure mixing.2–5At subcritical pressures, a liquid jet undergoes a convection driven process of

atomization and breakup. At supercritical pressure, the same liquid jet undergoes a diﬀusion driven mixing

—primarily due to the lack of surface tension, increased diﬀusivity and reduction in evaporation enthalpy.

∗Graduate Research Assistant, Department of Mechanical Engineering.

†Postdoctoral Research Fellow, Center for Turbulence Research.

‡Assistant Professor, Department of Mechanical Engineering.

§Assistant Professor, Department of Mechanical and Mechatronics Engineering.

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55th AIAA Aerospace Sciences Meeting

9 - 13 January 2017, Grapevine, Texas

AIAA 2017-0143

Copyright © 2017 by Peter C. Ma, Daniel T. Banuti, Matthias Ihme, Jean-Pierre Hickey. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

AIAA SciTech Forum

In the diﬀuse region between the liquid core and the gaseous outer ﬂow, large thermo-physical gradients

are observed and the dense core length is signiﬁcantly reduced under supercritical pressures.6Under these

conditions, a dense gas/gas mixing of the propellants and oxidizers is typically observed. The delineation

between atomization and diﬀusion driven mixing of multi-species mixtures has been characterized by Dahms

and Oefelein.7

In high pressure rocket combustion chambers, the turbulence time scale has a magnitude of 1µs in the

reactive shear layer while the length scale is 1µm near the exit lip.8The ﬂame thickness, for non-premixed

combustion, decreases with the inverse of the square root of pressure and strain rate.9In a follow-up study,

Lacaze and Oefelein10 suggested that the ﬂame thickness is proportional to the inverse of the square root of

pressure. The basis of the ﬂamelet formulation11 rests on a scale separation between the turbulence and the

chemical scales in both time and space. Ivancic and Mayer8inferred that the turbulent and chemical time

scales could be of similar magnitude —a result that could invalidate the laminar ﬂamelet assumption. To

address this issue, Zong et al.12 showed the appropriateness of the ﬂamelet assumptions in coaxial injectors

in the supercritical regime using scaling arguments. The validity of the ﬂamelet approach has lead to a

number of ﬂamelet-based numerically studied.9,10,13–16

Pre-tabulated ﬂamelet approaches have been successfully applied to a variety of combustion cases.11,17

The original look-up tables based on mixture fraction, e

Z, and its variance, g

Z002, tend to be inadequate

for describing topologically complex ﬂames. This is particularly true for lifted ﬂames where the mixture

fraction alone is insuﬃcient to model the full physical complexity.18,19 Careful experiments in high-pressure

combustors have revealed a multiplicity of ﬂame anchoring and stabilization mechanisms,20,21 sometimes with

a lifted ﬂame conﬁguration. In order to capture the intrinsic physics of these ﬂames using a computationally

tractable combustion modeling approach, a progress variable needs to be transported in addition to the

mixture fraction. The ﬂamelet progress variable (FPV) approach22,23 has been shown to better capture

the complex physics in detached ﬂames. The FPV approach has been used for supercritical simulations by

Cutrone et al.13 and more recently by Giorgi et al.24 in the context of Reynolds averaged simulations.

The extension of the FPV model for trans- and supercritical ﬂows in the fully compressible context with

large-eddy simulation (LES) remains, for the most part, unreported, speciﬁcally with regards to the pressure

and temperature coupling.

The use of LES for trans- and supercritical ﬂows has been initially explored using one-step chemistry25

and more recently extended for hydrogen combustion by Schmitt et al.26 using a thickened ﬂame approach.

Other groups have proposed an extension to the linear eddy model to account for combustion in high-

pressure systems.27 Further considerations have been proposed to account for subgrid-scale modeling under

supercritical conditions,28,29 sensitivity of the state equation,30 numerical stability issues,31–37 non-reﬂecting

boundary conditions,38,39 and compressibility eﬀects.40 While direct numerical simulations (DNS) have

been successfully applied to trans- and supercritical ﬂows in idealized conﬁgurations,41–44 the computational

limitations restrict the applications to academic problems.

In this work, we propose an extension to the classical FPV approach22,23 for trans- and supercritical

combustion simulations in the context of ﬁnite volume, fully compressible, explicit solvers. The novelty of

the present work lies in the ability to account for pressure and temperature variations from the baseline tab-

ulated values using a computationally tractable pre-tabulated combustion chemistry in a thermodynamically

consistent fashion. In addition, we show that the solution of the laminar ﬂamelets in mixture fraction space

and the chemistry tabulation requires special considerations in order to fully model the non-linear eﬀects in

transcritical ﬂows.

II. Thermodynamics

A. Equation of state

Cubic equations of state oﬀer an acceptable compromise between the conﬂicting requirements of accuracy and

computational tractability.45 The Peng-Robinson (PR) cubic EoS46 is used in this study for the evaluation

of thermodynamic quantities, which can be written as

p=RT

v−b−a

v2+ 2bv −b2(1)

where pis the pressure, Ris the gas constant, Tis the temperature, vis the speciﬁc volume, and the

attraction parameter aand eﬀective molecular volume bare dependent on temperature and composition to

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account for eﬀects of intermolecular forces. For mixtures, the parameters aand bare evaluated as47

a=

NS

X

α=1

NS

X

β=1

XαXβaαβ ,(2a)

b=

NS

X

α=1

Xαbα,(2b)

where Xαis the mole fraction of species α. Extended corresponding states principle and single-ﬂuid as-

sumption for mixtures are adopted.48,49 The parameters aαβ and bαare evaluated using the recommended

mixing rules by Harstad et al.:50

aαβ = 0.457236(RTc,αβ )2

pc,αβ 1 + cαβ 1−sT

Tc,αβ !!2

,(3a)

bα= 0.077796RTc,α

pc,α

,(3b)

cαβ = 0.37464 + 1.54226ωαβ −0.26992ω2

αβ ,(3c)

where Tc,α and pc,α are the critical temperature and pressure of species α, respectively. The critical mix-

ture conditions for temperature Tc,αβ , pressure pc,αβ , and acentric factor ωc,αβ are determined using the

corresponding state principles.47

B. Thermodynamic properties

Thermodynamic quantities in this study are evaluated consistently with respect to (w.r.t.) the EoS used

and no linearization is introduced.

1. Partial derivatives

Partial derivatives and thermodynamic quantities based on the PR EoS that are useful for evaluating other

thermodynamic variables are given as

∂p

∂T v ,Xi

=R

v−b−(∂a/∂ T )Xi

v2+ 2bv −b2,(4a)

∂p

∂v T ,Xi

=−RT

(v−b)2

1−2a"RT (v+b)v2+ 2bv −b2

v2−b22#−1

,(4b)

∂a

∂T Xi

=−1

T

NS

X

α=1

NS

X

β=1

XαXβaαβ Gαβ ,(4c)

∂2a

∂T 2Xi

= 0.457236R2

2T

NS

X

α=1

NS

X

β=1

XαXβcαβ (1 + cαβ )Tc,αβ

pc,αβ rTc,αβ

T,(4d)

Gαβ =

cαβ qT

Tc,αβ

1 + cαβ 1−qT

Tc,αβ ,(4e)

K1=Zv

+∞

1

v2+ 2bv −b2dv =1

√8bln v+ (1 −√2)b

v+ (1 + √2)b!.(4f)

2. Internal energy and enthalpy

For general real ﬂuids, thermodynamic quantities are typically evaluated from the ideal-gas value plus a

departure function that accounts for the deviation from the ideal-gas behavior. The ideal-gas enthalpy,

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entropy and speciﬁc heat can be evaluated from the commonly used NASA polynomials which have a

reference temperature of 298 K. The simple mixture-averaged mixing rule is used for ideal-gas mixtures.

The speciﬁc internal energy can be written as,

e(T, ρ, Xi) = eig(T , Xi) + Zρ

0"p−T∂p

∂T ρ,Xi#dρ

ρ2,(5)

where superscript “ig” indicates the ideal-gas value of the thermodynamic quantity, and Eq. (5) can be

integrated analytically for PR EoS:

e=eig +K1"a−T∂a

∂T Xi#,(6)

in which K1is computed through Eq. (4f). The speciﬁc enthalpy can be evaluated from the thermodynamic

relation h=e+pv, and we have

h=hig −RT +K1"a−T∂a

∂T Xi#+pv . (7)

Partial enthalpies for each species, hk, can be evaluated using the partial derivatives w.r.t. mole fractions.

The procedures for evaluating partial enthalpy for real ﬂuids are similar to those in Meng et al.51 and

therefore omitted here.

3. Speciﬁc heat capacity

The speciﬁc heat capacity at constant volume is evaluated as

cv=∂e

∂T v ,Xi

=cig

v−K1T∂2a

∂T 2Xi

,(8)

and the speciﬁc heat capacity at constant pressure is evaluated as

cp=∂h

∂T p,Xi

=cig

p−R−K1T∂2a

∂T 2Xi−T(∂ p/∂T )2

v,Xi

(∂p/∂v)T ,Xi

.(9)

4. Speed of sound

The speed of sound for general real ﬂuids can be evaluated as

c2=∂p

∂ρ s,Xi

=γ

ρκT

,(10)

where γis the speciﬁc heat ratio and κTis the isothermal compressibility, which is deﬁned as

κT=−1

v∂v

∂p T ,Xi

.(11)

C. Transport properties

The dynamic viscosity and thermal conductivity are evaluated using Chung’s high-pressure method.52,53

This method is known to produce oscillations in viscosity for multi-species mixtures that consist of species

with both positive and negative acentric factors.54,55 To solve this problem, a mass-fraction averaged or

mole-fraction averaged viscosity evaluated based on viscosity of each individual species can be used. In this

study, the negative acentric factor is set to zero only when evaluating the viscosity so that the anomalies in

viscosity can be removed. This approach has similar behavior to the mole-fraction averaged approach via

numerical tests.

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III. Transcritical Flamelets

The large Damk¨ohler number of the supercritical combustion12 supports the use of laminar ﬂamelet-

based combustion models. The basic assumption of the ﬂamelet model rests on the fact that the reaction

zone remains laminar and the diﬀusive transport is only important in the direction normal to the ﬂame.

By recasting the governing equations as a one-dimensional similarity solution in mixture fraction space, the

steady ﬂamelet equations for the species and temperature under unity Lewis number assumption can be

written as:

−ρχ

2

∂2Yk

∂Z 2= ˙ωk,(12a)

−ρχ

2

∂2T

∂Z 2=ρχ

2cp

∂cp

∂Z

∂T

∂Z +˙ωT

cp

,(12b)

where Ykis the mass fraction of species k,Zis the mixture fraction, ˙ωkis the reaction rate of species k, ˙ωTis

the heat release term, and χ= 2D|∇Z|2is the scalar dissipation rate where Dis the diﬀusion coeﬃcient. No

further assumptions are needed to the ﬂamelet equations for a generalized equation of state (for a constant

pressure combustion) apart from the modiﬁed thermodynamic relationship between temperature and density.

For a given proﬁle of the scalar dissipation (which accounts for the convective and diﬀusive terms normal

to the ﬂame front), the solution of the above equations represents the composition and temperature proﬁles

within the counterﬂow-diﬀusion ﬂame. In the low-Mach and low-pressure formulation, Peters56 derived a

scalar dissipation rate proﬁle for a shear layer under the assumption of a unitary Chapman-Rubesin parameter

that relates the local ratio of density and viscosity to its far-ﬁeld values. Although formally insuﬃcient for

many combustion cases, the modeling assumptions of the scalar dissipation rate proﬁle are robust. For

transcritical conditions, we retain the same scalar dissipation rate proﬁle.

The ﬂamelet equations of a counterﬂow-diﬀusion ﬂame, in mixture fraction space, are solved using the

FlameMaster solver.57 The original code was extended to incorporate the PR EoS, along with the ther-

ZH

0 0.2 0.4 0.6 0.8 1

Temperature [K]

0

500

1000

1500

2000

2500

3000

3500

4000

ZH

0 0.2 0.4 0.6 0.8 1

Mass fractions

0

0.2

0.4

0.6

0.8

1

O2 H2

H2O

ZH

10-5 10-4 10-3 10-2 10 -1 100

Density [kg/m3]

0

200

400

600

800

1000

1200

ZH

10-5 10-4 10-3 10-2 10 -1 100

Specific heat [kJ/(kg·K)]

0

5

10

15

Figure 1. Comparisons of temperature, mass fractions, density and speciﬁc heat between ﬂamelet (lines) and

DNS10 (symbols) results for a counterﬂow diﬀusion ﬂame with H2at 295 K and O2at 120 K at a pressure of

7.0 MPa.

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modynamically consistent departure functions and thermo-physical ﬂuid properties. The results of the

one-dimensional problem are compared with the two-dimensional direct numerical simulations performed

by Lacaze and Oefelein.10 In order to avoid inconsistencies in determining the scalar dissipation proﬁle,

only the near-equilibrium ﬂamelet solutions are compared. The setup consists of a counterﬂow diﬀusion

ﬂame with pure H2and O2streams, respectively, at 295 K and 120 K and a pressure of 7.0 MPa. The

strain rate, deﬁned as the velocity diﬀerence between both injectors, is set to 105s−1, which corresponds

to a scalar dissipation rate of 103s−1, following the classical relationship between the strain and the scalar

dissipation rate in laminar ﬂamelets derived by Peters.56 The high-pressure chemical mechanism by Burke

et al.58 is used, which accounts for 8 species and 27 reactions. The comparative results for temperature,

composition, density and speciﬁc heat capacity are shown in Fig. 1. Speciﬁcial attention needs to be paid to

the transcritical regions away from the ﬂame front, especiﬁcally in the oxidizer stream. In this region, large

variations of the thermophysical properties are observed which require an adaptive mesh reﬁnement based

on the thermophysical properties in addition to the usual gradient-based mesh adaptation techniques. With-

out these strong non-linear features, the full complexity of the transcritical behavior cannot be captured.

The grid adaptation is especially important to capture the pseudo-boiling point3,59 (PBP) which contains

a ﬁnite peak in the speciﬁc heat capacity, and this region acts as a proxy to phase change in subcritical

thermodynamics. Note that the location of the PBP region is typically at mixture fraction around 10−3

independent of the value of scalar dissipation rate. This requires special attention during the tabulation

process since a typical resolution in mixture fraction will completely miss the PBP region, and this will be

discussed in details later. As can be seen in Fig. 1, when the thermodynamic features are well resolved, the

low-dimensional ﬂamelet equations in mixture fraction space can capture the essential ﬂame structure.

IV. Numerical Methods

A. Governing equations

The governing equations are the Favre-averaged conservation equations of mass, momentum, total energy,

mixture fraction, mixture fraction variance, and progress variable, written as follows:

∂¯ρ

∂t +∂¯ρeuj

∂xj

= 0 ,(13a)

∂¯ρeui

∂t +¯ρeuieuj

∂xj

=−∂¯p

∂xi

+∂

∂xj(eµ+µt)∂eui

∂xj

+∂euj

∂xi−2

3δij

∂euk

∂xk,(13b)

∂¯ρe

E

∂t +¯ρeuje

E

∂xj

=∂

∂xj" f

λ

cp

+µt

Prt!∂e

h

∂xj−euj¯p+eui(¯τij + ¯τR

ij )#

+∂

∂xj"N

X

k=1 ¯ρe

Dk−f

λ

cp!e

hk

∂e

Yk

∂xj#,

(13c)

∂¯ρe

Z

∂t +¯ρeuje

Z

∂xj

=∂

∂xj"¯ρe

D+µt

Sct∂e

Z

∂xj#,(13d)

∂¯ρg

Z002

∂t +¯ρeujg

Z002

∂xj

=∂

∂xj"¯ρe

D+µt

Sct∂g

Z002

∂xj#+ 2 µt

Sct

∂e

Z

∂xj

∂e

Z

∂xj−¯ρeχ , (13e)

∂¯ρe

C

∂t +¯ρeuje

C

∂xj

=∂

∂xj"¯ρe

D+µt

Sct∂e

C

∂xj#+¯

˙ωC,(13f)

where uiis the ith component of the velocity vector, Eis the total energy including the chemical energy,

Cis the progress variable, µand µtare the laminar and turbulent viscosity, λis the thermal conductivity,

Dis the diﬀusion coeﬃcient for the scalars, ˙ωCis the source term for the progress variable, τij and τR

ij

are the viscous and subgrid-scale stresses which are assumed to take the form as the second term on the

right-hand side of Eq. (13b), Prtis the turbulent Prandtl number, and Sctis the turbulent Schmidt number.

An appropriate subgrid-scale model is needed for the computation of the turbulent viscosity µt. Under the

unity Lewis number assumption, the summation on the right-hand side of Eq. (13c) vanishes so that the

species mass fractions are not explicitly required for the energy equation. The system is closed with the PR

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EoS introduced in Section II and the ﬂamelet-based combustion model that will be discussed in the next

subsection. Moreover, the sub-grid terms associated with the EoS are neglected in this study.

B. Combustion model

A Flamelet/Progress Variable (FPV) approach22,23 is adopted in this study, in which the chemistry is pre-

computed and tabulated as a series of laminar ﬂamelet solutions. Flamelets are ﬁrst computed for diﬀerent

values of the scalar dissipation rate at a constant background pressure and speciﬁed constant fuel and air

temperatures, and then the ﬂamelets are parametrized by the mixture fraction and the reaction progress

variable. The resulting ﬂamelet table is used for the determination of the local temperature, species, density,

source term of the progress variable and other thermal-transport quantities needed by the solver. Presumed

PDFs are introduced to account for the turbulence/chemistry interaction. Typically, a β-PDF is used for the

mixture fraction and a δ-PDF for the progress variable, which was shown to be a reasonable approximation

in many studies. Since the reacting region is typically in the ideal gas regime even under transcritical

combustion conditions, the PDF closures are expected to perform similarly as in previous studies for ideal

gas reacting ﬂows.

In the low-Mach number ﬂamelet implementation, the temperature, species, and density are assumed

to depend only on the transported scalars. However, when compressibility eﬀects are taken into account,

an overdetermined thermodynamic state arises from the use of the ﬂamelet table with tabulated thermody-

namics. On the one hand, the full thermodynamic state, at constant pressure, is deﬁned within the ﬂamelet

table. On the other hand, the transport equations contain two thermodynamic variables, namely density

and internal energy, which are also suﬃcient to fully characterize the thermodynamic equilibrium state of

the ﬂuid if the compositions are given. In order to mend the over-determined thermodynamic states, a

strategy was developed by Saghaﬁan et al.60 in the context of ideal gas ﬂows to account for the pressure and

temperature variations arising in supersonic combustion using the FPV approach. The speciﬁc heat ratio

is linearized around temperature to eliminate the costly iterative procedure to determine temperature, and

also to obtain other thermodynamic quantities which are functions of temperature.

The proposed strategy for applying compressibility eﬀects in the FPV approach has been modiﬁed to

work with a generalized equation of state under transcritical conditions in the present work. The underlying

strategy rests on correcting the tabulated values with the transported quantities based on the EoS used.

Speciﬁcally, since PR EoS is used in this study, along with thermodynamic quantities needed for evaluation

of the ideal gas thermodynamic quantities, parameters a,b, and the ﬁrst and second derivatives of the

parameter aw.r.t. temperature are needed for the calculations of the partial derivatives in Eq. (4) that are

needed for the evaluation of the departure functions. The parameter bis a function of species composition only

and is independent of pressure and temperature. Therefore, it can be pre-tabulated within the ﬂamelet table

without any discrepancy in pressure and temperature. However, the parameter a, along with its derivatives,

is a function of both the species composition and the temperature, and thus may not be consistent with

the temperature corresponding to the transported variables. The following procedure is proposed for the

evaluation of the parameter aand its derivatives: the dependence of the parameter aon temperature is

assumed to be a quadratic function as follows,

a=C1e

T2+C2e

T+C3,(14)

where C1,C2, and C3can be determined from tabulated quantities,

C1=1

2∂2a

∂T 20

,(15a)

C2=∂a

∂T 0−2C1T0,(15b)

C3=a0−C1T02−C2T0,(15c)

where subscript 0 indicates the stored baseline quantities in the table. The ﬁrst and second derivatives of

parameter aw.r.t. temperature can be determined accordingly by taking derivatives of Eq. (14), and the

proposed model corresponds to a linear and a constant approximation to the ﬁrst and second derivatives,

respectively. Once the parameter aand its derivatives are obtained, along with the parameter band the gas

constant R, all the partial derivatives which are needed for computing other thermodynamic quantities can

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Temperature [K]

0 200 400 600 800 1000

Parameter a

-100

-50

0

50

100

150

200

Exact

Quadratic model

Linear model

Constant model

Temperature [K]

0 200 400 600 800 1000

First derivative of parameter a

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Exact

Quadratic model

Linear model

Constant model

Figure 2. Illustration of the quadratic, linear and constant model for parameter a. Results for parameter a

(left) and its ﬁrst derivative w.r.t. temperature (right) are shown. Exact solution is for O2at 100 bar. Black

dot indicates the reference point at 300 K.

be evaluated for a given mixture, and therefore, the thermodynamic state is determined. Note that similar

to the quadratic model as in Eq. (14), a linear model or a constant model for the parameter acan also be

constructed based on the stored values in the table. The performance of the quadratic, linear, and constant

model will be examined later.

As an illustration, Fig. 2shows the results of parameter aand its ﬁrst derivative w.r.t. temperature,

along with the approximations evaluated based on the quadratic, linear, and constant model. Pure oxygen

at 100 bar is considered for this example. The reference point is assumed to be at a temperature of 300 K,

as indicated by the black dot in Fig. 2. As can be seen from Fig. 2, the quadratic assumption works well in

the region within 200 K and 100 K from the reference temperature for the parameter aand its derivative,

respectively. The linear model yields a linear proﬁle for aand hence a constant proﬁle for its derivative. The

constant model for the parameter agives zero value for its derivative. Similarly, the behavior of the second

derivative of a, which is also important for the evaluation of real-ﬂuid thermodynamics, can be expected

for the three models considered. The quadratic model for ashows superior performance in predicting aand

its derivatives when temperature is away from the reference value, and its performance in predicting other

thermodynamic quantities will be examined in details to conﬁrm the validity of the proposed approach.

As an example, to calculate the internal energy including the chemical energy from the temperature

based on the proposed approach for PR EoS for a given mixture, i.e. ﬁxed e

Z,g

Z002, and e

C, the ideal gas part

and the departure function are calculated separately,

ee=eeig +eedep ,(16)

where eeig and eedep are the ideal-gas and departure function values of the speciﬁc internal energy. The

ideal-gas value including the chemical energy of the mixture is calculated with linearized speciﬁc heat ratio60

eeig =eeig

0+e

R

aig

γ

ln 1 + aig

γ(e

T−T0)

eγig

0−1!,(17)

where eeig

0,e

R,T0,eγig

0, and aig

γare all stored variables in the table for given e

Z,g

Z002, and e

C, in which aig

γis

the slope of the ideal-gas speciﬁc heat ratio w.r.t. the temperature. The departure function is determined

as

eedep =K1"a−e

T∂a

∂T Xi#,(18)

where Eqs. (14) and (15) are used for the evaluation of parameters needed for the PR EoS. Other thermo-

dynamic quantities, such as the speciﬁc heat and speed of sound, can be evaluated similarly.

To determine primitive variables from conservative variables, a secant method is used to obtain temper-

ature given the transported density and internal energy. With this, the pressure along with other thermody-

namic quantities is evaluated as an explicit function of the density and temperature. Since the parameters

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needed for the real ﬂuid EoS are pre-tabulated and approximated from the table, the computational overhead

of the iteration process is acceptable.

Transport quantities are evaluated based on the method due to Chung et al.52,53 Since the variation

of the viscosity and conductivity on pressure is small under transcritical conditions, a power-law is used to

approximate the temperature dependency as follows,

eµ

eµ0

= e

T

T0!aµ

,(19a)

e

λ

e

λ0

= e

T

T0!aλ

,(19b)

where eµ0and e

λ0are stored in the table along with their corresponding slopes, aµand aλ.

Note that the proposed approach is not limited to the PR EoS. All cubic EoS’s have similar structure

and a similar approach can be used for other types of cubic EoS, such as the Soave-Redlich-Kwong (SRK)

EoS.61

The developed FPV approach focuses on the thermodynamics and no special modiﬁcation is taken for

the chemistry part. The transcritical combustion dynamics were shown to have similar structures as for ideal

gases by several studies.9,15,16 Indeed, in typical engines, combustion takes place at high temperatures away

from the real-ﬂuid region which is characterized by cryogenic temperatures. Moreover, for the applications

considered in this study, combustion can be considered to be at low-Mach conditions where compressibility

eﬀects can be neglected. However, for the inert and equilibrium parts of the ﬂows without chemical source

terms, compressibility may play a critical role for predicting the behaviors of the system of interest, for

example ﬂows in the injectors and through the nozzles for rocket engines. Therefore, a strategy is proposed

here for practical simulations, that the ﬂamelet table can be generated at conditions of the combustion

chamber where combustion takes place, and discrepancies in temperature and pressure in other parts of the

ﬂows can be corrected by the FPV model described above. If supersonic combustion needs to be considered,

methodologies developed by Saghaﬁan et al.60 can be used.

C. Numerical schemes

The massively paralleled, ﬁnite-volume solver, CharLES x, developed at the Center for Turbulence Research,

is used in this study. A control-volume based ﬁnite volume approach is utilized for the discretization of the

system of equations, Eq. (13):

∂U

∂t Vcv +X

f

FeAf=X

f

FvAf+SVcv ,(20)

where Uis the vector of conserved variables, Feis the face-normal Euler ﬂux vector, Fvis the face-normal

viscous ﬂux vector which corresponds to the r.h.s of Eq. (13), Sis the source term vector, Vcv is the volume of

the control volume, and Afis the face area. A strong stability preserving 3rd-order Runge-Kutta (SSP-RK3)

scheme62 is used for time advancement.

The convective ﬂux is discretized using a sensor-based hybrid scheme in which a high-order, non-

dissipative scheme is combined with a low-order, dissipative scheme to minimize the numerical dissipation

introduced. A central scheme which is fourth-order on uniform meshes is used along with a second-order

ENO scheme for the hybrid scheme. A density sensor37,54 is adopted in this study. Due to the large density

gradients across the PBP region under transcritical conditions, an entropy-stable ﬂux correction technique,

developed by Ma et al.,37 is used to ensure the physical realizability of the numerical solutions and to dampen

the non-linear instabilities in the numerical schemes.

Due to the strong non-linearlity inherited in the real ﬂuid EoS, spurious pressure oscillations will be

generated when a fully conservative scheme is used,37,63 and severe oscillations in the pressure ﬁeld could

make the solver diverge which cannot be solved by adding artiﬁcial dissipation. A double-ﬂux method35–37

is extended to the transcritical regime to eliminate the spurious pressure oscillations. The eﬀective speciﬁc

heat ratio based on the speed of sound is frozen both spatially and temporally for a given cell when the

ﬂuxes of its faces are evaluated, which renders a local system as an equivalently ideal-gas system. The two

ﬂuxes at a face evaluated in the double-ﬂux method are not the same, yielding a quasi-conservative scheme

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and the conservative error in total energy was shown to converge to zero with increasing resolution.37 A

Strang-splitting scheme64 is applied in this study to separate the convection operator from the remaining

operators of the system.

V. Results and Discussions

A. Tabulation approach analysis

The validity and performance of the FPV model with the tabulation approach developed in the previous

sections will be assessed in the following through an a priori analysis. To this end, a ﬂamelet table is ﬁrstly

generated at certain reference conditions and then a ﬂamelet solution at diﬀerent reference conditions is

used as a probe for the assessment. In a compressible solver, initial and boundary conditions are typically

speciﬁed through primitive variables (temperature, pressure, velocity, and species). With the FPV approach,

species are determined from mixture fraction and progress variable. The solver then converts primitive

variables to conservative variables and takes time advancement. To be consistent with the compressible

solver, the primitive variables, namely temperature, pressure, mixture fraction and progress variable, from

the probing ﬂamelet solution are given, and thermodynamic parameters, such as gas constant, speciﬁc heat

ratio, parameter aand bfor the PR EoS, are interpolated from the table by ﬁxing mixture fraction and

progress variable. Then the species and thermodynamic quantities of interest are compared to the exact

values from the probing ﬂamelet to assess the performance of the current developed tabulation approach.

An extreme case is considered here to show the capabilities of the current model to recover real-ﬂuid

thermodynamics. The ﬂamelet table is generated at ideal-gas conditions with Tox = 300 K, Tf= 300 K,

p= 60 bar. A transcritical ﬂamelet in equilibrium at relatively low scalar dissipation rate at conditions

Tox = 100 K, Tf= 150 K, p= 100 bar is considered as the probing ﬂamelet. Note that the ﬂamelet table

contains only ideal-gas information and directly reading variables from the table is expected to give signiﬁcant

errors. This situation corresponds to the case when an ideal-gas table is used for transcritical simulations,

or when the table resolution in mixture fraction is insuﬃcient, so that the PBP region is not resolved.

Figure 3shows the results of the a priori analysis. Species and thermodynamic quantities of interest

evaluated from the current FPV model and directly read from the table are compared to the exact values

from the probing ﬂamelet. The current FPV model assumes that the composition is not changed with

perturbations on the reference conditions, or in other words, species are only functions of the mixture

fraction and the progress variable. Thus species from the FPV model are expected to have the same values

as in the ﬂamelet table. Quadratic, linear, and constant models for the parameter aare compared in terms

of real-ﬂuid thermodynamic quantities. In the following, we will go through diﬀerent aspects contained in

this analysis.

The species results, including major species (H2, O2, and H2O) and representative minor species (OH,

O, H, and HO2), are shown in the ﬁrst three subﬁgures in Fig. 3. It can be seen that the major species read

from the table exhibit negligible diﬀerence from the exact values even considering the dramatic diﬀerence

in the pressure. For minor species, small but noticeable discrepancies can be observed, but the inﬂuence

on the mixture properties is expected to be insigniﬁcant due to the low magnitude of their mass fractions.

These results are consistent with the ﬁndings by Saghaﬁan et al.60 where ﬂamelets at diﬀerent operating

conditions are extensively studied for ideal gases at pressures close to ambient conditions. Similar results

are found for the entire S-shaped curve with probing ﬂamelets at diﬀerent scalar dissipation rates which are

not shown here.

Density, speciﬁc heat at constant pressure, and speed of sound are used as examples for comparison of the

thermodynamic quantities in Fig. 3. Since the ﬂamelet table is generated at diﬀerent reference conditions

from the probing ﬂamelet, the quantities directly read from the table show signiﬁcant discrepancies from the

ﬂamelet. The table is at ideal-gas conditions, and therefore, the large density value of LOX and the sharp

change of density proﬁle in the PBP region are completely missed. Similarly, the peak in speciﬁc heat in the

PBP region cannot be read from the table. The behavior in speed of sound is not correct as compared to

the probing ﬂamelet in the real-ﬂuid region as indicated by the shaded area. In the ideal-gas region, which

is indicated by the white area in the last subﬁgures of Fig. 3(deﬁned by 5% deviation of compressibility

factor from unity), although the gas constant is well recovered by the species information, due to the drastic

diﬀerence in temperature and pressure, the table gives signiﬁcant errors especially in density. This is better

depicted in Fig. 4, where relative errors in density and speciﬁc heat from diﬀerent models are shown. As can

be seen in Fig. 4, the density directly interpolated from the table has errors exceeding 80% and 40% in the

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Mixture fraction

0 0.2 0.4 0.6 0.8 1

Mass fractions (H2, O2, H2O)

0

0.2

0.4

0.6

0.8

1

O2 H2

H2O Flamelet

FPV

Table

Mixture fraction

0 0.2 0.4 0.6 0.8 1

Mass fractions (OH, O)

0

0.05

0.1

0.15

OH

O

Flamelet

FPV

Table

Mixture fraction

0 0.2 0.4 0.6 0.8 1

Mass fractions (H, HO2)

#10-3

0

1

2

3

4

H

HO2

Flamelet

FPV

Table

Mixture fraction

10-4 10-3 10-2 10-1 100

Density [kg/m3]

0

200

400

600

800

1000

1200

1400 Flamelet

FPV, quadratic

FPV, linear

FPV, constant

Table

Mixture fraction

10-4 10-3 10-2 10-1 100

Specific heat [kJ/(kg"K)]

0

5

10

15

20 Flamelet

FPV, quadratic

FPV, linear

FPV, constant

Table

Mixture fraction

10-4 10-3 10-2 10-1 100

Speed of sound [m/s]

0

500

1000

1500

2000 Flamelet

FPV, quadratic

FPV, linear

FPV, constant

Table

Figure 3. Mass fractions (H2, O2, H2O, OH, O, H, HO2), density, speciﬁc heat and speed of sound predicted

by the current FPV approach with quadratic, linear, and constant models for parameter ain comparison with

exact values from a probing ﬂamelet under transcritical conditions (Tox = 100 K, Tf= 150 K, p= 100 bar).

Table used by the FPV approach is constructed using ﬂamelets under ideal-gas conditions at a diﬀerent pressure

(Tox = 300 K, Tf= 300 K, p= 60 bar). Temperature, pressure, mixture fraction and progress variables in

the FPV approach are obtained from the probing ﬂamelet. Black dotted line represents the exact values from

the probing ﬂamelet. Red lines represents predictions from the current FPV approach. Blue dashed line

corresponds to values stored in the table. Shaded area indicates real-ﬂuid region.

real-ﬂuid and ideal-gas regions, respectively.

In contrast, the current FPV model with a quadratic model for the parameter ashows superior per-

formance in recovering these quantities. In the ideal-gas region, a linearized speciﬁc heat ratio is used to

compensate the temperature discrepancy from the table, and this has been shown to yield good predictions

with a temperature discrepancy up to 500 K.65 Similar behavior can be seen in the current study. As shown

in Fig. 4that the relative error from the FPV model in the ideal-gas region is less than 5%. In the real-ﬂuid

region, despite the fact that extrapolation from the table is needed for recovering the thermodynamic quan-

tities (200 K diﬀerence of temperature for the oxidizer stream), the current FPV model shows signiﬁcant

improvements. The quadratic, linear and constant model yield increasingly more accurate results successively

towards the ﬂamelet solution as expected. The relative error of the quadratic model is below 10% and 20%

for the density and speciﬁc heat as shown in Fig. 4. Obviously when the reference table contains real-ﬂuid

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Mixture fraction

10-4 10-3 10-2 10-1 100

Error in density [%]

-100

-80

-60

-40

-20

0

20

FPV, quadratic

FPV, linear

FPV, constant

Table

Mixture fraction

10-4 10-3 10-2 10-1 100

Error in specific heat [%]

-80

-60

-40

-20

0

20

FPV, quadratic

FPV, linear

FPV, constant

Table

Figure 4. Relative error of density (left) and speciﬁc heat (right) in percentage of Fig. 3. Shaded area indicates

real-ﬂuid region and white area ideal-gas region.

information by lowering the oxidizer temperature, and with grid points in mixture fraction clustered in the

oxidizer side, the performance of the current FPV model can be further improved. The operating conditions

are deliberately chosen to challenge the current FPV model.

In summary, the assumption that the composition is a weak function of the reference conditions is valid

under transcritical conditions. The currently developed FPV approach with quadratic expression for the

attraction parameter can accurately recover the real-ﬂuid and ideal-gas thermodynamic quantities despite

variations in temperature and pressure.

B. Cryogenic LOX/GH2 mixing case

The proposed FPV approach is tested with a benchmark case proposed by Ruiz et al.55 for high-Reynolds

number turbulent ﬂows with large density ratios. A two-dimensional mixing layer of liquid-oxygen (LOX) and

gaseous-hydrogen (GH2) streams is simulated. Details of the conﬁguration and the computation domain for

the simulation can be found in Ruiz et al.55 The conﬁguration is representative of a coaxial rocket combustor,

in which dense LOX is injected in the center to mix with the surrounding high-speed GH2 stream. The

two streams are separated by the injector lip which is also included in the computational domain. The

computational domain has a dimension of 10h×10h, where h= 0.5 mm is the height of the injector lip.

A sponge layer of length 5his put at the end of the domain. A fully structured Cartesian mesh is utilized

in this case with 100 grid points across the injector lip. A uniform mesh is used in axial direction. For the

region within 3haround the injector lip in transverse direction, a uniform mesh is adopted and stretching is

applied with a ratio of 1.02 outside this region. The mesh is stretched in axial direction in the sponge layer.

This results in a mesh with a total of 5.2×105cells.

Adiabatic no-slip wall conditions are applied at the injector lip and adiabatic slip wall conditions are

applied for the top and bottom boundaries of the domain. A 1/7th power law for velocity is used for both

the LOX and GH2 streams. Pressure outlet boundary conditions are applied after the sponge layer where

acoustic waves are suppressed. The LOX stream is injected at a temperature of 100 K, and GH2 is injected

at a temperature of 150 K. The pressure is set to 10 MPa which is representative of rocket combustor

conditions. Note that the density ratio between LOX and GH2 is about 80. The hybrid scheme using the

double-ﬂux model is used with the RS sensor37,54 set to a value of 0.2. A ﬁrst-order upwinding scheme is

used in the sponge layer. The CFL number is set to 1.0 and no subgrid scale model is used to facilitate

comparisons with the reference solutions.

The ﬂamelet table is generated at same conditions, namely Tox = 100 K, Tf= 150 K, and p= 100 bar.

Transcritical ﬂamelets are calculated for the entire S-shaped curve at reference conditions. Water is used

as the progress variable. Resolutions in mixture fraction and progress variable are both 100 grid points. In

mixture fraction dimension, the mesh is uniform from zero to the stoichiometric value using one third of the

grid points and then stretches to one with the rest of the grid points. No special treatment is conducted

for the resolution in the PBP region for the reason that the current developed FPV model is insensitive to

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Figure 5. Instantaneous ﬁelds of axial velocity, density, and mass fraction of oxygen from top to bottom for

the cryogenic LOX/GH2 mixing case.

the table resolution at this region as demonstrated in the previous subsection. For the pure mixing case

considered in this subsection, the progress variable, e

C, is set to zero.

Figure 5shows results for instantaneous axial velocity, density, and oxygen mass fraction. Due to the

implementation of the proposed FPV model, for pure mixing case, the FPV model is expected to perform

almost the same as a multi-component model in which species mass fractions are explicitly solved. Indeed,

the mixture fraction for the mixing case considered here acts as the mass fraction of hydrogen. The only

diﬀerence comes from the evaluation of the thermodynamic quantities which is expected to be small as

demonstrated in the previous subsection. The results from the current FPV model is found to give almost

the same results both qualitatively and quantitatively as the multi-component model in Ma et al.37 As can

be seen in Fig. 5, the ﬂow ﬁeld is dominated by large vortical structures in the mixing layer and three large

vortical structures are separated by waves with a wavelength of approximately 5h. The predicted structures

of vortices are in good agreement with those reported in Ruiz et al.55 From the density ﬁeld, “comb-like” or

“ﬁnger-like” structures2can clearly be seen, which was also observed through experiments for transcritical

mixing under typical rocket engine operating conditions.3,4

The simulation results are averaged in time to facilitate quantitative comparisons and 15 ﬂow-through-

times are used for the averaging process after steady state is reached. One ﬂow through time corresponds to

0.125 ms.55 Figure 6shows the results of mean and root mean square (RMS) values for axial velocity, mass

fraction of oxygen, and temperature. Statistics at diﬀerent axial locations (x/h = 1, 3, 5, 7) are plotted

as a function of normalized transverse distance. Results from the current solver, CharLES x, are compared

to those obtained from two other solvers, namely AV BP and RAP T OR.55 The mean axial velocity is in

good agreement between the three diﬀerent solvers, while there are some discrepancies in the RMS values.

Results from CharLES xshow slightly lower RMS values on the GH2 side, especially for the axial location

of x/h = 3. This is probably due to the diﬀerent implementation of the sponge layer and outlet boundary

conditions adopted by the diﬀerent solvers. Results for the oxygen mass fraction are almost identical for the

three solvers except for the small diﬀerence seen at the GH2 side which could be related to the diﬀerence

seen in the velocity results. Appreciable diﬀerences are observed in the temperature statistics. The results

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(a) Axial velocity

(b) Mass fraction of oxygen

(c) Temperature

Figure 6. Mean and RMS results for axial velocity, mass fraction of oxygen, and temperature at diﬀerent

transverse cuts in comparison with Ruiz et al.55 for the cryogenic LOX/GH2 mixing case.

from CharLES xand AVBP are similar and show a narrower thermal mixing layer as compared to that of

RAPTOR. Overall, the results obtained from the three solvers are in good agreement, demonstrating the

capability of the proposed FPV model and the robustness of the numerical schemes for transcritical mixing

cases.

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Figure 7. Instantaneous temperature, mass fractions of H2, O2, and OH from top to bottom for the cryogenic

LOX/GH2 reacting case. Black curve corresponds to the stoichiometric mixture fraction, and pink curve

indicates the PBP location characterized by the peak of speciﬁc heat.

C. Cryogenic LOX/GH2 reacting case

The cryogenic LOX/GH2 mixing case described in the previous subsection is then ignited to further assess

the performance of the proposed FPV model for reacting cases. The computational domain, mesh resolution,

boundary conditions, and numerical schemes are kept the same as in the mixing case. The same ﬂamelet

table is adopted for the reacting case. As expected, large acoustic waves are generated right after the ignition

takes places, and ﬁrst-order upwinding scheme is used in the sponge layer to ensure that the pressure waves

exit the domain and eventually a stable ﬂame is obtained.

Figure 7shows instantaneous results for temperature, mass fractions of H2, O2, and OH. The black curve

in the temperature subﬁgure in Fig. 7indicates the stoichiometric value of mixture fraction. Pink curves in

Fig. 7correspond to the PBP location which is characterized by the peak of the speciﬁc heat capacity. As

can be seen from Fig. 7, the proposed FPV model is able to predict the transcritical ﬂame robustly due to

the double-ﬂux model and entropy ﬂux correction technique adopted in the numerical solver. No spurious

oscillations in pressure or velocity were observed. Due to the adiabatic boundary conditions applied at the

injector lip, the ﬂame is attached. The ﬂame exhibits laminar behavior close to the injector tip and is more

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(a) Mixing case

(b) Reacting case

Figure 8. Instantaneous density ﬁelds of the cryogenic LOX/GH2 mixing and reacting cases. Units in kg/m3.

Legends in logarithmic scale.

wrinkled and turbulent downstream. The structure of a transcritical ﬂame is found to be similar to that

under ideal-gas conditions through ﬂamelet studies. The PBP region is spatially separated from the reaction

zone and the real-ﬂuid pseudo-boiling process is found to take place in the region characterized by almost

pure oxygen.16,66 This can be seen from the current simulation results that the PBP location as indicated by

the pink curve has no interaction with the ﬂame. The transcritical process happens to almost pure oxygen

as can be seen from the oxygen mass fraction results in Fig. 7.

Figure 8compares instantaneous density ﬁelds from both the cryogenic mixing and reacting cases. The

legends are in logarithmic scale to emphasize the small structures at relatively low density values. As can

be clearly seen from Fig. 8, the density ﬁelds show considerable diﬀerences between the two cases. Lower

density can be seen in the reacting case due to the ﬂame in between the two streams. The LOX stream in

the reacting case shows suppressed vortical structures. Whereas in the mixing case, large vortical structures

can be seen and the dense LOX stream penetrates into the GH2 stream by a height of more than habove

the injector lip. The heat release from the ﬂame not only separates the two streams which results in an

almost pure species pesudo-boiling process, but also suppresses the development of the vortical structures in

the mixing layer. This is in consistent with the simulation results by Ruiz.67

The proposed FPV model has been demonstrated to have the capability for transcritical reacting ﬂow sim-

ulations. The current numerical scheme is directly applicable to LES of real applications under transcritical

conditions.

VI. Conclusions

A FPV approach is developed for trans- and supercritical combustion simulations in the context of ﬁnite

volume, fully compressible, explicit solvers. The PR cubic EoS is used for the consistent evaluation of

the thermodynamic quantities under transcritical conditions. Capabilities to calculate transcritical ﬂamelet

solutions have been implemented in the FlameMaster57 solver and validated against DNS results. The double-

ﬂux model developed for transcritical ﬂows37 is used to eliminate spurious pressure oscillations caused by the

nonlinearity inherent in the real-ﬂuid EoS. A hybrid scheme with entropy-stable ﬂux correction technique37 is

used to deal with the large density ratio in transcritical combustion cases. For the FPV approach, parameters

aand bin the cubic EoS are pre-tabulated for the evaluation of the departure functions and a quadratic model

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American Institute of Aeronautics and Astronautics

is used to recover the attraction parameter a. The ideal-gas values are calculated from a linearized speciﬁc

heat ratio model.60 The novelty of the proposed approach lies in the ability to account for pressure and

temperature variations from the reference tabulated values using a computationally tractable pre-tabulated

combustion chemistry in a thermodynamically consistent fashion. An a priori analysis was conducted to

assess the performance of the proposed transcritical FPV approach and it is shown that the assumption that

the composition is a weak function of the reference conditions is valid under transcritical conditions, and

the currently developed FPV approach can accurately recover the real-ﬂuid and ideal-gas thermodynamic

quantities despite perturbations in temperature and pressure. The solution of the laminar ﬂamelets in

mixture fraction space and the chemistry tabulation requires special considerations in order to account

for the full non-linear eﬀects in transcritical ﬂows. The transcritical FPV approach works robustly and

accurately even with an insuﬃcient resolution in mixture fraction in the table. Cryogenic LOX/GH2 mixing

and reacting cases are performed to demonstrate the capability of the proposed approach in multidimensional

simulations. The current combustion model and numerical schemes are directly applicable to LES of real

applications under transcritical conditions.

Acknowledgments

Financial support through NASA with award numbers NNX14CM43P and NNM13AA11G are gratefully

acknowledged. The authors would like to thank Dr. Guilhem Lacaze for sharing data for comparison.

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