Transcritical Diffuse-Interface Hydrodynamics of Propellants in High-Pressure Combustors of Chemical Propulsion Systems

Progress in Energy and Combustion Science 82 (2021) 100877

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Progress in Energy and Combustion Science

journal homepage: www.elsevier.com/locate/pecs

Transcritical diffuse-interface hydrodynamics of propellants in high-pressure combustors of chemical propulsion systems

∗ Lluís Jofre, Javier Urzay

Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA

a r t i c l e i n f o a b s t r a c t

Article history: Rocket engines and high-power new generations of gas-turbine jet engines and diesel engines often- Received 30 September 2019 times involve the injection of one or more reactants at subcritical temperatures into combustor envi-

Accepted 10 August 2020 ronments at high pressures, and more particularly at pressures higher than those corresponding to the critical points of the individual components of the mixture, which typically range from 13 to 50 bars for Keywords: most propellants. This class of trajectories in the thermodynamic space has been traditionally referred to Fuel sprays as transcritical. However, the fundamental understanding of fuel atomization, vaporization, mixing, and Transcritical ﬂows combustion processes at such high pressures remains elusive. In particular, whereas fuel sprays are rela-

Phase change tively well characterized at normal pressures, analyses of dispersion of fuel in high-pressure combustors

High-pressure thermodynamics are hindered by the limited experimental diagnostics and theoretical formulations available. The description of the thermodynamics of hydrocarbon-fueled mixtures employed in chemical propulsion systems is complex and involves mixing-induced phenomena, including an elevation of the critical point whereby the coexistence region of the mixture extends up to pressures much larger than the critical pressures of the individual components. As a result, interfaces subject to surface-tension forces may persist in multicomponent systems despite the high pressures, and may give rise to unexpected spray-like atomization dynamics that are otherwise absent in monocomponent systems above their critical point. In this article, the current understanding of this phenomenon is reviewed within the context of propulsion systems fueled by heavy hydrocarbons. Emphasis is made on analytical descriptions at mesoscopic scales of interest for computational fluid dynamics. In particular, a set of modifications of the constitutive laws in the Navier–Stokes equations for multicomponent flows, supplemented with a high-pressure equation of state and appropriate redefinitions of the thermodynamic potentials, are introduced in this work based on an extended version of the diffuse-interface theory of van der Waals. The resulting formulation involves re- visited forms of the stress tensor and transport fluxes of heat and species, and enables a description of the mesoscopic volumetric effects induced by transcritical interfaces consistently with the thermodynamic phase diagram of the mixture at high pressures. Applications of the theory are illustrated in canonical problems, including dodecane/nitrogen transcritical interfaces in non-isothermal systems. The results indicate that a transcritical interface is formed between the propellant streams that persists downstream of the injection orifice over distances of the same order as the characteristic thermal-entrance length of the fuel stream. The transcritical interface vanishes at an edge that gives rise to a fully supercritical mixing layer. ©2020 Elsevier Ltd. All rights reserved.

Contents

1. Introduction ...... 3 1.1. Transcritical conditions in combustors fueled by heavy hydrocarbons...... 3 1.2. Objectives and outline of the present review ...... 8 2. High-pressure transcritical ﬂow phenomena ...... 8 2.1. Transcriticality in monocomponent systems ...... 8

∗ Corresponding author. E-mail address: [email protected] (J. Urzay). https://doi.org/10.1016/j.pecs.2020.100877 0360-1285/© 2020 Elsevier Ltd. All rights reserved. 2 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

2.2. Transcriticality in bicomponent systems ...... 11 2.3. Transcriticality in aerospace propulsion systems...... 11 2.4. Additional research aspects of transcritical systems ...... 13 3. Thermodynamic phase diagrams of mixtures fueled by heavy hydrocarbons ...... 14 3.1. Equation of state...... 14 3.2. Critical-point elevation in hydrocarbon-fueled mixtures ...... 14 3.3. Conditions suppressing transcritical behavior ...... 18 4. Theoretical foundations of the diffuse-interface theory for transcritical flows ...... 18 4.1. Gradient-dependent thermodynamic potentials...... 19 4.2. Gradient-energy coefficients ...... 19 4.3. Open issues of experimentally calibrated models for gradient-energy coefficients ...... 20 4.4. Characteristic scales in the diffuse-interface theory ...... 21 4.5. The interface-continuum hypothesis ...... 22 4.6. The role of antidiffusion in transcritical flows...... 23 4.7. Conservation equations for transcritical flows ...... 23 5. The structure of transcritical interfaces ...... 24 5.1. Entropy-production sources ...... 24 5.2. Closure for the interfacial stress tensor ...... 25 5.3. Mechanical equilibrium conditions...... 25 5.3.1. Mechanical equilibrium normal to the interface...... 26 5.3.2. Mechanical equilibrium tangential to the interface ...... 28 5.3.3. The thermodynamic pressure near the interface ...... 28 5.3.4. Generalizations to multicomponent systems ...... 29 5.4. Closures for the interfacial transport fluxes of heat and species ...... 30 5.5. The Onsager coefficients in transcritical conditions ...... 31 5.6. Transport equilibrium condition ...... 32 5.6.1. The transport equilibrium condition in terms of a balance of species fluxes...... 32 5.6.2. The transport equilibrium condition in terms of a balance of heat fluxes...... 35 5.6.3. Combination of mechanical and transport equilibrium conditions in terms of chemical potentials ...... 35 5.6.4. Generalizations to multicomponent systems ...... 36 5.7. Remarks on transcritical interfaces in equilibrium and near-equilibrium conditions ...... 36 6. Transcritical interfaces in isothermal bicomponent systems...... 37 6.1. Formulation...... 37 6.2. Results: the steady transcritical interface ...... 37 6.3. The generalized Cahn-Hilliard equation ...... 40 7. Transcritical interfaces in non-isothermal bicomponent systems ...... 42 7.1. Formulation...... 42 7.2. Results: The evolution of a transcritical interface downstream of the injection orifice ...... 45 8. Concluding remarks ...... 47 Declaration of Competing Interest ...... 51 Acknowledgments ...... 51 Appendix A. Basic aspects of thermodynamic stability and phase equilibrium for transcritical systems ...... 51 A1. Thermodynamic-stability considerations...... 51 A2. Phase equilibrium...... 53 Appendix B. Transcritical behavior of thermophysical properties and molecular transport coefficients ...... 53 B1. Molar enthalpy ...... 53 B2. Constant-pressure molar heat ...... 55 B3. Chemical potentials ...... 55 B4. Molecular transport coefficients ...... 56 Appendix C. Supplementary thermodynamic expressions applicable to multicomponent systems at high pressures ...... 58 C1. Local thermodynamic relations ...... 58 C2. Coefficients of the Peng-Robinson equation of state ...... 58 C3. Parameters of the model for the gradient-energy coefficient ...... 58 C4. Molar Helmholtz free energy and chemical potential ...... 58 C5. Volume expansivity ...... 59 C6. Enthalpy departure function ...... 59 C7. Molar heats, adiabatic coefficient, and isentropic compressibility ...... 59 C8. Partial molar enthalpies and partial-enthalpy departure functions ...... 60 C9. Internal-energy departure function ...... 60 C10. Partial molar internal energies ...... 60 C11. Partial molar volume and isothermal compressibility ...... 60 C12. Ideal-gas Gibbs free energies ...... 60 C13. Fugacity coefficients ...... 61 C14. Speed of sound ...... 61 Appendix D. High-pressure interfaces in subcritical monocomponent systems...... 61 D1. Formulation...... 61 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 3

D2. Results...... 62 Appendix E. Transcritical bicomponent systems approaching mechanically unstable conditions...... 63 E1. Singularities of the theory at the limit of mechanical stability ...... 63 E2. Negative thermodynamic pressures...... 66 Appendix F. Supplementary expressions for molecular transport coefficients at high pressures ...... 66 F1. Dynamic viscosity ...... 66 F2. Thermal conductivity ...... 67 F3. Stefan-Maxwell forms of the standard fluxes of heat and species ...... 68 F4. Fickian diffusion coefficients ...... 68 F5. Binary diffusion coefficients ...... 69 F6. Thermal-diffusion ratios ...... 69 Appendix G. Numerical methods for the calculation of transcritical interfaces...... 69 G1. Spatial and temporal discretizations ...... 70 G2. Numerical considerations for cases involving equilibrium isothermal bicomponent systems ...... 70 G3. Numerical considerations for cases involving non-isothermal bicomponent systems...... 71 References ...... 71

1. Introduction (a) it summarizes relevant operating conditions leading to transcriticality; (b) it describes the state-of-the-art in the theoretical The utilization of high pressures for burning fuel and oxidizer and computational understanding of transcritical interfaces at the is a necessary requirement for the generation of thrust in chemical continuum level; (c) it provides a comprehensive diffuse-interface propulsion systems for ground, air, and space engineering applica- theory of the hydrodynamics of propellants under transcritical con- tions. In modern designs, the characteristic pressures in the com- ditions; and (d) it employs that theory to describe, for the first bustor range from 40 to 560 bars in rocket engines, to 20–60 bars time, the downstream evolution of a transcritical interface sepa- in gas-turbine jet engines at takeoff, and 10–30 bars in diesel en- rating the propellant streams. This review article also provides an gines. However, great difficulties arise when attempting to model analysis of the terminal structure of the transcritical interface at the dynamics of chemically reacting flows at high pressures. One an edge located at a supercriticalization distance downstream of of them, which influences the dispersion of fuel in the combus- the orifice that is calculated as part of the solution. The diffusional tor and represents the focus of the present work, is related to the critical point of the mixture of the propellants is shown to play a occurrence of transcritical conditions. crucial role in the description of the interface edge. Downstream Challenges associated with transcritical conditions have been of the interface edge, the flow becomes fully supercritical, the in- known since the development of high-pressure thrust chambers terface disappears, and the propellants mix by molecular diffusion for liquid-fueled rocket engines [1–4] . A number of controlled ex- across a growing compositional mixing layer. periments have been undertaken over the years to study flows at high pressures [5–14] , and many have revealed the presence of 1.1. Transcritical conditions in combustors fueled by heavy interfaces separating the propellants [15–23] . The high pressures, hydrocarbons however, limit the amount of information that can be obtained from state-of-the-art experimental diagnostics. Numerical simu- Although there may be several definitions of the char- lations have accompanied these endeavors [24–31] , but most of acter of transcritical conditions in the literature [1–4,6,7,9– these computations have been limited to high-pressure conditions 14,19,22,24,25,30,32,33,35–37] , one that is of practical relevance that did not warrant the existence of interfaces. Major reviews in for chemical propulsion systems corresponds to the thermody- the field have also focused on those conditions [32–34] . The field namic conditions attained in the injection configuration sketched has evolved rapidly in recent years spurred by a number of publi- in Fig. 1 . There, two streams of different propellants, denoted cations, including (a) the thermodynamic analysis of Qiu and Reitz by the subindexes F (for the fuel stream) and O (for the coflow [35] , which has provided detailed information about the thermo- stream), are injected in such a way that the temperature of the dynamic space transited by homogeneous hydrocarbon-fueled mix- coflow stream TO is higher than that of the fuel stream TF . In this tures at high pressures; (b) the molecular-dynamics simulations by configuration, transcriticality is attained when at least one of the Mo and Qiao [36] , which have described the structure of transcrit- propellant streams, often the fuel stream in practical applications, ical interfaces albeit limited to very small length and time scales enters the combustor at a subcritical temperature, but the com- imposed by the high computational cost of molecular methods; bustor pressure P ∞ is larger than the critical pressures of the two and (c) the continuum-level theoretical work of Dahms and Oe- separate components, denoted here by Pc ,F and Pc ,O . felein [37] , and Gaillard et al. [38,39] , which have provided insight The configuration depicted in Fig. 1 drives the discussion into the problem of transcriticality by using the diffuse-interface throughout the remainder of this manuscript. It is shown in theory of van der Waals. The work by Dahms and Oefelein [37] was Section 7 that the main characteristics of the flow field sketched limited to the analysis of mean thermodynamic states in the com- in Fig. 1 , including the interface separating the propellant streams, bustor and stationary interfaces in absence of flow effects. In con- can be described by the diffuse-interface theory presented in trast, Gaillard et al. [38,39] made significant progress in coupling Sections 4 and 5 even in a simpler canonical configuration where extensions of the diffuse-interface theory of van der Waals with complex but inessential factors such as turbulence, shear, and in- the Navier-Stokes equations to describe, for instance, steady coun- terface corrugation are not considered. This section provides a terflow hydrogen diffusion flames in transcritical conditions. qualitative summary of the main physical processes and the asso- This review article shows that fluid-mechanical effects are cen- ciated characteristic scales. tral to the description of transcritical phenomena, and that the The possible number of combinations of thermodynamic condi- problem cannot be treated solely by mean thermodynamic states tions in the propellant streams that may lead to transcritical phe- in the combustor or isolated stationary interfaces, as previously at- nomena are large and cannot be addressed in a single study. Their tempted [37] . This review article fulfills the following objectives: particularities depend on the propulsion system and the mixture 4 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Nomenclature Re Reynolds number s / s¯/ s˜ speciﬁc / molar / generalized entropy

Latin letters s˙prod entropy production rate a , b coeﬃcients of the Peng-Robinson equation of state t time A P / A μ characteristic oscillation amplitudes of the pressure T temperature

/ chemical potential across the interface Tc ,diff temperature of the diffusional critical point A , B parameters of the gradient-energy coeﬃcient model Tc ,F /Tc ,O critical temperatures of the propellants

b¯ volume expansivity divided by the corresponding Tc ,mix critical mixing temperature

ideal-gas value Tpb pseudo-boiling temperature

c speed of sound TTR supercriticalization temperature

cˆ ratio between real and ideal specific heats at con- UF / UO injection velocities of the propellants stant pressure v velocity vector c p / c¯ p constant-pressure specific / molar heat v molar volume c v / c¯ v constant-volume specific / molar heat V volume of the system d equivalent hard-sphere diameter VLE vapor-liquid equilibrium

Di , j Fickian diffusion coeﬃcient V¯ partial molar volume D i, j binary diffusion coeﬃcient W / W molecular weight / mean molecular weight of the

DT thermal diffusivity mixture e / e¯ specific / molar internal energy W ratio between fuel and coflow molecular weights E total energy We Weber number E m / E s mechanical- / transport-equilibrium parameters x streamwise spatial coordinate f / f¯ specific / molar Helmholtz free energy X molar fraction F Helmholtz free energy of the system y transversal spatial coordinate F vector of thermodynamic forces Y mass fraction f fugacity Z compressibility factor g / g¯ specific / molar partial Gibbs free energy G Gibbs free energy of the system Greek symbols h / h¯ specific / molar enthalpy α thermal-expansion ratio

H enthalpy of the system βs isentropic compressibility β Ji species diffusion ﬂux T isothermal compressibility J i interfacial species ﬂux βv volume expansivity

kT thermal-diffusion ratio γ adiabatic coeﬃcient δ KI local curvature of the interface I interface thickness K δ δ interfacial stress tensor T / Y thermal / compositional mixing-layer thickness KnI Knudsen number d e / d h internal energy / enthalpy departure function ca capillary length R / T large-scale / thermal Cahn number

Li , j Onsager coeﬃcients η dynamic viscosity L Onsager matrix θ reduced temperature

LTR supercriticalization length ϑ binary-interaction parameter Le Lewis number ι model parameter for calculation of thermal- M momentum-flux ratio diffusion ratio Ma Mach number κ gradient-energy coefficient n number of moles λ thermal conductivity nˆ spatial coordinate normal to the interface molecular mean free path N number of components μ / μ¯ / μ specific / molar / generalized chemical potential N nondimensional curvature ρ density N A Avogadro’s number σ surface-tension coefficient σ Oh Ohnesorge number 0 characteristic value of the surface-tension coeffi- OPR overall pressure ratio cient P pressure τ ratio between interface and fuel-stream tempera- P a air pressure at a given altitude tures

Pc ,F /Pc ,O critical pressures of the propellants τ viscous stress tensor P ∞ combustor pressure ϕ fugacity coefficient P ratio between the characteristic oscillation ampli- φ vector of thermodynamic currents tude of the pressure through the interface and the χ auxiliary variable combustor pressure Landau potential Pe Péclet number ψ auxiliary vector q heat flux q t heat flux (excluding heat transport by interdiffusion Main subscripts of species) c critical thermodynamic state Q interfacial heat flux diff diffusional critical point

RF radius of the oriﬁce e vapor-liquid equilibrium conditions

R 0 universal gas constant f formation value R ratio between fuel and oxidizer densities F fuel stream L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 5

The magnitude of UF relative to the coﬂow-stream velocity UO FF, O employed to denote the value of the binary diffusiv- is represented by the momentum-ﬂux ratio

ity in the fuel stream 2 ρO U F, OO employed to denote the value of the binary diffusiv- M = O , (3) ρ 2 ity in the coﬂow stream F UF

GD gradient-dependent thermodynamic potential or ρ with O being the density of the coﬂow stream. The momentum-

pressure ﬂux ratio is of order M ∼ 1–10 in rocket engines and gas-turbine jet

I interface-related quantity engines [43,46–48] , whereas M = 0 for diesel engines. In the high-

mix critical mixing conditions pressure operating conditions investigated herein, the density ratio O coﬂow stream pb pseudo-boiling conditions R = ρ /ρ (4) ∞ conditions in the combustor environment F O R = R = Main superscripts is within the range 5–40, as opposed to 500–1000 at at-

g gas-like state mospheric√ pressure. The corresponding velocity ratios are of order / = R ∼ IG ideal-gas conditions UO UF M 1–10.

liquid-like state In the model problem depicted in Fig. 1, the coﬂow tempera-

T species or heat ﬂux components dependent on tem- ture TO is large enough to heat up the fuel to its critical temper-

perature gradients ature downstream of the injection orifice. For coflows comprised species or heat flux components independent of of N2 , O2 , or air, the prevailing condition implies that TO must > temperature gradients be larger than the critical temperature of the coflow, TO Tc ,O , ∼ nondimensional variable since Tc ,O 120–150 K is in the range of cryogenic temperatures

0 reference or initial value for those components (see Table 1). As a result, the relative temperature difference between the propellant streams, defined as the thermal-expansion ratio of propellants considered, as discussed in Sections 2 and 3 . How- ever, the transcritical conditions illustrated in Fig. 1 are of general T O − T F α = , (5) relevance in combustors fueled by heavy hydrocarbons because of T F two factors: is an order-unity parameter. Additionally, the coflow is cooled (a) As observed in Table 1 , heavy hydrocarbons have relatively high down by the fuel stream and, to a much lesser extent, the fuel critical temperatures, and therefore require significant temper- stream is heated by the coflow. This discrepancy is driven by the ature increments of order unity relative to their injection tem- much higher thermal diffusivity of the coflow. The thickness of δ perature to become fully supercritical, namely the resulting thermal mixing layer, T , grows with distance downstream from small values compared to R near the orifice (a good T c, F − T F F = O (1) . (1) assumption for thin orifice rings), to values of order RF far from T F the orifice where the interface vanishes. Since the Prandtl num- − ∼ Specifically, Tc, F TF 200–400 K for heavy hydrocarbon fuels δ ber remains close to that of air at normal conditions, only T may of interest (e.g., C H , C H , C H , C H , Jet-A, and RP- 6 14 7 16 10 22 12 26 be considered, for simplicity, as the length scale representative of 1), where T ∼ 300–450 K and T ∼ 50 0–70 0 K are the typi- F c,F both momentum and thermal transport. cal ranges of the fuel injection temperature and the fuel critical The transcritical interface extends from the orifice ring to an temperature, respectively. edge at a supercriticalization distance x = L , where the fuel (b) As discussed in Section 3 , mixtures of heavy hydrocarbons with TR stream has been heated up to a supercriticalization temperature

molecular nitrogen (N2 ), molecular oxygen (O2 ), or air have T = T . The latter depends on the combination of propellants and much higher critical pressures than their separate components. TR is a solution of the thermodynamic phase diagram of the mixture.

Under the aforementioned transcritical conditions, which are For mixtures of heavy hydrocarbons with N2 , O2 , or air, it will be often found in practice as discussed in Section 2 , and despite the shown that prevailing high pressures, the flow field contains a region immedi- , ately downstream of the injector orifice where a thin transcritical TTR Tc, diff (6) interface is formed between the fuel and coflow streams because where Tc ,diff is the temperature of the diffusional critical point of of diffusion processes essential to the mixing that necessarily pre- the mixture at the combustor pressure P ∞ (see Appendix A for a cedes combustion. Specifically, as described in Section 5 , the the- formal definition of the diffusional critical point). The temperature ory presented here indicates that surface tension survives the high of the diffusional critical point Tc ,diff is smaller than the fuel critical pressures because of the intrinsic diffusional instability of the fluid temperature Tc ,F by small amounts, resulting from mixing the two propellant streams. This diffusional T , − T , instability, whose basic foundations are discussed in Appendix A, c F c diff 1 . (7) leads to separation of the propellants by a transcritical interface T c, F that bears large composition gradients across and is accompanied For instance, T , − T , ∼ 10–40 K at pressures P ∞ ∼ 50–200 bar by surface tension. c F c diff in C H /N mixtures, thereby yielding (T , − T , ) /T , = 0.02– Both thermal and compositional mixing layers sketched in 12 26 2 c F c diff c F 0.06. The value of T is also similar to the critical mixing tem- Fig. 1 are slender, since the characteristic Reynolds number c,diff

perature Tc ,mix deﬁned as the maximum temperature that can be Re F = ρF U F R F /ηF (2) attained by the mixture under conditions of phase coexistence at − is typically much larger than unity in practical applications. In Eq. P∞ , with Tc, mix Tc, diff being at most 5 K in C12 H26 /N2 mixtures at ρ η P ∞ = 50–200 bar. As a consequence, T is also larger than the fuel (2), UF , F , and F are the velocity, density, and dynamic viscos- TR injection temperature T by relative amounts of order unity, ity of the fuel stream, respectively, and RF is the fuel oriﬁce radius, F which is typically of order 0.1–1 mm in chemical propulsion sys- T − T TR F = ( ) . tems of interest [19,43–45] . O 1 (8) T F 6 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 1. Schematics of a transcritical flow near an injector orifice delivering a cold supercritical heavy-hydrocarbon fuel into a hot supercritical coflow in a combustor at a pressure higher than the critical pressures of the two separate components. The symbol R F denotes the radius of the orifice, and U F and U O are the characteristic injection velocities of the fuel and oxidizer streams, respectively. Additionally, σ is the surface-tension coefficient, P ∞ is the combustor pressure, Y F is the fuel mass fraction, and T F and T O correspond to the temperatures of the fuel and coflow streams, respectively. The critical temperatures and pressures of the fuel and oxidizer streams are denoted, respectively, by T c ,F and P c ,F , and by T c ,O and P c ,O . The rest of the symbols are defined in the main text.

Table 1 Critical temperatures and pressures for selected species [40–42] .

N 2 O 2 air H 2 O CO 2 H 2 CH 4 C 6 H 14 C 7 H 16 C 10 H 22 C 12 H 26 Jet-A RP-1

T c [K] 126 155 131 647 304 33 191 508 540 618 658 671 677 P c [bar] 34 50 36 220 74 13 46 30 27 21 18 24 22

The transcritical interface terminates at an edge crossed by the are both much smaller than unity in the transcritical region, where supercriticalization isotherm, where a fundamental change in the the interface survives. These represent fundamental characteris- diffusional characteristics of the mixture occurs. The structure of tics that simplify the theoretical description of the interface struc- the edge can be previewed in terms of spatial contours of the fuel ture. Further details and additional explanations of the results in mass fraction YF in Fig. 2 for a system consisting of C12 H26 injected Figs. 2 and 3 are deferred to Section 7. = = < < at TF 450 K into a coflowing N2 stream at TO 1000 K. The sys- In the transcritical region 0 x LTR , the interface survives tem is at P ∞ = 100 bar, which represents a supercritical pressure along with the surface-tension force it engenders. In general, the with respect to both components. The resulting distribution of sur- quantification of LTR requires numerical integration of the diffuse- face tension along the transcritical interface with distance down- interface formulation presented here. In laminar flows at moder- stream of the injection orifice is shown in Fig. 3. The results in ately large values of the Reynolds number (2), an estimate of LTR both of those figures are obtained by using the diffuse-interface can be obtained by an integral balance between the streamwise theory presented in Sections 4 and 5 . The resulting formulation variation of the enthalpy flux in the fuel stream and the total describes high-pressure flows with interfaces whose strength can amount of heat conduction received across the interface from the vary in space and vanish altogether. The results obtained by using hot coflow, δ this theory indicate that the interface thickness I remains small (T − T ) (T − T ) along a large portion of the interface despite the high pressures ρ 2 TR F ∼ λ O F , F cp, F UF RF O RF (11) involved. Consequently, the integration of the formulation is hin- L TR δT δ dered by the large disparity between the interface thickness I and δ λ other length scales, including T , RF , LTR , and the interface radius of where O is the thermal conductivity in the coflow stream. For δ ∼ curvature RI . In particular, it will be shown, in conditions of prac- axisymmetric jets, the approximation T RF can be made in Eq. tical relevance, that the large-scale Cahn number (11) in analogy to cold round jets at downstream distances ap-

proaching thermal-entrance conditions [49]. It follows that LTR de- = δ /R (9) R I F pends quadratically on the oriﬁce radius RF as and the thermal Cahn number 2 R − RF UF cp, F Tc, F TF L ∼ , (12) = δ /δ , TR T I T (10) D T , O c p, O T O − T F L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 7

Fig. 2. Numerical results of the present theory showing a zoomed view of the terminal structure of a transcritical interface at an edge. The interface separates two planar laminar streams at high Péclet numbers: one of dodecane (C 12 H 26 ) injected at T F = 450 K (below the interface), and one of molecular nitrogen (N 2 ) injected at T O = 10 0 0 K (above the interface). The system is at a supercritical pressure P ∞ = 100 bar with respect to both components. The figure shows the fuel mass fraction Y F (solid contours and dark thin dashed lines), temperature (colored thick dashed-line contours), diffusional critical point (purple diamond symbol), and diffusional spinodal (thick purple dashed lines). The vertical axis is the transversal distance from the jet axis y normalized with the orifice radius R F , whereas the horizontal axis is the streamwise distance x normalized with the estimate (13) for the supercriticalization length. The flow attains fully supercritical conditions downstream of the interface edge. There, both propellants resemble gas-like supercritical fluids and mix by molecular diffusion across a compositional mixing layer that emanates from the interface edge. Details about the calculations are provided in Section 7 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Evolution of the surface-tension coeﬃcient along the transcritical interface in Fig. 2 obtained from the present theory. Further details about these calculations are provided in Section 7 .

≈ ≈ δ ∼ 1/2 where the approximation TTR Tc ,diff Tc ,F has been made in ac- T (DT,O LTR /UF ) . This leads to the estimate cordance with Eqs. (6) and (7). In Eq. (12), DT,O is the thermal 2 diffusivity of the coflow stream, and c / c is the ratio of fuel 2 2 2 p,F p,O R U F R c p, F T , − T ∼ F c F F , and coflow specific heats, which is of order unity in the con- L TR (13) D , c , T − T ditions analyzed here, as discussed in Appendix B . Additionally, T O p O O F ( − ) / ( − ) Tc, F TF TO TF is an order-unity temperature-difference ratio, which is used for normalizing x in Figs. 2 and 3 . as prescribed by Eqs. (1) and (5). In contrast, two-dimensional The estimates (12) or (13) predict that the supercriticalization laminar jets are known to have much longer thermal-entrance length increases with decreasing values of the fuel injection tem- lengths than their round counterparts because the outer region, perature TF , coflow injection temperature TO , and combustor pres- where streamwise convection and transverse diffusion balance, sure P ∞ . Among these three quantities, the maximum sensitivity grows indefinitely with the square root of x, thereby yielding com- is to variations of TO . For instance, a decrease of 300 K in TO paratively smaller heat fluxes into the cold stream [49]. In this leads to a four-fold increment in LTR in the conditions analyzed case, a more appropriate approximation to utilize in Eq. (11) is in Figs. 2 and 3 . 8 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

The considerations above suggest that the injection oriﬁce ra- The theory outlined here provides a description of the struc-

dius RF has a decisive inﬂuence on the size of the transcritical re- ture of stationary transcritical interfaces in equilibrium, as well as

gion. In practice, LTR is a fraction of the thermal-entrance length, the downstream evolution of non-equilibrium transcritical inter- the latter delimiting the condition for which the hot coﬂow tem- faces separating propellant streams with different temperatures. In

perature TO (rather than Tc ,F ) is attained in the bulk of the fuel the latter case, this theory predicts that the interface ends at an stream. However, it will be shown that the transfer of heat from edge, whose structure elicits the central role played by the diffu- the coﬂow to the fuel stream is hindered in transcritical conditions sional critical point of the mixture in supercriticalizing the system, because of two factors: as shown in Fig. 2 . Emphasis is made on transcritical interfaces

in C12 H26 /N2 mixtures, but supplementary thermodynamic analy- (a) High pressures lead to small values of the thermal diffusivity in ses are provided for several other mixtures of interest that iden- the coflow stream. tify pressure and temperature conditions inducing transcritical in- (b) For a significant portion of the interface length, the specific terfaces in the flow field. A discussion is also included about op- heat attains a global maximum in the vicinity of the interface, erating conditions in combustors of aerospace propulsion systems thereby leading to a decrease in the local thermal diffusivity leading to transcriticality. there. The remainder of this manuscript is structured as follows. A qualitative description of transcritical phenomena is pro- These factors should be taken into account when interpret- vided in Section 2 for monocomponent and bicomponent sys- ing molecular-dynamics simulations of transcritical systems, since tems, along with a short survey of characteristic injection con- those techniques are typically limited to microscopic length scales ditions in aerospace propulsion systems. Phase diagrams for that may be too small compared to the supercriticalization length hydrocarbon-fueled binary mixtures are discussed in Section 3 . required to describe the macroscopic evolution of the interface Using the diffuse-interface theory, a formulation of the Navier- with distance downstream of the injection orifice. Stokes equations that incorporates high-pressure effects and vol- Additional complexities arise in realistic scenarios that are be- umetric terms representing embedded interfaces is outlined in yond the scope of this study. For instance, in turbulent transcriti- Section 4 . Entropy-based closures of interface-related terms in cal flows at large values of the Reynolds number (2) , the molecular the augmented Navier-Stokes equations, which enable the descrip- thermal diffusivity D in Eq. (12) should be replaced by a turbu- T,O tion of the structure of transcritical interfaces, are derived in lent eddy diffusivity of order D ∼ U R , which is lager than D turb F F T,O Section 5 . Readers who are not interested in the development of by a factor of order Re 1. Consequently, the ratio of the super- F the formulation can skip Sections 4 and 5 and move directly to criticalization length to the orifice radius decreases from L /R ∼ TR F Sections 6 and 7 , in which calculations of simplified canonical Re R in laminar flows to L /R ∼ R in turbulent flows. There is F TR F problems are provided. In Section 6 , transcritical interfaces in equi- however little work to date that have addressed closures in high- librium isothermal systems are analyzed. Section 7 is devoted to pressure turbulent flows [50] , and therefore the effects of turbu- non-equilibrium transcritical interfaces in non-isothermal systems lence will not be further pursued here. Similarly, hydrodynamic and their evolution downstream of the injection orifice. Lastly, con- instabilities may develop as a result of the competition between cluding remarks are given in Section 8 . Additionally, seven appen- aerodynamic stresses and the interface-restoring surface-tension dices are included that contain supplementary details and funda- force in the transcritical region. These force imbalances may roll mental concepts closely connected with the formulation. In partic- the interface leading to a finite local radius of interface curvature ular, Appendix A reviews basic concepts of thermodynamic stabil- R , and eventually to the formation of ligaments and droplets, as I ity and phase equilibrium. Appendix B provides a description of sketched in Fig. 1 . Configurations different from that in Fig. 1 may thermophysical properties and transport coefficients at high pres- therefore exist in which the supercriticalization isotherm never sures. Appendix C outlines supplementary thermodynamic expres- crosses the interface before it breaks up, but instead crosses the sions for systems at high pressures. Appendix D focuses on numer- ensuing fuel spray cloud. ical results obtained for near-critical interfaces in monocomponent systems. Appendix E discusses the behavior of the present the- 1.2. Objectives and outline of the present review ory near mechanically unstable conditions. Appendix F lists supplementary expressions for transport coefficients at high pressures, This review focuses on theoretical aspects of transcritical phe- and Appendix G provides methods for the numerical integration of nomena at a continuum level and in a manner amenable to com- the formulation presented here. putational fluid dynamics (CFD). The key question to be explored is whether transcritical systems can be computed by integrating the 2. High-pressure transcritical flow phenomena Navier-Stokes equations uniformly over the entire flow field while subject to high-pressure equations of state along with the standard A significant challenge for the predictive calculation of tran- transport theory for multicomponent mixtures. The key concern scritical flows lies in the complexity of the transitional and that impedes that approach is related to the antidiffusive character composition-dependent character of the interface formation and of the mass transport predicted by the standard theory in tran- breakup, and of the subsequent dispersion and mixing of fuel in scritical conditions, which renders the problem ill-posed. To ad- the combustor. To address these aspects, this section begins by dress that, a formulation of the Navier-Stokes equations with revis- introducing the main problems associated with transcriticality in ited constitutive laws, supplemented with (a) proper redefinitions monocomponent flows, followed by a description of more complex of thermodynamic potentials, (b) a high-pressure equation of state, phenomena in bicomponent flows. In addition, a survey of charac- and (c) high-pressure thermophysical and multicomponent trans- teristic injection conditions in aerospace chemical propulsion sys- port properties, is introduced that regularizes the problem and en- tems is provided at the end of this section that serves to familiar- ables the description of volumetric effects of transcritical interfaces ize the reader with practical configurations. in multicomponent flows. The formulation is based on extensions of the diffuse-interface theory of van der Waals [108] –a useful 2.1. Transcriticality in monocomponent systems formalism originally designed for monocomponent isothermal systems at high pressures, when the interface broadens significantly Consider a closed vessel filled with a monocomponent liquid in relative to intermolecular distances. phase equilibrium with its own vapor. The pressure and temper- L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 9

Fig. 4. Thermodynamic phase diagram for pure N 2 calculated with the Peng-Robinson equation of state introduced in Section 3.1 . Included in the plot are the mechanical spinodals (dot-dashed lines), vapor-liquid equilibrium lines (thick red dashed lines), the pseudo-boiling line (thick green dot-dashed line), and the critical point (square symbol). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) ature in the vessel are denoted by P and T , respectively. At nor- using interface-capturing methods such as the volume-of-fluid mal pressure, the liquid and vapor are separated by an interface or level-set formulations, both of which volumetrically incorpo- whose thickness is clearly not in the continuum range, and which rate the surface-tension force in the momentum equation through is subject to a surface-tension force. Below the critical pressure P c , a surface-tension coefficient σ multiplied by the interface cur- the presence of such interface is warranted since each thermody- vature and by a Dirac-delta function localized at the interface namic state ( P , T ) is associated with two densities [51] correspond- [62–66] . As the ambient pressure P ∞ increases, the liquid-to-

ρg, ρ ing to the phase-equilibrium values for vapor, e and liquid, e . gas density ratio decreases. In particular, the interface becomes This is illustrated in the P − v thermodynamic phase diagram in thicker as the critical pressure is approached, both surface ten- Fig. 4 for N , where = W /ρ is the molar volume and W is sion and vaporization enthalpy decrease, and the atomization 2 v N2 N2 the molecular weight. The resulting ratio of densities of the phases, process becomes a mixing between gas-like ﬂuids without any

R = ρ /ρg, 3 e e is typically of order 10 at normal conditions. breakup. As the temperature of the vessel is increased, the liquid evapo- Entering the supercritical region, it is important to distinguish rates and a new phase-equilibrium condition is attained in which between supercritical gas-like and supercritical liquid-like fluids as the pressure increases as well in accordance with the Clausius– depicted in Fig. 4 . As discussed in Appendix B.4 , the factors that Clapeyron relation. During this process, the interface thickens and separate the behavior of liquid-like from gas-like supercritical flu- the density ratio decreases to a value close to unity near the crit- ids are: ical pressure P c , at which the two phases become visually indis- (a) A supercritical liquid-like fluid is one whose density is large, tinguishable. Above the critical pressure, P > P c , the fluid is super- whose isothermal compressibility is small, and whose transport critical with a uniquely defined density ρ for each thermodynamic coefficients behave in a manner that is reminiscent of a liquid, state ( P , T ), thereby preventing the existence of an interface. Mean- with decreasing viscosities and thermal conductivities found at while, the surface tension necessarily vanishes at P ≥ P c ; the limit increasingly larger values of temperature, whereas the mass dif- P → P c is taken here in combination with T → T c [53,54] . fusivity remains mostly constant or increases slowly with tem- The transition from subcritical to supercritical conditions de- perature. scribed above can be visualized in laboratory experiments involv- (b) In contrast, the density of supercritical gas-like fluids is rel- ing the thermodynamic characterization of pure substances by us- atively smaller, their isothermal compressibility is larger, and ing equilibrium cells [52,55] . A sequence of photographs that il- their viscosity, thermal conductivity, and mass diffusivity in- lustrates the vanishing distinction between the two phases in a crease with temperature, thus resembling the behavior ob- system approaching and crossing the critical point of propane is served in gases. At high temperatures, T T , and despite the provided in Fig. 5 (a). c high pressures, the intermolecular distances are sufficiently In contrast with closed systems, combustors involve continuous large for the supercritical gas-like fluid to behave like an ideal flows of propellants. Consider a cold liquid jet injected at tempera-

gas. ture TF and atmospheric pressure into its own hot vapor at temper- ≤ ≤ ature TO , where TF Te TO , with Te being the vapor-liquid equilib- At pressures not too large compared to P c , a clear tran- rium (VLE) or boiling temperature at the corresponding pressure. sition from liquid-like to gas-like supercritical behavior occurs 3 The resulting liquid-to-gas density ratio ρ / ρ is of order 10 . The F O at the pseudo-boiling, or Widom, temperature Tpb denoted by liquid jet breaks up due to unbalances between aerodynamic and the thick green dot-dashed line in Fig. 4 (see Refs. [67,68] and surface-tension forces [45,56–60] as illustrated in Fig. 5 (b). A rela- Appendix B.4 for related discussions). This temperature depends on tively large amount of energy, comparable to the thermal enthalpy ∼ pressure and is not too different from Tc (e.g., Tpb 130 K at 40 bar of the hot environment, must be transferred to the spray to heat it ∼ = and Tpb 150 K at 60 bar, both for N2 , for which Tc 126 K). In par- up and vaporize it [61] . ticular, Tpb corresponds approximately to the temperature where Similarly to the closed vessel operating at normal pressure, a second-order phase transition occurs, with the density, viscosity the thickness of the liquid-gas interface in the jet flow men- and thermal conductivity plunging as transition from a liquid-like tioned above is several orders of magnitude smaller than any of to a gas-like state occurs, and with the specific heat at constant the relevant hydrodynamic scales involved. Nonetheless, predic- pressure displaying a characteristic finite-amplitude spike [69,70] . tive simulation capabilities have been developed in recent years As the pressure becomes increasingly larger than P c , the distinction 10 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 5. Experimental visualizations of high-pressure flow systems. (a) Equilibrium cell filled with pure propane at pressure and temperature below the critical point (left panel), at pressure and temperature near the critical point (center panel), and at pressure and temperature above the critical point (right panel) (adapted from Ref. [52] ). (b) Cold nitrogen at 105 K (dark stream) injected into a hot nitrogen environment at 300 K at subcritical pressure 10 bar (left panel), near-critical pressure 30 bar (center panel), and supercritical pressure 40 bar (right panel) (adapted from Ref. [6] ). (c) N-hexadecane injected at subcritical temperature 300 K (dark stream) into a hot supercritical nitrogen environment at temperature 907 K and pressure 79 bar (i.e., higher than the critical pressures of the individual components). The different panels in (c) correspond to different instants of time after injection cutoff, with insets showing magnified views of fuel droplets (reprinted from Ref. [19] with permission from Elsevier). between liquid-like and gas-like supercritical behavior is increas- ever, surface-tension effects are irrelevant in monocomponent sys- ingly more difficult to ascertain. Consequently, the dynamical rele- tems above the critical point [53,71–74] as corroborated, for in- vance of the pseudo-boiling line diminishes away from the critical stance, by the absence of a spray in the right panel in Fig. 5 (b). Fur- point. thermore, the decrease of surface tension with pressure and tem- The chamber pressure in the jet flow shown in the right panel perature is continuous, and as a result its value is always of little to in Fig. 5 (b) is supercritical, but the jet and ambient temperatures no dynamical relevance even at subcritical conditions nearing the are, respectively, smaller and larger than Tpb . Many studies have critical point in monocomponent systems. As a result, the liquid- characterized those conditions as transcritical [5,8,26–29,31]. How- like supercritical N2 jet in the right panel in Fig. 5(b) mixes with L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 11

the gas-like supercritical N2 environment similarly to pure gaseous As noted in Section 1.1, the disparity between the critical (Tc ,F ) mixing. and injection (TF ) temperatures of the fuel stream plays an important role in the onset of transcriticality in combustors fueled by heavy hydrocarbons. The larger this disparity, the larger the su-

2.2. Transcriticality in bicomponent systems percriticalization length LTR , and the more favored are transcritical thermodynamic trajectories across the elevated coexistence dome.

Bicomponent systems behave fundamentally different from As indicated in Table 1, Tc ,F is typically 200–400 K larger than TF , monocomponent ones and can lead to spray formation at com- since the latter is typically limited to 300–450 K in order to min- bustor pressures that are supercritical with respect to the sepa- imize carbon deposition by thermal decomposition of the fuel in > rate components, P∞ max{Pc ,F , Pc ,O } [15–19,22,23]. As shown in feeding systems and engine wall-cooling channels [77]. At those Section 3.2 , an important physical characteristic enabling the for- injection temperatures, heavy-hydrocarbon fuels resemble super- mation of the spray is that binary mixtures of heavy hydrocarbons critical liquid-like fluids. That tendency increases with the number with typical oxidizers, including pure oxygen (for rockets) and air of carbon atoms due to the corresponding increase in the fuel crit- = (for gas-turbine jet engines and diesel engines), have much higher ical temperature: Tc, F 191 K for CH4 , 508 K for C6 H14 , 540 K for critical pressures than those of the individual components. For C7 H16 , 618 K for C10 H22 , and 658 K for C12 H26 . However, this is mixtures of n-alkanes with O2 , this effect is easily recognizable as counterbalanced by an increasing supercriticality in pressure with = a promontory in the coexistence region of the phase diagram along increasing number of carbon atoms: Pc, F 46 bar for CH4 , 30 bar the pressure axis. For mixtures of n-alkanes with N2 or air, the for C6 H14 , 27 bar for C7 H16 , 21 bar for C10 H22 , and 18 bar for maximum pressure of the coexistence region diverges [21,75,76] . C12 H26 . Consequently, even at supercritical pressures with respect to both In contradistinction, the critical temperature of the coflow = components, the system can traverse the coexistence region when stream is much smaller: Tc 155 K for O2 , 126 K for N2 , and 131 K the injection temperatures of the two propellant streams are op- for air. As a result, the coflow stream often resembles a gas-like su- positely placed on either side of it. These incursions in the coexis- percritical fluid in propulsion systems operating at high pressures, tence region, which engender transcritical behavior in combustors, including (a) gas-turbine jet engines, where the air coflow entering ∼ are particularly likely when the injection temperature of at least the combustor is at TO 700–1000 K; (b) diesel engines, where the one of the propellant streams (typically the heavy hydrocarbon fuel enclosed air near the top-dead center attains similar temperatures; stream) is smaller than its critical temperature. and (c) thrust chambers of O2 -rich staged-combustion rocket en- ∼ An innermost zone is encountered within the coexistence re- gines, where the oxidizer stream is preheated to TO 600–700 K gion of the thermodynamic phase diagram that is bounded by the in a preburner. All these systems operate in the transcritical con- spinodal lines similarly to Fig. 4 for a pure component. In binary ditions discussed in Section 1.1 . systems, this zone can exist above the critical pressures of the separate components due to the elevation of the critical pressure 2.3. Transcriticality in aerospace propulsion systems of the mixture, as mentioned above. This zone is thermodynamically unstable, in that no stable thermodynamic state exists that Developments in aviation jet-propulsion technologies have been describes a spatially homogeneous mixture there. On the contrary, characterized over the last five decades by ever-growing overall in those conditions, the components tend to stay separated into pressure ratios (OPR) as a means of increasing thermal efficiency two fluids bounded by an interface where capillary forces become and reducing emissions of CO, CO2 , and unburnt hydrocarbons. important. This phenomenon is observed in experiments (e.g., see This trend is illustrated in Fig. 6 (a). The overall pressure ratio of Refs. [15–23] ), but not necessarily in computations unless an ap- a jet engine can be approximated as OPR ≈ P ∞ / P a , where P ∞ is the propriate formulation of the problem, such as the diffuse-interface combustor pressure and P a is the ambient pressure at the corre- theory developed in this review article, is employed that accounts sponding flight altitude. At sea level, P a = 1 bar and OPR is ap- for interfaces consistently with the thermodynamic stability of the proximately equal to P ∞ in bars. Modern engine designs reach mixture. OPR ∼ 50–60, which translate into fully supercritical combustor In bicomponent systems at high pressures, the fluids on each pressures P ∞ ∼ 50–60 bar at takeoff. At these pressures, the fuel is side of the interface, or on each side of the coexistence region injected as a supercritical liquid-like fluid into a supercritical gas- of the thermodynamic phase diagram, cease to be different ordi- like air coflow, thereby leading to the transcritical conditions de- nary phases such as liquid and vapor, and instead become distinct picted in Fig. 1 . Transcriticality in the combustor may persist up supercritical fluids whose composition depends on the particular to flight altitudes of 2.7 km for OPR = 50, and 4.2 km for OPR = 60, configuration. For instance, in the problem depicted in Fig. 1 , the above which the combustor pressure P ∞ becomes smaller than the two participating fluids are a liquid-like supercritical mixture rich critical pressure of air, P c = 36 bar. in a heavy hydrocarbon on the fuel side of the interface, and a Whereas diesel and gas-turbine jet engines are now mature gas-like supercritical mixture rich in coflow species (e.g., O2 , N2 , technologies that operate with air and derivatives of kerosene at or air) on the coflow side of the interface. Fundamental differences standardized injection temperatures, the propellant combinations between gas-like and liquid-like supercritical C12 H26 /N2 mixtures and injection conditions are much more numerous in rocket en- are analyzed in Appendix B.4 . gines due to big leaps in designs over the last 80 years, as sum- In the present theory, the transcritical interface is a physical- marized in Table 2 . Early developments in rocket propulsion dur- space representation of the coexistence region. References to VLE ing the V-2 program involved the utilization of a steam-catalyst- lines at high pressures should therefore be understood as the generator cycle at subcritical pressure P ∞ ∼ 15 bar [79] . Motivated loci of points where gas-like and liquid-like supercritical fluids – by the need of higher thrust to lift heavier payloads to orbit rather than classic vapor and liquid states – coexist. Pseudo-boiling and accomplish longer-range mission profiles in space exploration conditions are attained within the transcritical interface, these be- and military applications, the operating pressures in rocket en- ing understood in multicomponent systems as approximately the gines have increased significantly over recent decades, as indicated conditions where the constant-pressure molar heat of the mixture in Fig. 6 (b), and they have reached values much larger than the attains a local maximum, and where transition between liquid- critical pressures of the propellants. Typical pressures range from like and gas-like supercritical behavior takes place, as discussed in P ∞ ∼ 70 bar in the F-1 gas-generator rocket engine employed dur- Appendices B.2 and B.4 . ing the Apollo program, to P ∞ ∼ 250–350 bars in thrust cham- 12 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 6. (a) Overall pressure ratio (or equivalently, combustor pressure P ∞ in bars at sea-level takeoff) for gas-turbine jet engines as a function of time (adapted and extended with data beyond year 20 0 0 from the original version in Ref. [78] ). The different colors indicate manufacturers, including Rolls Royce (green), General Electric (red), CFM (blue), and Pratt & Whitney (yellow). The upper and lower dashed horizontal lines mark the OPRs required to reach supercritical combustor pressure with respect to both air

( P c, O = 36 bar) and Jet-A ( P c, F = 24 bar [41] ) during takeoff at sea level. (b) Pressures in gas generators, preburners, and main thrust chambers of rocket engines as a function of the production year. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

bers of modern staged-combustion-based rocket engines such as injection in the gas generator and thrust chamber. Much higher the RD-180 and Raptor, whose preburners operate at much higher pressures are achieved in RP-1-fueled closed cycles such as the pressures of about 50 0–60 0 bar. RD-170 and RD-180 engines. The preburners of these engines op- In rockets fueled by RP-1 (a highly reﬁned kerosene whose erate with RP-1 and LOX injected at very high combustor pres- critical point can be approximated as P c = 22 bar and T c = 677 K sures P ∞ > 500 bar but at cold near-storage temperatures, thereby

[42]), the fuel is injected at temperatures ranging from near- resembling supercritical liquid-like fluids. The O2 -rich hot com- ∼ storage values TF 280 K in gas generators and preburners, to bustion products from the preburner are expanded in the tur- ∼ ∼ TF 320 K in thrust chambers after passing through the wall cool- bine and injected at temperatures TO 600–700 K along with the ing channels. In RP-1-fueled open-cycle systems such as the F- cold RP-1 stream into the thrust chamber at pressures larger than 1 and Merlin engines, whose gas generators and thrust cham- 200 bar. The resulting transcritical conditions in the thrust chambers operate at 60–100 bar, the liquid oxidizer (LOX) is injected ber are analogous to those depicted in Fig. 1 , with RP-1 and ∼ at near-storage cryogenic temperatures TO 90–100 K. As a re- oxidizer entering as liquid-like and gas-like supercritical fluids, sult, the RP-1 and LOX resemble liquid-like supercritical fluids at respectively. L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 13

Table 2 Characteristic propellant injection conditions in rocket engines. Gas-generator rocket-engine cycles Gas Generator Thrust Chamber F-1 (Saturn-V ﬁrst stage) P ∞ = 62 bar P ∞ = 66 bar

[80,81] O 2 : T O = 94 K (supercritical liquid-like) O 2 : T O = 94 K (supercritical liquid-like) RP-1: T F = 305 K (supercritical liquid-like) RP-1: T F = 315 K (supercritical liquid-like) MC-1 P ∞ = 40 bar P ∞ = 44 bar

[82] O 2 : T O = 94 K (subcritical liquid) O 2 : T O = 94 K (subcritical liquid) RP-1: T F = 300 K (supercritical liquid-like) RP-1: T F = 300 K (supercritical liquid-like) STME (Space Transportation Main Engine) P ∞ = 152 bar P ∞ = 153 bar

[81] O 2 : T O = 101 K (supercritical liquid-like) O 2 : T O = 100 K (supercritical liquid-like) H 2 : T F = 43 K (supercritical gas-like) H 2 : T F = 110 K (supercritical gas-like) Merlin (Falcon-9 ﬁrst stage, approx.) P ∞ = 89 bar P ∞ = 100 bar

[83] O 2 : T O = 105 K (supercritical liquid-like) O 2 : T O = 105 K (supercritical liquid-like) RP-1: T F = 303 K (supercritical liquid-like) RP-1: T F = 313 K (supercritical liquid-like) Fuel-rich staged-combustion rocket-engine cycles Preburner Thrust Chamber SSME (Space Shuttle Main Engines) P ∞ = 382 bar P ∞ = 208 bar

[84,85] O 2 : T O = 118 K (supercritical liquid-like) O 2 : T O = 104 K (supercritical liquid-like) H 2 : T F = 154 K (supercritical gas-like) H 2 -rich products: T F = 810 K (supercritical gas-like) Oxidizer-rich staged-combustion rocket-engine cycles Preburner Thrust Chamber RD-170 (Energia-Buran ﬁrst stage) P ∞ = 563 bar P ∞ = 244 bar

[81] O 2 : T O = 101 K (supercritical liquid-like) O 2 -rich products: T O = 703 K (supercritical gas-like) RP-1: T F = 288 K (supercritical liquid-like) RP-1: T F = 414 K (supercritical liquid-like) RD-180 (Atlas-V ﬁrst stage) P ∞ = 537 bar P ∞ = 256 bar

[86] O 2 : T O = 118 K (supercritical liquid-like) O 2 -rich products: T O = 689 K (supercritical gas-like) RP-1: T F = 318 K (supercritical liquid-like) RP-1: T F = 480 K (supercritical liquid-like) Expander rocket-engine cycles RL-10 (upper stages of Atlas-V and Delta-IV) P ∞ = 33 bar

[87] O 2 : T O = 100 K (subcritical liquid) H 2 : T F = 200 K (supercritical gas-like) Vinci (Ariane-6 upper stage) P ∞ = 61 bar

[88] O 2 : T O = 94 K (supercritical liquid-like) H 2 : T F = 225 K (supercritical gas-like) Steam-catalyst-generator rocket engines V-2 P ∞ = 15 bar (subcritical)

[79] O 2 : T O = unavailable Water/alcohol mixture: T F = unavailable A-7 (Mercury-Redstone) P ∞ = 21 bar (subcritical)

[89] O 2 : T O = unavailable Water/alcohol mixture: T F = unavailable

In H2 -fueled rocket engines, the role of the fuel and coflow tures behave differently from mixtures of heavy hydrocarbons with streams is opposite to that shown in Fig. 1, in that the LOX O2 , in that the coexistence region in CH4 /O2 mixtures does not stream occupies the center core, whereas the H2 stream is the display a significant elevation above the critical pressures of the coflow [43,84,85,89] . As discussed in Section 3.2 , the thermody- separate components. As a result, the diffuse-interface theory pre- namic phase diagram of H2 /O2 mixtures is qualitatively similar to sented here predicts no interface formation in CH4 /O2 systems those of mixtures of n-alkanes with N2 or air. Since the H2 is at supercritical pressures with respect to both propellants above typically recirculated through the wall cooling channels, or is in- P ∞ ࣡ 50 bar. jected via fuel-rich combustion products exhausted from the preburner, its temperature is always much larger than the critical 2.4. Additional research aspects of transcritical systems value T c = 33 K, and often larger than the critical temperature of oxygen. Correspondingly, the H2 and LOX streams resemble, re- There exist several other aspects that may be of technological spectively, gas-like and liquid-like supercritical fluids at injection in and scientific relevance that have been intentionally left out of the thrust chambers of rocket engines such as the SSME or Vinci. The remainder of this paper because of their complexity: resulting transcritical conditions may involve supercriticalization of the oxidizer stream upon undergoing relatively small temperature (a) The autoxidation and pyrolysis of the heavy-hydrocarbon fuel as increments of order T c, O − T O ∼ 50 K by heat transfer from the hot it progressively heats up flowing downstream of the injection hydrogen stream. However, at elevated pressures, when the com- orifice in transcritical configurations similar to that depicted in bustor is ignited and the orifice ring is thick, results from exist- Fig. 1 . The breakdown of the fuel may cause variations in its ing numerical simulations suggest that the ensuing diffusion flame physical properties (including surface tension) that could influ- may be anchored so close to the orifice that it would rapidly super- ence atomization [90] . criticalize the oxidizer, thereby diminishing the practical relevance (b) The release of chemical heat as a result of combustion reactions of the transcritical region [25] . between fuel and oxidizer, which may contribute through con- Some of the most recent rocket engines currently under de- duction and radiation to prompt supercriticalization of the flow sign or testing, such as BE4 (oxidizer-rich staged-combustion cy- upstream and around flames [10,25,32] . cle, P ∞ ∼ 134 bar in thrust chamber) and Raptor (full-flow staged- (c) The role of the complex thermodynamic-state spaces of tertiary combustion cycle, P ∞ ∼ 300 bar in thrust chamber) operate with and higher-degree mixtures of heavy hydrocarbons with oxy-

CH4 fuel instead of RP-1. As shown in Section 3.2, CH4 /O2 mix- gen, nitrogen, and combustion products at high pressures, the 14 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

understanding of which remains mostly incomplete in the literature. (d) The transition from subcritical to supercritical combustor pressures during ignition sequences, including in-space ignition of upper stages of rocket space launchers or spacecraft reaction- control thrusters [91] . In particular, as observed in the phase diagrams introduced in Section 3 , hydrocarbon fuels such as < CH4 injected in subcritical gaseous state at TF Tc ,F early during engine ignition may undergo complex processes until fully supercritical pressures are attained in the combustor, including retrograde condensation when the combustor pressure becomes

similar to the phase-equilibrium pressure at TF .

3. Thermodynamic phase diagrams of mixtures fueled by heavy hydrocarbons

The physical-space solution of the diffuse-interface formulation developed in this review article has a direct correspondence to thermodynamic states in the phase diagram. In this section, quantitative descriptions of phase diagrams of relevant mixtures are

provided. Fig. 7. Iso-composition cross-sections of the phase-equilibrium surface projected on

the P − T plane for C 12 H 26 /N 2 mixtures, including (a) a qualitative description of = . , 3.1. Equation of state the diagram for dodecane mass fraction YF 0 5 along with (b) a more detailed diagram for multiple values of Y F . Experimentally measured critical points obtained from Ref. [75] are denoted by triangles in panel (a). The thermodynamic space of solutions for the state variables

ρ } –inconditions of coexistence –themultivalued character of the {P, , T, and Y1 , Y2 ...YN−1 of a single substance or a mixture of N

density (in monocomponent flows) and the fugacity (in multicom- components is described by its phase diagram. In this notation, Yi − = − N 1 ponent flows), as discussed in Section 4. Like many other mem- is the mass fraction of species i, with YN 1 = Yi . A first con- i 1 straint on the state variables that narrows down all of their possi- bers of the family of cubic equations of state, Eq. (14) provides

ble combinations is provided by an equation of state. One popular three values of the density for the same pressure P in the co-

choice for systems at high pressures, which is used in this study, existence region in monocomponent ﬂows. Correspondingly, every

is the Peng-Robinson equation of state [92] isotherm passing through the coexistence region contains a single oscillation or Maxwell loop, as shown in Fig. 4 . This behavior 0 R T a is consistent with thermodynamic stability theory, which predicts P = − , (14) v − b v 2 + 2 bv − b2 two neutral-stability points, namely the spinodal points. The latter can be unambiguously identiﬁed with the maximum and min- where R 0 is the universal gas constant, v = W /ρ is the molar vol-

ρ = imum of the looping isotherm. Several other cubic equations of ume of the mixture, is the density of the mixture, and W −1 N state, including the classic interpolating formula of van der Waals ( = Y /W ) = = X W is the mean molecular weight, with W i 1 i i i 1 i i i [51] , are available in the literature that have similar characteris- and X = W Y /W the corresponding values of the molecular weight i i i tics and have been ubiquitously employed for treating real-gas ef- and molar fraction of species i , respectively. The coefficients a and fects at high pressures [54,96–99] . Enhanced accuracy in the pre- b take into account real-gas effects related to attractive forces and diction of thermodynamic states outside the coexistence region can finite packing volume, respectively, and depend on the critical tem- be achieved by using multiparameter equations of state. However, peratures, critical pressures, and acentric factors of the individual these equations tend to contain multiple Maxwell loops within the mixture components, as well as on the local temperature and mix- coexistence region that lead to artificial intermediate pseudo-stable ture composition. In particular, a and b are obtained by first com- states [99] . Improved multi-parameter equations of state with de- puting the individual values of the coefficients for each species, a i pressed or no intermediate loops in the coexistence region have and b , as specified in Ref. [92] and also provided in Eqs. (C.10) – i been recently developed that may be amenable to coupling with (C.12) in Appendix C . The individual coefficients a and b are com- i i diffuse-interface formulations [100] . bined using the empirical van-der-Waals mixing rules

N N N 3.2. Critical-point elevation in hydrocarbon-fueled mixtures a = X i X j a i, j , a i, j = 1 − ϑ i, j a i a j , b = X i b i , (15) = = = i 1 j 1 i 1 This section focuses on numerical calculations of thermody- ϑ where i , j are binary-interaction parameters listed in Appendix C. namic phase diagrams of selected mixtures. Basic concepts of ther- Volume-translating corrections to the Peng-Robinson equation modynamic stability and phase equilibrium, including formal def- (14) exist that provide a higher accuracy of the density near the initions of the VLE lines, along with mechanical and diffusional critical point [93–95] , but are not considered in the present study critical lines for mixtures, are required to follow this discussion. since they tend to lead to a mismatch of the specific heat, which Those are provided in Appendix A . In particular, the calculations is an undesirable characteristic for describing the transfer of heat in this section employ Eq. (A.3) for the mechanical spinodals, across the transcritical interface. As a result, all supplementary Eqs. (A .3) and (A .4) for the mechanical critical line, Eq. (A .9) for thermodynamic relations written in Appendix C have been partic- the diffusional spinodals, Eqs. (A .9) –(A .11) for the diffusional crit- ularized for the standard version of the Peng-Robinson equation ical line, along with Eqs. (A .12) , (A .13) and (A .15) for the coexis- (14) . tence envelope, with the fugacity being defined in Eq. (B.21) from The particular choice of Eq. (14) as equation of state is not Appendix B and particularized for the Peng-Robinson equation of fundamental to the theory developed here. The only indispens- state in Eq. (C.47) from Appendix C . These constraints are supple- able characteristic of Eq. (14) is that it is capable of reproducing mented with the equation of state (14) , where the coefficients a L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 15

Fig. 8. Iso-composition cross-sections of the phase-equilibrium surface colored by pressure for (a) C 12 H 26 /N 2 mixtures, (b) C 12 H 26 /air mixtures, and (c) C 12 H 26 /O 2 mixtures. Pure-substance boiling lines for C 12 H 26 (dark dashed lines) and N 2 /O 2 /air (red dotted lines) are shown, along with their corresponding critical points (CP; squares) and the critical line of the mixture (green dotted lines). Experimentally measured critical points obtained from Ref. [75] are denoted by green triangles in panel (a). 2D projections of the critical line (dark dotted lines) and experimental data (dark triangles) on the T − Y F plane are also provided. Arrows indicate diverging behavior of isocomposition lines along the pressure axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 16 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 and b are computed using the mixing rules (15) in conjunction Table 3 with Eqs. (C.10) –(C.12) . Auxiliary quantities required for the cal- Pressure, temperature, and composition at diffusional critical points and critical mixing conditions in C H /N mixtures. culations include the critical pressures and temperatures listed in 12 26 2 Table 1 , along with the acentric and binary-interaction parameters Diffusional critical point Critical mixing conditions listed in Appendix C. Combustion chemical reactions are neglected P∞ [bar] Tc ,diff [K] YF c ,diff Tc ,mix [K] YF c ,mix here when constructing phase diagrams involving mixtures of fuel 20 658 0.99 658 0.99 and oxidizer because of the relatively low temperatures considered 50 650 0.93 651 0.94 within the visualization windows. Practical methods for the cal- 100 639 0.85 641 0.86 culation of VLE and diffusional critical lines are available in Refs. 200 615 0.72 620 0.76 [101–105] . Depending on the physical properties of the mixture components, binary mixtures can be classified in types that display vastly pressure is observed in the coexistence region up to approximately different forms of the phase diagram. Details on such classifica- 700 bar at ambient temperature. tion can be found in Ref. [106] , which employs the van der Waals Additional phase diagrams of relevant binary mixtures are pro- equation of state subject to phase-equilibrium and critical-point vided in Fig. 9(c–f). Mixtures of N2 with C7 H16 , or with a ma- constraints such as those in Appendix A (see also Ref. [107] for jor combustion product such as CO2 , along with mixtures of H2 a historical perspective on this subject). Specifically, the analysis with O2 or N2 , all lead to a very similar behavior to the one de- and associated simplifications carried out in Ref. [106], including scribed above for C12 H26 /N2 systems. However, the coexistence re- ≈ the assumption of equal excluded volumes bF bO (or equivalently, gion in H2 -fueled mixtures exists only for a narrow interval of ≈ Tc ,O /Tc ,F Pc ,O /Pc ,F ), pointed at the difference between the critical cryogenic temperatures, whose width decreases rapidly with in- | − | , temperatures of the two components, Tc, F Tc, O as an important creasing pressure. In contrast, mixtures of O2 with C7 H16 or CO2 factor in determining the structure of the coexistence region of the lead to bounded coexistence regions similar to that observed in mixture at high pressures. Several other factors may however par- Fig. 8(c) for C12 H26 /O2 mixtures. ticipate when more complex equations of state are used because of Important practical systems that do not feature any significant the additional parameters involved. In addition, in the mixtures fu- elevation of the critical point are mixtures of CH4 with O2 , and CH4 eled by heavy hydrocarbons examined here, the less volatile com- with air or N2 , which are relevant for new generations of rocket ponent (i.e., the heavy-hydrocarbon fuel) does not always have the engines (e.g., BE-4 and Raptor) and for land-based gas turbines, re- largest critical pressure. Despite these shortcomings, some of the spectively. In those mixtures, the pressure of the diffusional crit- qualitative observations made in Ref. [106] , including the occur- ical line is never too far above the critical pressures of the sepa- rence of coexistence regions that are unbounded in pressure, hold rate components. As a result, those systems are mostly dominated true in the present analysis. by interfaceless, homogeneously mixed states when the pressure is

In C12 H26 /N2 mixtures described by the Peng-Robinson equa- supercritical with respect to both components ( P ࣡ 50 bar). tion of state (14) and the mixing rules (15), the phase diagram does Constant-pressure cross-sections of the P − T − Y F phase dia- not contain a continuous critical line joining the critical points of gram are provided in Fig. 10 for C12 H26 /N2 mixtures. As the pres- the individual components, and the coexistence region becomes sure increases, Fig. 10 indicates that the mixture composition and unbounded in pressure. This phenomenon is shown in the thermo- temperature corresponding to the diffusional critical point be- dynamic phase diagram in Fig. 7, where isocomposition projections comes fuel-leaner and colder, respectively. For C12 H26 /N2 mixtures, of the phase-equilibrium surface on the P − T plane are provided. and within the interval of pressures reported in Fig. 10 , the tem-

A three-dimensional visualization of the elevation of the critical perature of the diffusional critical point Tc ,diff does not differ much − ∼ point in the thermodynamic space formed by P, T, and YF is shown from the fuel critical temperature Tc ,F , with Tc, F Tc, mix 43 K be- in Fig. 8(a) also for C12 H26 /N2 mixtures. There, the coexistence re- ing the largest difference observed at 200 bar. Similarly, Tc ,diff is gion is enclosed under the surface enveloping the isocomposition approximately equal to the critical mixing temperature Tc ,mix corre- curves and reaches pressures of order 1500 bar at ambient temper- sponding to the maximum temperature attained along the VLE line − ature. at a given pressure. Table 3 shows that the difference Tc, mix Tc, diff The computation of critical points in complex mixtures involves increases as the pressure increases, although the maximum value a number of assumptions and model-parameter values that ﬁnd is only 5 K at 200 bar. The system may sample the coexistence re- ≤ ≤ no clear justiﬁcation on physical grounds. For instance, the mixing gion within the very small gap of temperatures Tc ,diff T Tc ,mix , rules (15) correspond to molar weighting of the individual coef- but the thermodynamic states there are metastable, and for all

ficients ai and bi , whose expressions depend on calibrated inter- practical purposes, they can be considered as fully supercritical. action parameters and measured critical points of the pure sub- Although no equilibrium trajectories connecting the two injec- stances (see Appendix C ). It is however remarkable that the critical tion states of the propellants exist in general flows, it is convenient line calculated in Figs. 7 and 8 (a) agrees well with values experi- to locate along a trajectory on the phase diagram the local thermo- mentally obtained in Ref. [75] , including the divergent trend of the dynamic state of points normal to the transcritical interface in the critical pressure. configurations that will be addressed in Sections 6 and 7 . In this

Since N2 is a major component of air, both C12 H26 /N2 and study, these trajectories are referred to as pseudo-trajectories to

C12 H26 /air mixtures lead to similar thermodynamic behavior, as emphasize that they are not thermodynamic trajectories followed shown in Fig. 8 (b). Note that the mechanical critical point of by the system by quasi-statically changing its state, as traditionally air provided in Table 1 is very similar to its diffusional critical meant in the theory of thermodynamics of closed systems. A no- point (P c = 37 bar, T c = 131 K ) obtained in Fig. 9 (a) using Eqs. tional envelope of transcritical pseudo-trajectories is provided in

(A.9) and (A.11). This is because the coexistence envelope of N2 /O2 Fig. 10 that joins the characteristic injection-temperature ranges ∼ ∼ mixtures does not exceed the critical pressures of the separate TF 300–450 K and TO 700–1000 K of the propellant streams. components. It will be shown in Sections 6 and 7 that transcritical pseudo-

Mixtures of heavy hydrocarbons with O2 do not generally trajectories are characterized by crossing the coexistence region of lead to coexistence envelopes unbounded in pressure. However, a the thermodynamic phase diagram of the mixture. Note, however, bounded elevation of the critical point still occurs in those mix- that any pseudo-trajectory plotted on constant-pressure cross- tures, as shown in Figs. 8 (c) and 9 (b), where a promontory in sections of the thermodynamic phase diagram, such as those in L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 17

Fig. 9. Iso-composition cross-sections of the phase-equilibrium surface projected on the P − T plane for (a) N 2 /O 2 mixtures, (b) C 12 H 26 /O 2 mixtures, (c) C 7 H 16 mixed with O 2 (left) or N 2 (right), (d) CO 2 mixed with O 2 (left) or N 2 (right), (e) CH 4 mixed with O 2 (left) or N 2 (right), and (f) H 2 mixed with O 2 (left) or N 2 (right). The diagrams include diffusional critical lines (green-dotted lines), and pure-substance VLE lines (dark dashed and red-dotted lines) with their corresponding critical points (CP; squares). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 10 , should be interpreted with caution, since signiﬁcant vari- vided in Fig. 11 for two different conﬁgurations: (a) one in which

ations of the thermodynamic pressure occur across the interface the fuel is cold and the coflow (N2 or O2 ) is hot (a configura- away from the edge, as discussed in Sections 4 and 5 . tion relevant for propulsion systems fueled by heavy hydrocarbons The considerations above suggest that the selection of particu- as in Fig. 1 ), and (b) one in which the fuel is hot and the oxi- lar values of injection temperatures and combustor pressures may dizer is cold (a configuration relevant for rocket engines fueled by

lead to operating conditions without transcritical interfaces. For in- hydrogen). Speciﬁcally, Fig. 11 provides Tc ,diff as a function of P∞

stance, if the injection temperature of C12 H26 in Fig. 10 is increased for different mixtures. The point of intersection between the diffu-

to values larger than Tc ,diff while maintaining constant the hot sional critical temperature and the horizontal line corresponding to temperature of the nitrogen coflow, the pseudo-trajectories will the chosen value of the injection temperature of the cold propel- not cross the coexistence region, and the flow will not bear any lant stream defines the combustor pressure above which interfaces transcritical interfaces. This circumvention of the coexistence re- cannot exist because the pseudo-trajectories of the system will cir- gion may be achieved dynamically because of supercriticalization cumvent the coexistence region. For instance, considering an injec- = of the fuel stream by heat transfer from the coflow, as shown in tion temperature TF 300 K for a fuel stream of C12 H26 or C7 H16

Fig. 2 and discussed in Section 7. into a coﬂow of N2 or O2 , the limiting pressures that lead to ab- ࣃ ࣃ The limiting pressure above which interfaces cannot exist at a sence of interfaces are P∞ 1500 bar (for C12 H26 /N2 ), P∞ 690 bar ࣃ ࣃ given injection temperature of the cold propellant stream is pro- (for C12 H26 /O2 ), P∞ 820 bar (for C7 H16 /N2 ), and P∞ 410 bar (for 18 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 10. Constant-pressure T − Y F cross-sections of the phase diagram, including diffusional critical points (diamond symbols with coordinates) and vapor-equilibrium lines (thick red dashed lines) bounding the coexistence region of C 12 H 26 /N 2 mixtures at different pressures, along with the corresponding diffusional spinodal lines (thin purple dashed lines) and the loci of the maximum of the molar heat at constant pressure (thick green dot-dashed line) at each pressure. The mechanical critical point (solid circle) and the mechanical spinodal lines (thin solid lines) are provided only for the lowest pressure case. For pressures larger than 30 bar, the mechanically unstable region lies at

much lower temperatures than those shown in the ﬁgure. Solid ellipses denote characteristic injection conditions of N 2 (left) and C 12 H 26 (right), while shaded regions are notional envelopes of pseudo-trajectories joining thermodynamic states along the normal to the interface. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

= C7 H16 /O2 ). Similarly, for a cryogenic O2 stream at TO 110 K sur- tence region or will traverse it across its largest pressure values,

rounded by a hot coaxial stream of H2 , the corresponding limiting where the density gradients are small.

pressure above which the ﬂow cannot bear transcritical interfaces (c) Increasing the temperature of the fuel stream: The higher TF , is P ∞ ࣃ 290 bar. the less likely the pseudo-trajectories will cross the coexistence region.

3.3. Conditions suppressing transcritical behavior These observations have been ratified by recent experiments [18,19] . The combustor pressure and injection temperatures can be var- ied in the following manner in order to suppress transcritical in- 4. Theoretical foundations of the diffuse-interface theory for terfaces in the flow: transcritical flows

(a) Increasing the combustor pressure: The higher P ∞ , the smaller Crossings of the thermodynamically unstable regions in

Tc ,mix and Tc ,diff are, and the narrower the coexistence region is Fig. 10 require modiﬁcations of the constitutive laws in the Navier- along the temperature and composition axes of the thermody- Stokes equations. This section introduces basic assumptions, pa- namic phase diagram. rameters, and scalings of the diffuse-interface theory, and outlines

(b) Increasing the temperature of the coﬂow stream: The higher TO , a set of conservation equations that serve as groundwork for the the more likely pseudo-trajectories will circumvent the coexis- derivation of constitutive laws in Section 5 . L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 19

sults in a gradient-dependent free energy [108] . This is represented by a correction to the bulk free energy that involves a coeﬃcient multiplied by the square of the density gradient [111] . The equivalent description of thermodynamic potentials in multicomponent systems is based on the gradients of the partial densities [116,117] . For instance, the speciﬁc values of the gradient- dependent Helmholtz free energy f , internal energy e , and entropy s are [38,39,110]

N N 1 f = f + κ , ∇(ρY ) · ∇(ρY ) , (16) GD 2 ρ i j i j i =1 j=1

N N 1 = + κ e ∇(ρ ) · ∇(ρ ) , eGD e Y Y (17) 2 ρ i, j i j i =1 j=1

N N 1 Fig. 11. Combustor pressures and injection temperatures of the cold propellant = + κ s ∇(ρ ) · ∇(ρ ) , sGD s i, j Yi Yj (18) stream leading to absence of interfaces. The lines denote the diffusional critical 2 ρ i =1 j=1 temperature Tc ,diff of the corresponding mixture. For a given value of TF (in configuration a ) or TO (in configuration b ), the intersection with the diffusional critical where f is defined as temperature provides the limiting combustor pressure P ∞ above which interfaces cannot exist. f = e − T s. (19)

κ κe , κs In this notation, N is the number of species, and i , j , i, j and i, j are gradient-energy coeﬃcients discussed in Section 4.2 . In this

4.1. Gradient-dependent thermodynamic potentials formulation, f , e and s are evaluated at local conditions { P , ρ, T , } and Y1 , Y2 ...YN−1 within the interface. The diffuse-interface theory was first formulated by van der Waals in Ref. [108] , where the finite-thickness structure of an in- 4.2. Gradient-energy coefficients terface between liquid and vapor phases in isothermal hydrostatic monocomponent systems was described by performing spatial in- Expressions relating κe , κs and κ can be easily derived by tegrations of the intermolecular potential in the presence of a i, j i, j i , j mean density gradient. The reader is referred to seminal treatises substituting Eqs. (16)–(18) into Eq. (19), which gives based on the molecular theory of liquids for in-depth expositions ∂κ ∂κ of the topic [53] . This formalism was later extended to study bi- e i, j s i, j κ = κ , + T and κ , = − , i, j i j ∂ i j ∂ nary mixtures near the critical point by Cahn and Hilliard [109] . T ρ, T ρ, Yk | k =1 , ... N Yk | k =1 , ... N More recently, notable work has been done to couple consistently (20) the diffuse-interface theory of van der Waals with conservation equations of fluid motion [110–115] . Developments of the diffuse- where the definition s = −(∂ f /∂ T ) ρ, | = , ... − has been used. The Yi i 1 N 1 interface theory for treating the mechanics of interfaces in complex determination of the gradient-energy coefficients, which are di- mixtures have been applied mostly to 1D phase-transitioning sys- rectly related to the interface thickness and surface tension, consti- tems in hydrostatic equilibrium at uniform temperature to provide tutes a major source of uncertainty in the formulation, since they predictions of the surface-tension coefficients [116–118] . The inves- either require knowledge of the underlying molecular structure of tigation of hydrodynamic and thermal effects in the formulation is the fluid, or must be tuned by comparing theoretical results with a discipline that is currently in its infancy. experimental data. The diffuse-interface theory of van der Waals addresses mono- Models for the gradient-energy coefficients often rely on corre- component thermodynamic systems traversing the coexistence relations such as gion as they become macroscopically separated into phases. Specif- / κ 2 2 3 i,i Wi NA ically, the diffuse-interface theory describes the mechanics of the = G(θ ) . (21) 2 / 3 i transition layer, which gives rise to familiar surface-tension forces a i b i emerging from the resistance of the interface to get deformed. Al- In this formulation, Tc , i is the critical temperature of species i, though the predicted interfaces are hydrodynamically thin, and the NA is the Avogadro’s number, and ai and bi correspond to coeffi- theory is formulated at the continuum level, the results agree well G(θ ) cients of the equation of state. The function i depends on the with experiments and molecular dynamics simulations [99,116– θ = − / reduced temperature i 1 T Tc,i and is calibrated from exper-

119]. imental data. One class of experiments used for calibration con-

On average, molecules in ideal gases are separated by long dis- sists in enclosing a monocomponent fluid in an equilibrium cell tances that make the intermolecular forces negligible. In contrast, [e.g., see Fig. 5 (a)] and measuring its surface-tension coefficient σ the increased molecular packing at high pressures requires con- [15,72,116–118] . As discussed in Section 5.3 (see also Ref. [117] ), the sideration of intermolecular forces, which are manifested by the diffuse-interface theory for monocomponent flows predicts that σ second term in the equation of state (14) involving the coefficient is given by a . The latter is proportional to the zeroth-moment of the intermolecular potential in the far-field attraction range [53] . In highly ∞ ∂ρ 2 σ = κ , density-stratified conditions as those found across interfaces in dnˆ (22) −∞ ∂ nˆ monocomponent systems, the effect of long-range attraction forces is compounded by the fact that the integration of the spherico- where nˆ is a spatial coordinate locally normal to the interface. Val- symmetric intermolecular potential across the stratified zone re- ues of κ as a function of temperature can be inferred from Eq. 20 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

4.3. Open issues of experimentally calibrated models for gradient-energy coeﬃcients

Gradient-energy coefficients computed from experimentally calibrated relations such as Eq. (23) are contrived by the manner that experiments are carried out. As the equilibrium cell is heated, and starting from a phase-separated state in vapor-liquid equilibrium, both P and T reach their critical values simultaneously because of vapor-liquid equilibrium. As a result, the pressure dependency on Eq. (23) , and similar ones encountered in the literature, is artificially concealed in the temperature. Relevant steps toward improved models of gradient-energy coefficients have been recently κ undertaken by Dahms [100] using parametrizations of i , j in terms of the local partial-density gradients instead of temperature. Contrary to theoretical predictions [53,120] , which suggest that the gradient-energy coefficient diverges near the critical point of the component, there are experimentally calibrated models such as Eq. (23) that remain finite there, as observed in Fig. 12 . How- ever, the singularity predicted by the theory is weak and in practice is overruled by the rapid decrease in the density gradients on approach to the critical point, where both σ and d σ / dT vanish. Additional complications arising from the utilization of Eq. (23) or analogous ones [118] is that the experimentally calibrated Fig. 12. Gradient-energy coefficients κ i , i for monocomponent systems composed of > (a) C 12 H 26 and (b) N 2 , as obtained by using expression (23) in conjunction with the gradient-energy coefficients are not defined for T Tc , i . Recall that equation of state (14) . a monocomponent system cannot engender any surface tension for T > T c , since, as indicated in Fig. 4 , only a single homogeneous thermodynamic state is possible above the critical point. As a re- (22) by using measurements of σ in the equilibrium cell. The result, the calibration of κ must necessarily stop at T > T . In the G(θ ) i,i c,i sulting function i can be represented as [120] > > > transcritical conditions in Fig. 1, in which TO Tc ,F TF Tc ,O , the 2 / 3 κ 2 gradient-energy coefficients given by Eq. (23) are not defined in i,i Wi NA T = A 1 − + B . (23) 2 / 3 i i regions of the flow where the local temperature is larger than the a b Tc,i i i critical temperature of the corresponding component. This has the A B The dimensionless coefficients i and i are provided in Eqs. following implications: (C.17) and (C.18) (see also Refs. [116,120] ). The resulting depen- ∼ dency of the gradient-energy coefficient on temperature is shown (a) Since the injection temperatures considered here are TF 300– ∼ 450 K and TO 700–1000 K, the temperature is everywhere in Fig. 12 for C12 H26 and N2 . Validations of the model (23) are pro-

vided in Section 6 and Appendix D . larger than the critical temperatures of typical coﬂowing

Three approximations in the gradient-energy coeﬃcients are species, including N2 , O2 , and air. Consequently, the only

made in the theory developed here: gradient-energy coefficient that is defined and explicitly appears in the formulation is that of the fuel, denoted here by κ (a) For simplicity, the gradient dependency of s is neglected in F,F . κ comparison with similar long-range effects on f and e, or equiv- (b) The fuel gradient-energy coefficient F,F ceases to be defined on alently > the coflow side of the interface where T Tc ,F . However, these limitations do not have significant consequences ∂κi, j T κ , , (24) ∂ i j on the theory developed here. As discussed in Section 1, the in- T ρ, Yk | k =1 , ... N terface vanishes at an edge when its temperature nears the tem- − κe ≈ κ | κs | κ perature of the diffusional critical point Tc ,diff, which is 10 40 K which gives i, j i, j and T i, j i, j . = (b) The long-range interaction between components in mixtures, smaller than Tc ,F in the pressure range P∞ 50–200 bar. Corre- κ > κ = spondingly, regions where F,F is not defined because T Tc ,F are measured by the cross-influence parameters i , j (i j) are mod-

eled as located either far away from the interface on the coflow side, or downstream of the interface edge, as schematically indicated in κ = ( − υ ) κ κ = κ , i, j 1 i, j i,i j, j j,i (25) Fig. 13 . Since composition gradients in those regions are small compared to those encountered across the interface, the gradient- with υ , = 0 chosen for simplicity [117] , although no rigorous i j dependent corrections of the local thermodynamic potentials are justification for such geometric averaging appears to exist yet dynamically irrelevant there regardless of whether κ is made to other than experimental observations of indirect quantities that F,F artificially plunge to zero at T = T , , or is extrapolated for T > T . ratify this choice [121] . c F c ,F From a molecular-scale perspective, κ can be shown to be pro- (c) As shown in Section 5.6 , the largest variations of the tem- i , j portional to the second moment of the intermolecular potential perature occur across distances much larger than the interface (e.g., see Ch. 4 in Ref. [53] for details). It can also be expressed thickness. Correspondingly, it is assumed herein that κ varies i,j in terms of the direct correlation function in both single- and mul- slowly in space around the interface, and therefore commutes ticomponent systems [122,123] . It is conceivable that future devel- with spatial derivatives. Temperature-induced variations in κ i,j opments in hydrodynamic applications of the diffuse-interface the- are included explicitly by evaluating κ at the local tempera- i,j ory for high-pressure interfaces would benefit from models of κ ture in all expressions. i , j that could be cognizant of the microscale structure of supercritical These approximations were also implicitly used in the original fluids and their mixtures. Multiscale modeling strategies based on diffuse-interface theory of van der Waals [108] . combination of molecular dynamics and continuum-level theories L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 21

Fig. 14. Schematics of the thermodynamic-to-physical space mapping (31) built in the diffuse-interface theory for transforming the variations of the chemical potential into a sharp monotonic density proﬁle in a monocomponent isothermal system.

= which can be combined with Eq. (C.7) giving Fig. 13. Schematics of the region enclosed by the fuel critical isotherm T Tc, F (gold colored zone), where the fuel gradient-energy coefficient κ is defined ac- F,F ∂ cording to the model (23) . μ = (ρ ) . ∂ρ f (30) T or simulations may bear some potential for on-the-fly calculations By substituting P from Eq. (14) into Eq. (29) and integrating the re- κ sulting equation, an expression for f can be found and be later sub- of i , j based on direct correlation functions [124,125]. stituted into Eq. (30) to obtain one for μ. The resulting expressions μ 4.4. Characteristic scales in the diffuse-interface theory for f and are not essential here and are deferred to Appendix C. Equations (27) –(30) are standard expressions in thermodynam-

δ ics that are useful here for the following reason. Since the pres- The disparity between the interface thickness I and the macro- ρ δ sure P is an oscillatory function of across the coexistence re- scopic flow scales T , RI , and RF plays a central role in the dynam- μ δ gion of the thermodynamic phase diagram (e.g., see Fig. 4), f and ics of the interface. The value of I is closely connected with the ρ μ gradient-energy coefficient and with the surface tension σ . In par- are also oscillatory functions of there. In particular, satisfies a Maxwell’s construction rule [126] , in that the oscillation of μ − μ ticular, Eq. (22) indicates that the gradient-energy coefficient scales ρ e F in the coexistence region encloses zero net area, ρ (μ − μe ) dρ = as O , 0 as observed by using Eqs. (30) and (C.7). κ ∼ σ δ /ρ2, 0 I F (26) A fundamental result of the diffuse-interface theory for mono- σ ρ component isothermal systems is the equation where 0 is a characteristic value of the surface tension, and F is the density of the fuel stream. In performing the estimate (26), ∂ 2 ρ the characteristic density difference across the interface has been μ − μe = κ , (31) ∂ nˆ 2 approximated as ρF (1 − 1 / R ) ≈ ρF , with R 1 the density ratio defined in Eq. (4) . which relates μ, ρ, and the normal coordinate to the interface nˆ . To understand the role of the gradient-energy coefficient in the The derivation of Eq. (31) is deferred to Section 5.3 . It will also present theory, it is important to note that any thermodynamic be shown in Section 5.6 that an equation similar to (31) exists for variable that oscillates across the coexistence region in the phase multicomponent isothermal systems. The boundary conditions of diagram can be mapped into physical space within an interface- Eq. (31) are ρ = ρF at nˆ → −∞ and ρ = ρO at nˆ → + ∞ . The right- like profile of an indicator function such as ρ( nˆ ) in monocompo- hand side of Eq. (31) vanishes far away on both sides of the inter- nent flows, or the partial densities in multicomponent flows. Al- face, where the density gradients are negligible and the chemical though the details of the mapping depend on the particular prob- potential attains the phase-equilibrium value μe . lem and thermodynamic quantity under consideration, it is illus- As illustrated in Fig. 14 , the expression (31) is responsible trative to momentarily sideline the problem in Fig. 1 and focus on for mapping the oscillations of the chemical potential in the the most idealized case that can be addressed with the diffuse- two-phase region of the thermodynamic space μ = μ(ρ) into an interface theory, namely that of a flat interface in a monocompo- interface-like profile ρ = ρ( nˆ ) , in which ρ undergoes a rapid vari- ρ δ nent isothermal flow. In this case, the mapping is simpler and the ation of order F within a very short distance of order I . To obtain free-stream densities correspond to those in phase equilibrium, i.e., an estimate of δ , Eq. (31) can be rewritten in terms of a nondi- I ρ = ρ ρ = ρg F e and O e . mensional density

As discussed in Appendix A.2, in phase equilibrium, the chemi- ρ = ρ/ρ cal potentials in both streams are the same and equal to the equi- F (32) μ librium value e [see Eq. (A.16)]. Similarly, since the temperatures and a nondimensional chemical potential μ = (μ − μe ) /A μ, with of the streams must be equal in phase equilibrium, Eqs. (C.4) and A μ the characteristic amplitude of the oscillation of μ in the co- (C.5) simplify, respectively, to existence region. In these variables, the first integral of Eq. (31) is dμ = dP/ρ (27) ρ 2 κρF ∂ρ and μ dρ = . (33) 1 / R A μ ∂ nˆ df = −P d(1 /ρ) . (28) Equation (33) evaluated far away from the interface, where In particular, Eq. (28) provides the definition of the pressure in ρ → 1 and ∂ ρ /∂ nˆ → 0 , is consistent with Maxwell’s construction κ ρ terms of the free energy rule. The parameters , F , and Aμ can be scaled out of Eq. (33) by defining a dimensionless normal coordinate nˆ = nˆ/ δI , with ∂ f = ρ2 , P (29) 1 / 2 ∂ρ δI = (κρF /A μ) . (34) T 22 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

The combination of Eqs. (26) and (34) yields In this formulation, dF and dO are the equivalent hard-sphere di- ameters of the fuel and coﬂow species, respectively, with d FO = A μ ∼ σ0 / (δI ρF ) . (35) (d + d ) / 2 the mean diameter (i.e., d = d = 0 . 712 nm, d = F O F C12 H26 O In particular, Eq. (35) connects the oscillation amplitude A μ, which d = 0 . 362 nm, and d , = 0 . 537 nm for C H /N mixtures N2 F O 12 26 2 can be obtained independently from the thermodynamic relations [130] ). δ (14) and (30), with the intrinsic properties of the interface I and The direct application of Eq. (41) to the present problem σ 0 , the latter being a measurable quantity. A cornerstone of the is problematic because of the strong composition gradients as- κ diffuse-interface theory is that the order of magnitude of is set sociated with transcritical interfaces. In the calculations of KnI δ such that I estimated from (34) is small enough to yield values of shown in Section 6 , the partial densities in Eq. (41) are ap- σ ρ ∼ 0 ρ( − ) ∼ ( − 0 from Eq. (22) comparable to the experimentally observed ones proximated as YF XF,max P∞ WF /(ZF,max R Te ) and 1 YF 1 − − σ ∼ δ ∼ κ ∼ 15 17 ) / ( 0 ) , 0 0.001–0.1 N/m. This gives I 1–100 nm and 10 –10 XF , max P∞ WO ZO , max R Te where Te is a characteristic interface

Jm 5/kg 2 (see Fig. 12 ). temperature, and the subindex max denotes quantities evaluated The characteristic amplitude of the pressure oscillation within at the location of maximum density gradient across the interface. the interface, denoted by A P , is obtained from Eqs. (29) , (30) , and With these approximations in mind, Eq. (41) becomes (35) giving k T ≈ B e , ∼ ρ ∼ σ /δ . √ (42) AP F Aμ 0 I (36) 2 2 ( − , ) + / XF , max dF 2 1 XF max dFO 1 WF WO π + Equation (36) yields A ∼ 10–100 bar when the estimates for σ P∞ P 0 Z F , max Z O , max δ and I introduced above are employed. A dimensionless pressure parameter P can be deﬁned as the ratio of the characteristic pres- where k B is the Boltzmann constant. For the monocomponent sys- sure variation A across the interface to the combustor pressure P tems calculated in Appendix D , Eq. (42) is used with X F , max = 1 , P ∞ , Z F , max = Z max , and d F = d. σ Small Knudsen numbers (40) are often found in monocompo- P = 0 , (37) δ δI P ∞ nent systems as the critical point is neared, when I increases which is related by Eq. (36) to A μ as and decreases. For instance, the interface thickness calculated in Appendix D for a monocomponent N system at P ∞ = 0 . 9 P , ≈ 2 c N2 A μW δ ∼ P ∼ F , 31 bar is I 5 nm. This translates into approximately 32 N2 (38) 0 ∼ ZF R TF molecules across, along with a Knudsen number KnI 0.05 (see where Fig. D.3 in Appendix D ). However, it is widely accepted that the requirement KnI 1 is not always satisﬁed by the diffuse-interface P ∞ W F Z = (39) F 0 theory of van der Waals as the pressure becomes increasingly ρF R T F smaller than P c [53,131,132] . This can be seen in the same mono- is the compressibility factor. The dimensionless pressure parameter component N system mentioned above at P ∞ = 0 . 4 P , ≈ 14 bar, 2 c N2 P δ ∼ ∼ increases with decreasing pressures. At near-atmospheric pres- for which I 2 nm and KnI 0.15, thereby indicating a larger rar- sures, P is much larger than unity, and the pressure oscillation efaction and a weaker foundation for the continuum character of predicted by the equation of state (14) can induce large negative the theory. Despite these shortfalls, the diffuse-interface theory of pressures oftentimes observed in the form of tensile strength in van der Waals tends to reproduce well numerical results obtained liquids brought to metastable states by superheating them above from molecular dynamics (e.g., see Fig. 6 in Ref. [99] ). the boiling temperature, or by stretching them below the vapor The considerations above indicate that the continuum assump- σ δ pressure [104,127,128]. As P∞ increases, 0 decreases and I in- tion across the interface is often on the borderline of invalidity. creases, thereby rendering smaller values of A P and P. In the hydrocarbon-fueled mixtures analyzed in Figs. 7–9 (b–e), the

The numerical results presented in Sections 6 and 7 indicate pressures that can be reached before KnI becomes much smaller that the dimensionless pressure parameter P is of order unity in than unity and are much larger than those in monocomponent sys- transcritical conditions. As a result, despite the large values of P ∞ tems because of the elevation of the critical point caused by mix- involved, the present theory predicts that mechanically subcritical ing. It is however remarkable that no excessively large values of pressures, and even negative pressures, may occur within the in- KnI that would indicate a flagrant violation of the continuum as- terface. These aspects are further discussed in Appendix E as they sumption are encountered at the high pressures studied here. Field may appear controversial in the description, although they do not formulations of fluid motion are well-known for performing cor- have any significant effect on the results presented here. rectly even in flows where the continuum hypothesis may not be strictly satisfied [129] . The continuum treatment of diffuse inter- 4.5. The interface-continuum hypothesis faces is reminiscent of the description of the internal structure of weak shock waves by using the Navier–Stokes equations despite The approximation of operating with a continuous field across the small shock thicknesses involved [133] . However, one impor- the transcritical interface while keeping only the gradient terms in tant difference between shocks and interfaces is that the dynam- the series expansion of the thermodynamic potentials, as in Eqs. ics outside the shock discontinuity are independent of its internal δ (16)–(18), is justified when the interface thickness I is large com- structure, whereas the transport across the interface and its me- pared to the characteristic mean free path , or equivalently, when chanical coupling with the flow fundamentally depend on the in- the Knudsen number ternal structure of the interface, as discussed in Section 5 . The appropriateness of Eq. (41) as the relevant length scale to Kn I = /δI (40) evaluate the continuum character of transcritical interfaces is also is much smaller than unity. In multicomponent mixtures described under suspicion. Recent molecular dynamics simulations [134] in- by the hard-sphere model, is given by [129] dicate that confined gases undergo higher rates of collisions than 1 those predicted by Eq. (41) . It could be plausible that an interface = . √ (41) 2 may lead to an effect similar to confinement for the fluids on each Y d 2 2 (1 − Y ) d 1 + W /W F F F FO F O side and within the interface itself. Similarly, a recent study by ρπN A + W F W O Dahms [135] has found that the effective mean free path near the L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 23

Fig. 15. Constant-pressure distributions of the effective Fickian diffusivity of dodecane in C 12 H 26 /N 2 mixtures at (a) P = 50 bar and (b) 100 bar as a function of temperature and mass fraction of dodecane Y F (refer to the legend in the left panel). The plots include the diffusional critical point of the mixture at the corresponding pressure (purple diamond symbol) and the diffusional spinodals (thick purple dashed lines). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) interface should be 0.55 times smaller than the classic deﬁnition in Eq. (41) . Whether interfaces at high pressures fully satisfy the continuum requirement KnI 1 remains a subject open to debate.

4.6. The role of antidiffusion in transcritical flows Fig. 16. Schematics of diffusional processes across a notional transcritical interface An important characteristic of the mass transport across tran- separating fuel and coflow streams at high pressures. The sketch includes (a) diffu- scritical interfaces at sufficiently high pressures is the prevailing sional processes in physical space, and (b) associated evolution of the fugacity of the fuel species f in thermodynamic space. Note that the fugacity profile in panel (b) antidiffusion of matter that is predicted by the standard transport F does not necessarily correspond to one at uniform pressure such as those presented theory in conjunction with the equation of state (14). To under- in Fig. A.1 (a), since the pressure generally varies within the coexistence region. stand this, notice that the effective Fickian diffusion coefficient participating in the standard species diffusion flux of fuel at high pressures is not the binary diffusion coefficient D F , O but [136,137] composition gradients because of the diffusional instability of the mixture, as sketched in Fig. 16 . The antidiffusion of matter pre- ∂ ln fF dicted by the standard transport theory, which would lead to un- D F , F = D F , O , (43) ∂ ln X physical results, can be regularized by a an extra diffusion flux de- F P,T rived from the diffuse-interface theory, as described in Section 5 . where fF is the fuel fugacity defined in Eq. (B.21) as a function of the fugacity coefficient (B.22) , the latter being particularized in 4.7. Conservation equations for transcritical flows Eq. (C.47) for the Peng-Robinson equation of state (14) . A formal derivation of Eq. (43) is deferred to Section 5 . The consideration of gradient-dependent thermodynamic po- D , In contrast to F , O which is always positive (see Fig. B.3 and tentials, as in Eqs. (16) –(17) , leads to interface-related transport related discussion in Appendix B.4 ), the effective Fickian diffusion fluxes and mechanical stresses in the conservation equations. In coefficient DF,F can be negative for intermediate compositions at this study, a phenomenological approach described in Section 5 is temperatures smaller than Tc ,diff, as shown in Fig. 15 for C12 H26 /N2 followed based on linear augmentations of the deviatoric part of mixtures. The change in sign can be explained by the fact that the the stress tensor τ, and the heat and species diffusion fluxes q and fuel fugacity oscillates across the coexistence region (and therefore K Q Ji , with the interfacial stress tensor and transport fluxes and across the interface) similarly to the fuel chemical potential, as dis- J K, Q, J i . Expressions for and i are derived in Section 5. In terms cussed in Appendix A.1 and illustrated in Fig. A.1(a,b) for C12 H26 /N2 of the material-derivative operator D/Dt = ∂ /∂ t + v · ∇, the result- mixtures. ing conservation equations for mass, momentum, species, and total At moderate pressures or high temperatures approaching ideal- energy are gas conditions, the non-ideal diffusion prefactor becomes unity, ρ ∂ ∂ → D D ( ln fF / ln XF )P , T 1. Therefore DF , F F , O and traditional forward = −ρ( ∇ · v ), (44) molecular diffusion (i.e., diffusion in the direction of decreasing Dt fuel concentration) prevails. At high pressures, forward diffusion dominates in the compositional mixing layer downstream of the D v ρ = −∇P + ∇ · ( τ + K ), (45) interface edge. In contrast, near the injection orifice, where the fuel Dt GD ∂ ∂ is cold, the non-ideal prefactor ( ln fF / ln XF )P , T departs from unity and its sign changes depending on the local composition. In par- DY ρ i = −∇ · ( + J ), = , . . . , , ∂ ∂ Ji i i 1 N (46) ticular, ( ln fF / ln XF )P , T varies from positive values on the coflow Dt and fuel sides of the interface, where forward diffusion prevails, to negative values within the transcritical interface, where mass is DE ρ = −∇ · ( ) − ∇ · + Q + ∇ · τ + K · , transported in an antidiffusive manner in the direction of positive PGD v ( q ) [( ) v] (47) Dt 24 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 which describe the continuum mechanics of a multiphase, multi- macroscopic lengths, a local formulation in the moving frame at- component fluid of N species that moves at a mass-averaged veloc- tached to the interface is developed that provides mechanical ity v and has a density ρ and total energy E . The latter is related and transport equilibrium constraints applicable to interfaces in to the specific gradient-dependent internal energy (17) by the def- isothermal systems. Moderate temperature gradients across the in- inition terface, such as those encountered in Fig. 1 , are shown to require the consideration of small deviations from equilibrium. = + | | 2/ . E eGD v 2 (48)

Additionally, q is a molecular heat ﬂux deﬁned as 5.1. Entropy-production sources

N−1 The derivation of K, Q, and J is performed here using the q = qt + (h i − h N ) J i . (49) i

i =1 method of irreversible thermodynamics and Onsager’s reciprocal relations. The premise of this method is that the constitutive rela- The first term q on the right-hand side of Eq. (49) comprises t tions must lead to non-negative entropy production in accordance Fourier and Dufour mechanisms of heat transfer, while the sec- with the second principle of thermodynamics. This method paral- ond term corresponds to heat transport by interdiffusion of species lels the one utilized to determine the analytical form of the trans- with different partial specific enthalpies h , which are discussed in i port fluxes in the standard theory [138–140] . However, this method Appendices B.1 and C.8 . In addition, the symbol P is a gradient- GD cannot give expressions for the transport coefficients in terms of dependent pressure given by molecular properties. Those are provided separately in Appendices N N 1 B.4 and F. = − κ ∇ ρ ∇ ρ , PGD P i, j i j (50) The analysis begins by substituting Eqs. (17) and (50) into the 2 i =1 j=1 first principle (C.1) , and using the assumption of constant and sym- κ ρ metric ij coefficients, which leads to with P being related to , T and Yi through the equation of state (14) . The convenience of delocalizing the pressure as in Eq. N−1 N (50) will become clearer upon deriving an expression for the inter- T ds = de + P d(1 /ρ) − (μ − μ ) dY − ψ · [ d(∇(ρY )] /ρ, facial stress tensor K in Section 5 . GD GD i N i i i i =1 i =1 An alternative form of the energy Eq. (47) that will be employed (55) in the analysis is the enthalpy conservation equation

Dh DP where ρ = GD − ∇ · ( q + Q ) − ( τ + K ) : ∇v . (51) Dt Dt N ψ = κ , ∇(ρY ) (56) Equation (51) is obtained by subtracting the momentum conser- i i j j j=1 vation Eq. (45) multiplied by v from the total-energy conservation equation (47) , and by making use of the relation is an auxiliary vector. A transport equation for the specific entropy s can be derived by substituting Eqs. (44) –(47) and the definition h = e + P /ρ (52) GD GD (49) into the expression resulting from taking the material deriva- obtained from the definition of the local specific enthalpy h tive of Eq. (55) , which yields (C.3) along with Eqs. (17) and (50) . Methods for calculation of h N at high pressures are discussed in Appendices B.1 and C.6 . Ds 1 ρ + ∇ · qt + Q − ψ (ρY ∇ · v ) The integration of the conservation equations Eqs. (44) –(47) is Dt T i i i =1 K, J , Q, complicated by the additional terms i and which will − be shown to depend on the square of gradients and higher-order N1 − ( ψ − ψ ) ∇ · ( J + J ) derivatives of the partial densities across the interface. For in- i N i i i =1 stance, a full numerical integration of the momentum Eq. (45) in the flow sketched in Fig. 1 would incur significant numerical stiff- N−1 + ( − ) − ( μ − μ ) J = , ness, because the ratio of the interfacial terms to the convection [T si sN Ji i N i ] s˙prod (57) terms would be a large parameter, i =1 2 where ||∇ · (−P I + K ) || σ0 /δ 1 GD ∼ I ∼ , 1 (53) || ρv · ∇v || ρ 2/ 2 = + (∇ · ψ ) / μ = μ − ∇ · ψ O UO RF R WeO si si i T and i i i (58) where = are generalized versions of the partial specific entropy si (∂ /∂ ) μ = ρ 2 /σ S ni P,T,n || j =1 , ...N ( j = i ) and specific chemical potential i , respec- WeO O UO RF 0 1 (54) j

tively. In addition, s˙prod is an entropy source given by is a Weber number, and R is a large-scale Cahn number deﬁned

= ( 2 − 3 ) N in Eq. (9). Typical orders of magnitude are WeO O 10 10 and 1 = ( −4 − −5 ) s˙ = τ + K − ρY ∇ · ψ I − ∇(ρY ) ⊗ ψ : ∇v R O 10 10 in the present conditions, thereby making the prod T i i i i / (2 ) = ( 6 − 7 ) i =1 ratio (53) a large number of order 1 R WeO O 10 10 due δ − to the much smaller value of I compared to RF . N N1 + + Q − ψ (ρ ∇ · ) − ( ψ − ψ ) ∇ · + J qt i Yi v i N ( Ji i ) i =1 i =1 5. The structure of transcritical interfaces − N1 1 This section focuses on the derivation of the additional terms + [ T ( s − s ) ( J + J ) − (h − h ) J ] · ∇ i N i i i N i T K, Q, J i =1 and i in the constitutive laws of the Navier–Stokes Eqs. − (44)–(47) by using the diffuse-interface theory of van der Waals N1 ∇( μ − μ ) i N extended to multicomponent flows. Using those expressions and − ( J + J ) · , (59) i i T exploiting the small thickness of the interface relative to all other i =1 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 25 with I being the identity matrix and ⊗ denoting the dyadic prod- The quantities (67) are drivers of molecular transport, and conse- uct. The definition quently of entropy production. Different transport processes induced by different thermody- μ = h − T s , (60) i i i namic forces interfere with one another. For instance, in addi- the mass-conservation constraint tion to causing heat conduction, temperature gradients also induce mass transport through the Soret effect. More generally, the N N−1 transport fluxes in the conservation equations can be written as ( J i + J i ) = J N + J N + ( J i + J i ) = 0 , (61) a combination of all thermodynamic forces, and such combina- i =1 i =1 tion is linear in the first approximation [138–140] . The derivation and the vector identity of these expressions is facilitated by assuming that the transport D D fluxes and thermodynamic forces of different tensorial character ψ · [∇ (ρY ) ] = ψ · ∇ (ρY ) − [ ∇(ρY ) ⊗ ψ ] : ∇v (62) i Dt i i Dt i i i do not couple [141]. This simplification implies that there are no crossed effects between the mechanical stresses, which are driven ρ have been used in deriving Eq. (59), with D( Yi )/Dt being replaced by the velocity-gradient tensor, and the fluxes of heat and species, −∇ · ( + J ) − ρ ∇ · by Ji i Yi v in accordance with Eq. (46). To separate which are driven by gradients of temperature and chemical poten- transport by temperature gradients, the decomposition tials. Based on these considerations, a closure is formulated for K in Section 5.2 , followed by a discussion in Section 5.3 on the ef- ∇( μi − μN ) = ∇ T ( μi − μN ) fective surface-tension coefficient σ arising from this formulation. ∂ Onsager’s methodology will be used in Section 5.4 to determine − − + ∇ · ( ψ − ψ ) ∇ si sN i N T (63) Q J ∂ the interfacial fluxes of heat and species i . T , P Yj | j =1 , ...N −1 is performed, where use of Eq. (58) and of the reciprocity re- 5.2. Closure for the interfacial stress tensor = −(∂ μ /∂ ) lation for the partial specific entropy si i T P,Y | i =1 , ...N −1 i In the case of Newtonian fluids under zero bulk viscosity, the have been made. In the formulation, the operator ∇ T indicates spatial differentiation at constant temperature. Substitution of Eq. expression for theviscous stress tensor is

(63) into Eq. (59) yields τ = η ∇v + ∇v T − (2 η/ 3)(∇ · v ) I , (68) N η 1 where is the dynamic viscosity modeled in Appendix F.1. Vari- s˙ = τ + K − ρY ∇ · ψ I − ∇(ρY ) ⊗ ψ : ∇v η prod T i i i i ations of with temperature and composition are studied in i =1 force Appendix B.4 for C12 H26 /N2 mixtures at high pressures. Equa- τ ∇ ≥ current tion (68) implies that the viscous dissipation : v 0 is a positive quadratic and therefore leads to positive entropy produc- N N−1 + + Q − ψ (ρ ∇ · ) − ( ψ − ψ ) ∇ · + J tion. In contrast, the interfacial stress tensor K is assumed to be qt i Yi v i N ( Ji i )

i =1 i =1 elastically restoring, such that the production of entropy by the diffuse-interface terms multiplying the velocity-gradient tensor in current Eq. (64) is zero, namely N−1 1 + (χ − χ ) + J − ( − ) J · ∇ N [ i N ( Ji i ) hi hN i ] T K − ρ ∇ · ψ − ∇(ρ ) ⊗ ψ ∇ = , i =1 Yi i I Yi i : v 0 (69) force i =1 current(cont . )

− which gives N1 ∇ μ − μ T ( i N ) − ( J + J ) · , (64) N i i T = K = ρ ∇ · ψ − ∇(ρ ) ⊗ ψ . i 1 Yi i I Yi i (70) currents forces i =1 K where The interfacial stress tensor was originally introduced by Ko- K ∂ rteweg in a phenomenological manner [142]. The effects of are χ = ∇ · ψ − T ∇ · ψ (65) concentrated in the vicinity of the interface, where the partial- i i ∂ i T , | = , ... − P Yj j 1 N 1 density gradients are the largest. In that region, the first of the two terms on the right-hand side of Eq. (70) represents a hydrostatic is an auxiliary variable. The last term on the right-hand side of Eq. stress. As shown in Section 5.3 , this term gives rise to a curvature- (65) can be expressed as dependent pressure jump across the interface. The second term is N ∂ an anisotropic stress that acts primarily in the direction normal to 2 T ∇ · ψ = − κ T ∇ ρβ Y (66) ∂ i ij v j the interface. T , | = , ... − = P Yj j 1 N 1 j 1 In this formulation, there is no explicit surface-tension coeffi- σ σ by making use of the equation of state (14) and the definition of cient appearing in Eqs. (44)–(47) and (70). However, resur-

ψ β faces when the integral form of the momentum conservation equa- i in Eq. (56). In this formulation, v is the volume expansivity defined in Eq. (B.7) and particularized for the Peng-Robinson equation (45) is considered, as shown in the next section. tion of state (14) in Eq. (C.22) . 5.3. Mechanical equilibrium conditions Following an approach similar to that provided by Onsager [138] (see also Ch. 11.2 in [140] , and Ch. 6 in Ref. [139] ), the en- This section focuses on the derivation of mechanical equilib- tropy source (64) is interpreted as consisting of three production rium conditions across the interface. The analysis is first particu- terms that have a similar form. Specifically, each term consists of a larized for the bicomponent flow depicted in Fig. 1 . The problem current multiplied by one of the following thermodynamic forces: is best illustrated by analyzing a zoomed-up view of a segment of an interface embedded in a much wider thermal mixing layer, 1 ∇ T ( μ1 − μN ) ∇ T ( μN−1 − μN ) ∇ v , ∇ , , . . . . (67) δ T T T as sketched in Fig. 17. The thickness of the mixing layer T grows 26 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

and ∂ 2 − + K = − + ρκ (ρ ) PGD s,s P , Y Y F F F ∂ 2 F nˆ 2 1 ∂ ∂ + κ , (ρY ) + ρκ , K Y (ρY ) , (73) 2 F F ∂ nˆ F F F I F ∂ nˆ F

where use of Eqs. (50) and (70) has been made in conjunction with the assumption that only the fuel gradient-energy coeﬃ- cient participates in the conditions addressed in this study, as discussed in Section 4.2 . The curvature-related terms in Eqs. (72) and (73) , which arise from the divergence operator in the curvilinear system, are much smaller than the terms involving squared and second-order gradients normal to the interface. Equations (72) and (73) can be combined to give 2 ∂ K s,s = K n,n + κ , (ρY ) . (74) F F ∂ nˆ F

In the reference frame { nˆ , sˆ} , the continuity Eq. (44) can be written as ∂ρ ∂ Fig. 17. Zoomed-in schematics of a transcritical interface embedded in a thermal + (ρv n ) + ρv n K I = 0 , (75) mixing layer. ∂t ∂ nˆ whereas the normal and tangential components of the momentum conservation equation (45) can also be expressed as slowly with distance downstream and mostly coincides with the shear-layer thickness because of the near-unity Prandtl numbers ∂ v ∂ v ∂ ρ n + ρ n = (− + K + τ ) vn PGD n,n n,n and the order-unity relative differences of velocities and temper- ∂t ∂ nˆ ∂ nˆ atures between the free streams, as discussed in Section 1. The ∂ v I , n + K (K n,n − K s,s + τn,n − τs,s ) − ρ (76) thermal mixing layer is slanted toward the coflow side because I ∂t D / D 1 (e.g., D / D ࣃ 51 in C H 6/N systems at P ∞ = 50 T,O T,F T,O T,F 12 2 2 and bar, T F = 350 K, and T O = 10 0 0 K). ∂v ∂ ∂τ , ∂ v , The asymptotic limit considered in this analysis corresponds to ρ s = (− + K ) + n s + τ − ρ I s , vn PGD s , s 2KI n , s (77) thin interfaces compared to the thermal mixing layer, or equiva- ∂ nˆ ∂ sˆ ∂ nˆ ∂t respectively, where lently, T 1, with T being a thermal Cahn number defined in Eq. (10) . As the pressure increases, δ decreases and δ increases, T I 2 η ∂v 2 η ∂v n n and therefore T becomes increasingly larger. However, despite the τn,n = 2 − K v n , τs,s = − − 2K v n , ∂ I ∂ I 3 nˆ 3 nˆ high pressures, T always remains small because of the criticalpoint elevation phenomenon discussed in Section 3.2. ∂v s τn,s = η − K v s (78) This analysis is limited to two dimensions, although the formu- ∂ nˆ I lation can be easily extended to a third dimension by incorporat- = · = ing a second curvature. More general multiscale treatments, albeit are viscous stresses. In this formulation, vI , n vI en and vI , s · based on a significantly simpler incompressible diffuse-interface vI es are, respectively, the normal and tangential components of = ( − ) · formulation, can be found in Ref. [143] . the interface absolute velocity, and vn v vI en is the normal Two different reference frames are sketched in Fig. 17 that are component of the relative fluid velocity, with e n and e s being unit relevant for this analysis: the Cartesian laboratory frame { x , y } and vectors in the normal and tangential directions, respectively. The the curvilinear orthogonal frame { nˆ , sˆ} . The latter moves with the interface does not slip on the fluid in the tangential direction be- interface at its local absolute velocity vI . The local curvature inter- cause of the continuity of the tangential velocity. As a result, the = / , > = ( − ) · face is denoted as KI 1 RI with KI 0 for interfaces concave to- tangential component of the relative velocity vs v vI es van- ward the coflow (i.e., for interfaces such as the one sketched in the ishes in the vicinity of the interface, as depicted in Fig. 17 . upper panel in Fig. 17 ). In all cases of practical interest, the radius In general, the absolute velocity of the interface vI, n is a func- δ of curvature is much larger than the interface thickness, RI I . In tion of time, and consequently the momentum Eqs. (76) and δ contrast, the magnitude of RI relative to T depends on the partic- (77) require a fictitious acceleration given by the last term on their ular problem under consideration. This scaling analysis considers corresponding right-hand sides. Following experimental analyses of the case in which interface acceleration in liquid jets at normal pressure [144] , it is assumed herein that the characteristic value of the interface accel- N = δ / T RI (71) eration is a fraction of order 1 / R 1 of the convective accelera- 2 /δ is an order-unity parameter. tion UO T in the shear layer. N = ( ) , Under the conditions T 1 and O 1 the interface is slender and the largest variations in all variables are along the nor- 5.3.1. Mechanical equilibrium normal to the interface The transfer of heat into the fuel stream decreases the fuel mal nˆ . Consequently, −P GD I + K is diagonally dominant with diagonal components given by density near the interface and produces an outwards flow- displacement velocity whose order of magnitude is given by a bal- 2 ∂ − + K = − + ρκ (ρ ) ance between convection and heat conduction across the thermal PGD n,n P , Y Y F F F ∂ 2 F nˆ mixing layer, 2 1 ∂ ∂ (T − T ) (T − T ) − κ (ρ ) + ρκ (ρ ) ρ O e ∼ λ O e . F , F YF F , F KI YF YF (72) O UT cp, O O (79) 2 ∂ nˆ ∂ nˆ δ δ2 T T L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 27

The interface temperature Te is much closer to TF than to TO be- 2 cause DT,O /DT,F 1. The symbol Te, which coincides with the phase- ∂ ∂ K , = ρ κ , Y (ρY ) + N ρ κ , Y (ρ Y ) (92) equilibrium temperature in Eq. (A.13), is purposely chosen here be- s s F F F ∂ nˆ 2 F T F F F ∂ nˆ F cause it will be shown in Section 5.6 that the conditions in the as the nondimensional interfacial stresses, and vicinity of the interface are near phase equilibrium when the tem- η ∂ η ∂ perature gradients are not too large. The balance (79) yields the 2 vn 2 vn τn,n = 2 − N v , τs,s = − − 2 N v , thermal-expansion velocity scale 3 ∂ nˆ T n 3 ∂ nˆ T n

∼ /δ , (93) UT DT , O T (80) as the nondimensional viscous stresses, where with U T /U O = 1 /Pe T 1 being the inverse of a large Péclet number. In order to normalize Eq. (76), consider the dimensionless nor- η = η/ηO (94) mal coordinate is the nondimensional dynamic viscosity. Additionally, P r O = = / ( δ ) , ν / nˆ nˆ T T (81) O DT , O is the Prandtl number based on the kinematic viscosity ν = η /ρ = η / ρ σ δ and the dimensionless relative velocity O O O in the coﬂow, and OhT O O 0 T is an Ohnesorge

number. = / . vn vn UT (82) Characteristic parameters are provided in Table 4 for typical Based on (81) and (82) , the nondimensional time coordinate is transcritical conditions. Following those, the orders of magnitude defined as of the dimensionless groups multiplying the last four terms on 2 / R 2 = ( −2 ) , N / R = the right-hand side of Eq. (89) are T Pe O 10 PrO = / , T t t tI (83) ( −1 ) , N 2 / ( 2 R ) = ( 2 ) , / ( R ) = ( 2 ) O 10 PrO OhT O 10 and PrO T O 10 . Of par- where ticular interest is the much larger value of the inverse of the mechanical-equilibrium parameter E m multiplying the first term on = δ / tI T T UT (84) the right hand side of Eq.(89), represents a flow transit time across the interface. The pressure is 2 R ρ 2 T Oh F D , δI − E = T = T O = ( 6 ) . normalized by its variation (36) through the interface m O 10 1 (95) 2 σ δ δ PrO 0 T T δ P = T T , P (85) These estimates indicate that the derivative of P − K , along σ GD n n 0 the normal direction in Eq. (89) is zero in the first approximation. whereas the gradient-energy coefficient is nondimensionalized us- Retaining the higher-order effects of the normal derivative of the ing Eq. (26) as normal viscous stress and the curvature terms arising from the di- 2 vergence of the interfacial stress tensor, Eq. (89) simplifies to ρ κF , F κ = F . (86) F , F δ σ 2 T T 0 ∂ Oh − + K + T τ + N (K − K ) = . PGD n,n n,n T n,n s,s 0 (96) The unit of interfacial stresses is obtained by utilizing Eq. ∂ nˆ P r O ∂ (ρ ) /∂ ˆ ∼ ρ /δ (86) and the estimate YF n F I in Eq. (70), which gives The integral of Eq. (96) yields the mechanical equilibrium con-

K δ K K δ K dition in the normal direction to the interface n,n T T n,n s,s T T s,s K = = , K = = . 2 n,n s,s (87) 2 A P σ0 A P σ0 ∂ 1 ∂ − P + τ , + ρκ , Y (ρY ) − κ , (ρY ) n n F F F 2 F F F F The rest of the variables are nondimensionalized as ∂ nˆ 2 ∂ nˆ ∂ T δT τn,n δT τn,s T δT τs,s τ , = , τ , = , τ , = + ρκF , F K I Y F (ρY F ) = −P O + τn,n O n n η n s η s s η ∂ O UT O UO O UT nˆ + ∞ 2 ∂ v , R δ ∂ v , ∂ I n = T I n . − κ (ρ ) , (88) KI F , F YF dnˆ (97) ∂t 2 ∂t ∂ nˆ UO nˆ In the dimensionless variables (81) –(88) , the normal compo- where use of Eqs. (72) , (74) , and (90) has been made, and where nent of the momentum conservation equation (76) becomes dimensional variables have been recovered in the notation. In this formulation, P and τ are the local thermodynamic pressure ∂K O n,nO ∂ v ∂ v 1 ∂P , ρ n + ρ n = − GD − n n and normal viscous stress away from the interface at intermediate vn ∂t ∂ nˆ E ∂ nˆ ∂ nˆ m distances δI nˆ δT on the coﬂow side. 2 ∂τ Equation (97) is a fundamental mechanical equilibrium con- N P r P r , + O K − K + O n n n,n s,s dition that relates – locally through the interface –the partial- Oh 2R R ∂ nˆ T T density gradients, the thermodynamic pressure, the interface cur- N 2 ∂ PrO T PeT vI , n vature, and the normal viscous stress. It corresponds to the ﬁrst + τ , − τ , − ρ , (89) R n n s s R 2 ∂t integral of the normal component of the momentum Eq. (76) in the moving frame. Evaluation of Eq. (97) at large negative nˆ into with the fuel stream yields the familiar Young-Laplace jump condition κ 2 , ∂ = − F F (ρ ) P P YF (90) P − P + τ , − τ , = σ K , (98) GD 2 ∂ nˆ F O n nO n nF I where P and τ are the local thermodynamic pressure and nor- as the nondimensional gradient-dependent pressure, F n,nF mal viscous stress away from the interface at intermediate dis-

∂ 2 tances −δT nˆ −δI on the fuel side. Additionally, σ is the sur- K , = ρ κ , Y (ρY ) n n F F F ∂ 2 F face tension coeﬃcient nˆ 2 + ∞ 2 ∂ ∂ ∂ − κ (ρ ) + N ρ κ (ρ ) , σ = κ (ρ ) , F , F YF T F , F YF YF (91) F , F YF dnˆ (99) ∂ nˆ ∂ nˆ −∞ ∂ nˆ 28 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Table 4 Characteristic dimensional and nondimensional parameters for a typical high-pressure two-stream ﬂow similar to that depicted in Fig. 1 , where the coﬂow and fuel streams consist of N 2 and C 12 H 26 , respectively.

Dimensional parameters

δ δ m m ρ kg ρ kg kg kg RF [m] RI [m] T [m] I [m] UO UF P∞ [bar] TO [K] TF [K] Te [K] O F WO WF s s m 3 m 3 mol mol

− − − − − 5 · 10 −4 5 · 10 5 5 · 10 5 5 · 10 9 50 10 50 1000 350 500 16.67 640.62 28 · 10 3 170 · 10 3 W W kJ kJ m 2 m 2 m 2 m 2 η [Pa · s] η [Pa · s] λ λ c c D D D , D , O F O mK F mK p,O kgK p,F kgK T,O s T,F s F OO s FF O s

− − − − − 4 . 11 · 10 −5 4 . 76 · 10 4 0.07 0.10 1.17 2.25 3 . 53 · 10 6 6 . 93 · 10 8 6 . 20 · 10 7 1 . 03 · 10 8 m m m m 1 1 N Jm 5 IG β β σ κ UT cO cO cF v O v F 0 F,F tI [s] ca [m] s s s s K K m kg 2

− − − − 0.07 642 630 1,100 0 . 99 · 10 −3 0 . 64 · 10 3 0.01 1 . 22 · 10 16 7 . 15 · 10 8 2 . 40 · 10 7

Nondimensional parameters γ IG N O PrF PrO ReF ReO WeF WeO R T PeT OhT MaF MaO − 1.34 10.38 0.70 33,646 10,140 3300 2084 10 −5 10 4 1.0 714 0.01 0.01 0.08

R Le F α Z O Z F b¯ O b¯ F cˆ O PWτ

38.42 5.69 1.85 1.01 0.44 0.99 0.22 1.01 0.40 6.07 1.42

which represents the extension of Eq. (22) to bicomponent of Eqs. (74) and (99) provides the mechanical equilibrium condi- systems. Physical interpretations of σ based on the excess of tion in the tangential direction to the interface the Landau’s potential energy of the interface are discussed in ∂σ Section 5.6.3 . τn,s O − τn,s F = − , (105) ∂ sˆ 5.3.2. Mechanical equilibrium tangential to the interface where dimensional variables have been recovered in the notation. Additional considerations are required in order to normalize Eq. Equation (105) is commonly known as the Marangoni effect, and

(89) . Close to the interface, the tangential relative velocity v s van- describes the onset of ﬂuid motion along the interface due to tan- ishes and its normal gradient ∂ v s /∂ nˆ is locally a constant of order gential variations of the surface tension. δ UO / T . This motivates the introduction of the dimensionless variable 5.3.3. The thermodynamic pressure near the interface The analyses in Sections 5.3.1 and 5.3.2 highlight the multiscale v = v / ( U ) . (100) s s T O nature of the problem and the corresponding challenges associ- The tangential coordinate is normalized with the thermal ated with the integration of the momentum Eq. (45) because of mixing-layer thickness as the mostly balanced, spatially localized behavior of the gradient- dependent pressure and the interfacial stress tensor in the vicinity sˆ = sˆ/ δ , (101) T of the interface. The role of the thermodynamic pressure, which whereas the units of interface acceleration in the tangential and participates in this balance by linking the mechanical equilibrium normal directions are assumed to be the same and equal to that conditions with the equation of state, is studied in this section. used in Eq. (88) . In the dimensionless variables (81) –(88) and Consider the schematics in Fig. 18 (a) depicting the spatial dis- (100) and (101) , the tangential component of the momentum contribution of the thermodynamic pressure P around the interface at δ servation Eq. (77) becomes scales much larger than I . The Young-Laplace condition (98) states

∂ 2 ∂ that the local thermodynamic pressures PF and PO on each side vs PrO ρ v = (−P + K , ) of the curved interface in Fig. 18 (a) differ from one another by n ∂ 2 R ∂ GD s s nˆ T Oh PeT sˆ σ T amounts of order 0 /RI . However, this pressure difference is much ∂τ P r , 2 N P r P e ∂ v , smaller than the combustor pressure P∞ because RI is typically + O n s + O τ − T ρ I s , (102) n,s 2 much larger than σ / P ∞ even at high pressures (e.g., σ /P ∞ = T R ∂ nˆ R R ∂t 0 0 1 nm at P ∞ = 100 bar in the conditions addressed in Table 4 .). with The interfacial stresses and the gradient component of the pres- ∂ sure are negligible at distances | n | δ on both sides away from vs I τ , = η − N v (103) n s ∂ nˆ T s the interface, where the momentum Eq. (45) becomes ∂v the nondimensional shear stress. Upon substituting (92) into Eq. ρ + ρv · ∇v = −∇ p + ∇ · τ. (106) ∂ (102) , making use of Eqs. (74) and (97) , neglecting the tangential t gradients of thermodynamic pressure, and retaining the two most In Eq. (106) , the thermodynamic pressure P has been replaced by dominant terms multiplied by the largest dimensionless groups in a hydrodynamic pressure p by assuming P P ∞ + p. This decom- accordance with Table 4 , the simpliﬁed tangential momentum bal- position requires the free-stream Mach numbers Ma O = U O /c O and = / ance MaF UF cF to be small compared to unity. In the notation, cO and

2 ∂τ cF are the speed of the sound waves in the coﬂow and fuel free ∂ Oh P e T , K − K + T n s = s,s n,n 0 (104) streams, respectively. An expression for the speed of sound c at ∂ sˆ P r ∂ nˆ O high pressures, which generally differs from that of an ideal gas, 2 / = ( −1 ) is obtained, with OhT PeT PrO O 10 in the conditions ad- is provided in Eq. (C.48). At small Mach numbers, the local de- dressed in Table 4 . The integration of Eq. (104) and the substitution partures of P from its corresponding mean values on each side of L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 29

2 with relative errors of order MaO 1. With these simpliﬁcations in mind, and neglecting also the viscous stresses, Eq. (97) becomes 2 ∂ 2 1 ∂ P ∞ − P + ρκ , Y (ρY ) − κ , (ρY ) = 0 . (107) F F F ∂ nˆ 2 F 2 F F ∂ nˆ F

Equation (107) can be integrated once yielding ∞ 2 + σ = (P ∞ − P ) d nˆ , (108) 3 −∞ which indicates that the surface tension is originated by a net underpressure within the interface. Speciﬁcally, Eq. (108) predicts strongly localized variations of the thermodynamic pres- ∼ σ δ sure of order AP 0 / I within the interface, as anticipated in Section 4.4 and shown schematically in Fig. 18 (b). These variations are always much larger than those engendered by curvature by a factor of order 1 / (N T ) 1 , and they are also much larger than the hydrodynamic pressure variations by a factor of order 1/( R WeO ) 1. The boundary conditions employed in integrating Eq. (107) correspond to the fuel partial densities away from the interface. In the limit T 1, it will be shown in Section 5.6 that the temperature gradient across the interface is small compared to the composition gradient. To leading order in T , the interface is in phase equilibrium and consequently the fuel partial densities tend to their phase-equilibrium values on both sides of the interface, ρ → ρ → −∞ , YF e YFe at nˆ (109)

ρ → ρg g → + ∞ , YF e YFe at nˆ (110) → ±∞ δ where the limit nˆ denotes distances much larger than I but δ much smaller than T . In this limit, the relative temperature variations across the interface are of order T , and as a result the temperature becomes approximately uniform in the vicinity of the interface. The interface temperature T e evolves with distance down- ∼ ∼ stream, with Te TF near the injector oriﬁce, and Te T near c,diff g the interface edge. Both phase-equilibrium values YFe and YFe are therefore evaluated at P ∞ and T e . Because of the locally uniform temperature, the normal directional derivative of the Gibbs-Duhem relation (C.9) simpliﬁes to

∂P ∂μF ∂μO = ρY + ρ(1 − Y ) . (111) ∂ nˆ F ∂ nˆ F ∂ nˆ Upon substituting Eq. (111) into Eq. (107) differentiated with respect to nˆ , the alternative form of the mechanical equilibrium condition (107)

3 ∂ ∂μF ∂μO ρY κ , (ρY ) = ρY + ρ(1 − Y ) (112) F F F ∂ nˆ 3 F F ∂ nˆ F ∂ nˆ Fig. 18. Schematics of the pressure variations (a) near the interface, (b) within the is obtained. At long distances from the interface, Eq. (112) is sub- interface, and (c) in thermodynamic space, the latter being particularized for a ﬂat ject to conditions (109) and (110) , to the phase-equilibrium values interface in a monocomponent isothermal ﬂow. for chemical potentials,

μF → μFe and μO → μOe at nˆ → + ∞ , (113) the interface P F and P O = P ∞ are small. The condition that must be satisfied for these departures to be larger than the Young-Laplace and to vanishing composition gradients, σ pressure jump 0 /RI is that RI must be larger than the capillary ∂ (ρY ) → 0 at nˆ → + ∞ . (114) = σ / (ρ 2 ) F length ca 0 O UO . This condition tends to be always satis- ∂ nˆ fied at high pressures, since ca becomes very small because of the large values of density encountered in conjunction with the small 5.3.4. Generalizations to multicomponent systems values of surface tension, as suggested by the estimates provided The formulation above can be straightforwardly generalized in Table 4 . to multicomponent flows. Specifically, the definition (90) for P GD , The above considerations suggest that the curvature-related along with general forms of Eqs. (72) –(74) , namely terms in the mechanical equilibrium condition (97) can be ne- N N ∂ 2 σ glected with relative errors of order 0 /(P∞ RI ) 1, and the local − + K = − + ρ κ (ρ ) PGD n,n P Yi i, j Yj ∂ nˆ 2 coflow pressure PO can be replaced by the combustor pressure P∞ i =1 j=1 30 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 1 ∂ ∂ ∂ 5.4. Closures for the interfacial transport fluxes of heat and species − κ , (ρY ) (ρY ) + K ρY κ , (ρY ) , (115) 2 i j ∂ nˆ i ∂ nˆ j I i i j ∂ nˆ j Upon substituting Eq. (70) into Eq. (64) , the entropy production source can be expressed as N N ∂ 2 ( τ : ∇v ) − + K = − + ρ κ (ρ ) = + φ · PGD s,s P Y , Y s˙ F (124) i i j ∂ nˆ 2 j prod T i =1 j=1 in terms of the scalar product of a vector of N thermodynamic ∂ ∂ ∂ 1 forces + κ , (ρY ) (ρY ) + K ρY κ , (ρY ) , (116) 2 i j ∂ nˆ i ∂ nˆ j I i i j ∂ nˆ j 1 ∇ ( μ − μ ) ∇ ( μ − − μ ) F = ∇ , − T 1 N , ···− T N 1 N (125) and T T T

N N and a vector of N currents ∂ ∂ − K s,s = K n,n + κ , (ρY ) (ρY ) , (117) N N1 i j ∂ nˆ i ∂ nˆ j i =1 j=1 φ = qt + Q − ψ i (ρY i ∇ · v ) − ( ψ i − ψ N ) ∇ · ( J i + J i ) i =1 i =1 can be substituted into Eq. (96) yielding N−1 N N + [ (χ − χ ) ( J + J ) − (h − h ) J ], ∂ 2 i N i i i N i = − P + τn,n + ρY κ , (ρY ) i 1 i i j ∂ nˆ 2 j i =1 j=1 ( J + J ), . . . ( J − + J − ) . (126) 1 ∂ ∂ ∂ 1 1 N 1 N 1 − κ , (ρY ) (ρY ) + K ρY κ , (ρY ) 2 i j ∂ nˆ i ∂ nˆ j I i i j ∂ nˆ j

In dissipative systems, the second principle of thermodynamics + ∞ N N ∂ ∂ requires s˙ ≥ 0 , which translates into prod = −P O + τn,n O − K I κi, j (ρY i ) (ρY j ) d nˆ, nˆ ∂ nˆ ∂ nˆ i =1 j=1 ( τ : ∇v ) s˙ − = φ · F ≥ 0 , (127) (118) prod T because the viscous dissipation τ: ∇v in Eq. (124) is a positive which represents the mechanical equilibrium condition normal quadratic. The transport fluxes in φ must be such that Eq. (127) is to the interface for multicomponent flows. Evaluation of Eq. satisfied. The linear relation (118) away from the interface yields the Young-Laplace jump con- φ = L dition (98) , with σ given by the general expression F (128)

+ ∞ is the simplest approach that satisfies Eq. (127) . In Eq. (128) , L = L σ = (K s,s − K n,n ) d nˆ [ i,k ] is a matrix of Onsager phenomenological coefficients, with −∞ i, k = 1 , . . . N. Specifically, in order to satisfy Eq. (127) , L must be + ∞ N N L ∂ ∂ symmetric and positive semidefinite, in that its elements ik must = κ (ρ ) (ρ ) . i, j Yi Yj dnˆ (119) satisfy the conditions [138–140] −∞ ∂ nˆ ∂ nˆ i =1 j=1 L = L , L ≥ , L 2 ≤ L L , L ≥ . i,k k,i i,i 0 i,k i,i k,k det i,k 0 (129)

It can be shown by using Eqs. (116)–(118) that variations of the Using Eq. (128) in Eq. (124) , the expression surface tension coefficient along the interface in multicomponent N N flows give rise to the same tangential mechanical equilibrium con- ( τ : ∇v ) s˙ − = L F F dition (105) as in bicomponent flows. prod T ik i k i =1 k =1 Under the same assumptions used in Section 5.3.3 with regard − − − to small curvatures and Mach numbers, Eq. (118) simplifies to N1 N1 N1 = + + + Lq,q F1 Lq,k F k +1 F 1 Li,q F 1 Li,k F k +1 Fi +1 N N ∂ 2 1 ∂ ∂ k =1 i =1 k =1 − + ρ κ (ρ ) − κ (ρ ) (ρ ) = , P∞ P Yi i, j Yj i, j Yi Yj 0 (130) ∂ nˆ 2 2 ∂ nˆ ∂ nˆ i =1 j=1 is obtained. In Eq. (130) , the symbols F , F , and F + denote com- (120) i k k 1 ponents of F , with F 1 = ∇(1 /T ) and which represents the multicomponent version of Eq. (107) . ∇ ∇ · ( ψ − ψ ) ∇ ( μ − μ ) ∇ ( μ − μ ) T i N For 1, the boundary conditions employed to integrate Eq. T i N T i N T F i +1 = − = − + (120) are analogous to (109) and (110) for every component. In that T T T limit, the generalization of Eq. (112) to multicomponent mixtures is (131) for i = 1 , . . . , N − 1 . In writing Eq. (131) , use has been made of the N N 3 N μ ∂ ∂μ definition (58) for the generalized chemical potential i . Addition- ρ κ (ρ ) = ρ k . k k, j Yj k (121) ally, in Eq. (130) , the Onsager coefficients have been renamed as ∂ nˆ 3 ∂ nˆ k =1 j=1 k =1 = L , = L , = L , = L , Lq,q 1 , 1 Lq,k 1 ,k +1 Li,q i +1 , 1 Li,k i +1 ,k +1 (132) For k = 1 , . . . N, Eq. (121) is subject to equilibrium values of the with i, k = 1 , . . . N − 1 . A physical interpretation of these coeffi- chemical potentials, cients will be provided in Section 5.5 . The first bracketed term in the second line in Eq. (130) corre- μ → μ → + ∞ , k ke at nˆ (122) sponds to the first element of the vector of currents (126) . Equat- ing these quantities gives and to vanishing composition gradients, N N−1 ∂ q + Q − ψ (ρY ∇ · v ) − ( ψ − ψ ) ∇ · ( J + J ) (ρ ) → → + ∞ . t i i i N i i Yk 0 at nˆ (123) ∂ nˆ i =1 i =1 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 31

N−1 In practical implementations of the transport fluxes (136) – + (χ − χ ) ( + J ) − ( − ) J μ ψ [ i N Ji i hi hN i ] (139), the constant-temperature gradients of i and i are elim- i =1 inated in favor of the full gradients ∇ by making use of Eqs. N−1 (63) and (65) . The resulting expressions = + . Lq,q F1 Lq,k F k +1 (133) ∇(μ − μ ) = ∇ (μ − μ ) − (s − s ) ∇T , (140) = i N T i N i N k 1 Similarly, the second bracketed term in the second line in Eq. ∇ ∇ · ( ψ i − ψ N ) = ∇ T ∇ · ( ψ i − ψ N ) (130) corresponds to the i + 1 element of the vector of currents ∇ T (126), + ∇ · ( ψ − ψ ) − (χ − χ ) (141) i N i N T N−1 ∇ (μ − μ ) ∇(μ − μ ) , + J = + , canbe readily used to exchange T i N for i N and Ji i Li,q F 1 Li,k F k +1 (134) ∇ T ∇ · ( ψ − ψ N ) for ∇ ∇ · ( ψ − ψ N ) , respectively. k =1 i i with i = 1 , . . . N − 1 . 5.5. The Onsager coefficients in transcritical conditions In Eqs. (133) and (134) , terms on the left-hand side can be associated with those on the right-hand side depending on whether The fluxes (136) –(139) , along with the viscous and interfacial they are functions of the gradient-energy coefficients. For instance, stresses (68) and (70) , can be substituted into the entropy produc- on the left-hand side of Eq. (133) , q t can be matched with those tion rate (64) , yielding terms on the right-hand side that are independent of the gradient- 2 energy coefficients, ( τ : ∇v ) |∇T | s˙ = + L , prod q q 2 − T T N1 L q,q Lq,k = − ∇ − ∇ ( μ − μ ) . N−1 N−1 qt T T k N (135) T 2 T Li,k k =1 + ∇ ( μ − μ ) · ∇ ( μ − μ ) T 2 T k N T i N Substituting Eq. (135) into Eq. (49) gives the standard heat flux i =1 k =1 N−1 L , q i + 2 ∇ T ( μi − μN ) · ∇T . (142) − − 3 N1 N1 T L q,q Lq,k i =1 = − ∇ + ( − ) − ∇ ( μ − μ ) . q T hi hN Ji T k N T 2 T It is shown in this section that the right-hand side of Eq. (142) is i =1 k =1 Fourier conduction a positive quadratic in the transcritical conditions studied here. interdiffusion Dufour effect 2 = In Eqs. (136)–(139) and (142), the N coefficients Lq , q , Lq , i (i , . . . − , = , . . . − (136) 1 N 1), and Li , k (i k 1 N 1) are arranged in the Onsager

matrix as Similarly, on the left-hand side of Eq. (134), Ji can be matched with those terms on the right-hand side that are independent of the gradient-energy coefficients, giving the standard species diffusion flux − N1 Li,k Lq,i J = − ∇ ( μ − μ ) − ∇T . (137) (143) i T T k N T 2 k =1 Soret effect (thermal diffusion) Fickian diffusion and barodiffusion In Eqs. (136) and (137) , the constant-temperature gradient of the chemical potential can be expanded in terms of gradients of The first subset of L consists of the single coefficient L , which composition and pressure, as shown in Appendix F.3 , leading to q,q participates in the Fourier conduction component of the standard the more familiar representation of these fluxes in Stefan–Maxwell heat flux q . By simple inspection of Eqs. (136) or (F.24) , the coeffi- form. cient L can be written as q,q The interfacial corrections to q and Ji are derived by matching 2 the remainders on both sides of Eqs. (133) and (134) , respectively. L q,q = λT , (144) This procedure gives the interfacial heat flux with λ being a thermal conductivity modeled in Appendix F.2 . − N1 The second subset of L corresponds to the coefficients L [i.e., Lq,k i,k Q = ∇ ∇ · ψ − ψ T dot-dashed line rectangle in Eq. (143) ], which are related to the T k N k =1 J interfacial species flux i in Eq. (139) and to the Fickian compo- N N−1 nent of Ji in Eq. (F.25). The calculation of Li , k is summarized in + ψ i (ρY i ∇ · v ) + ( ψ i − ψ N ) ∇ · ( J i + J i ) Appendix F.4 and involves the utilization of Eq. (F.34) relating the = = D i 1 i 1 coefficients Li , k with the binary diffusion coefficients i, j modeled N−1 in Appendix F.5 . For binary mixtures, Eq. (F.34) becomes a scalar − [ (χi − χN ) ( J i + J i ) − (h i − h N ) J i ] (138) equation that yields i =1 ρD , W W X (1 − X ) = F O F O F F . and the interfacial species flux L F , F (145) R 0 W − N1 L Li,k The third subset of are the coefficients Lq , i [i.e., solid line J = ∇ ∇ · ψ − ψ , (139) i T T k N rectangles in Eq. (143) ], which are involved in the computation of = k 1 the Soret and Dufour effects appearing, respectively, in the stan- ψ χ with i and i being defined in Eqs. (56) and (65), respectively. dard diffusion fluxes of heat q in Eq. (136) and species Ji in Eq. Similarly to the interfacial stress tensor K, the species and heat (137) [see also Eqs. (F.24) and (F.25) for the corresponding Stefan– J Q fluxes i and are also highly localized at the interface, where Maxwell forms of these fluxes]. In particular, the coefficients Lq , i the partial-density gradients multiplied by the gradient-energy coefficients are dynamically relevant quantities. 32 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

(149) and (150) , the difference of partial speciﬁc entropies can be calculated as

s F − s O = ( h F − h O + μO − μF )/ T (151) by making use of the deﬁnition (60) . Alternatively, based on the analysis outlined in Section F.3 , q and J F can be recast in traditional

Stefan–Maxwell form as 0 ρD F , O R T W k F , T ∂ ln f F q = − λ∇ T − ∇ X F W W X (1 − X ) ∂ ln X F O F F F P,T X vW + F V − F ∇ + ( − ) ¯F P hF hO JF (152) R 0T W and ∂ ln f X vW = − ρD F ∇ + F V − F ∇ JF F , O XF ¯F P ∂ ln X R 0T Fig. 19. Constant-pressure distributions of the determinant of the Onsager matrix F P,T W for C 12 H 26 /N 2 mixtures at (a) P = 50 bar and (b) 100 bar as a function of temper- − ρD , k , (∇T /T ) , (153) ature and mass fraction of dodecane Y F (refer to the legend in the left panel). The F O F T plots include the diffusional critical point of the mixture at the corresponding pres- = sure (purple diamond symbol). (For interpretation of the references to color in this as shown by evaluating Eqs. (F.24) and (F.25) for N 2 and utiliz- figure legend, the reader is referred to the web version of this article.) ing Eqs. (145) and (146) . Despite the fact that the Stefan-Maxwell forms (152) and (153) are more insightful than their unexpanded can be related through Eq. (F.43) to the diagonal coefficients Li , i counterparts (149) and (150) , their utilization in transcritical flows and to the thermal-diffusion ratio ki ,T , the latter being modeled in is not advantageous when the combustor pressure is not suffi- Appendix F.6 . For binary mixtures, Eq. (F.43) simplifies to ciently high and the system crosses the mechanical spinodals. In those conditions, singularities arise in the non-ideal diffusion pref- L , = ρT D , k , . (146) q F F O F T ∂ ∂ V actor ( ln fF / ln XF )P , T and in the fuel partial molar volume ¯F [de- λ D , The distributions of , F , O and kF,T with temperature and compo- fined in Eq. (B.18) and particularized for the Peng-Robinson equa- sition are studied in Appendix B.4 for C12 H26 /N2 mixtures at high tion of state in Eq. (C.43)] that prevent from splitting the constant- pressures. temperature gradient of chemical potential into barodiffusion and The size of the Onsager matrix L in binary mixtures is 2 × 2 Fickian diffusion, as discussed in Appendix E.1 .

( L ) = , − 2 and its determinant is given by det Lq qLF , F Lq, F . The requirement that L be positive semidefinite, and therefore that the 5.6. Transport equilibrium condition entropy production source (142) be zero or positive, implies the condition det ( L ) ≥ 0 . Numerical evaluations for 15 bar ≤ P ≤ 100 bar In a similar way as the gradient-dependent pressure and in- ( L ) ≥ indicate that the condition det 0 is satisfied for C12 H26 /N2 terfacial stresses are balanced when the interface is in mechanical mixtures at temperatures 300 K ≤ T ≤ 1000 K as long as the param- equilibrium, a balance between interfacial fluxes and standard dif- ι eter constant participating in the calculation of kF,T is recalibrated fusion fluxes exists when the interface is in transport equilibrium, within 10% of its standard value (see Appendix F.6 for details). as discussed in this section. These considerations are illustrated in Fig. 19 for two representative pressures. 5.6.1. The transport equilibrium condition in terms of a balance of Upon substituting the Onsager coefficients (144) –(146) into Eqs. species fluxes (136) –(139) particularized for N = 2 , the expressions Consider again the slender interface separating two propellant 2 2 streams in Fig. 17 . The analysis begins by recalling that the char- Q = ρκF , F D F , O k F , T ∇ ∇ (ρY F ) + ∇ ( ρβv Y F )∇ T acteristic length of the variation of the temperature downstream + κ ∇(ρ ) ρ ∇ · + ∇ · ( + J ) F , F YF [ YF v JF F ] of the injection orifice is the thickness of the thermal mixing 2 2 layer δ . In contrast, the composition across the interface under- − κF , F ( J F + J F ) ∇ (ρY F ) + T ∇ ( ρβv Y F ) + (h F − h O ) J F (147) T goes rapid changes along much smaller distances of the same or- and der as the interface thickness δI = T δT δT . This large composi-

ρκ , D , W W X (1 − X ) tion gradient is due to the suppression of molecular diffusion in J = F F F O F O F F F thermodynamically unstable conditions at high pressures, where R 0 T W 2 2 the standard transport theory predicts antidiffusion of fuel and an × ∇ ∇ (ρY ) + ∇ ( ρβ Y )∇ T (148) F v F ever-increasing composition gradient, as discussed in Section 4.6 . are obtained for the interfacial ﬂuxes of heat and species, respec- It will be shown here that a mutual cancellation of the standard tively, along with and interfacial species ﬂuxes occurs at the interface in the limit T 1 corresponding to small temperature gradients compared to q = −λ∇T + (h F − h O ) J F − ρD F , O k F , T [∇ ( μF − μO ) + (s F − s O ) ∇T ] the composition gradients across the interface. The resulting trans-

(149) port equilibrium condition provides a quasi-steady thermochemical description of the interface structure that is shown schematically and in Fig. 20 and is elaborated in the remainder of this section. The normal components of the species fluxes (148) and (153) ca ρD F , O W F W O X F (1 − X F ) = − ∇ ( μ − μ ) + ( − ) ∇ JF [ F O sF sO T] be decomposed as R 0 T W − ρD (∇ / ) , = + T J = J + J T. F , O kF , T T T (150) JF , n JF , n JF , n and F , n F , n F , n (154) for the standard diffusion fluxes of heat and species, respectively, In this notation, the prime symbols refer to portions of the stan- where additional use of Eqs. (140) and (141) has been made. In Eqs. dard and interfacial species fluxes (153) and (148) that are inde- L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 33

Fig. 20. Schematics of the quasi-steady thermochemical structure of a transcritical interface near transport equilibrium in a non-isothermal system at small thermal Cahn numbers, T 1. pendent of the temperature gradients: The chemical-potential variations through the interface A μ, de- μ ﬁned in Eq. (35), also correspond to the variations of F through ∂ ln f ∂X X vW ∂P = −ρD F F + F V¯ − F , the coexistence region in thermodynamic space. These variations JF , n F , O F (155) ∂ ln X ∂ nˆ R 0T ∂ nˆ F P,T W are related by Eq. (F.23) to the variations of the logarithm of the fugacity with respect to composition as 3 2 ρκ , D , W W X (1 − X ) ∂ ∂ F F F O F O F F ∂ ln f ∂ ln f A μW J , = (ρY F ) + K I (ρY F ) . F F F F n 0 ∂ 3 ∂ 2 ∼ ∼ ∼ PZ (161) R T W nˆ nˆ ∂ ∂ 0 F ln XF , XF , R TF (156) P T P T

across the interface, where the fuel molar fraction XF is of order The superindex T refers to the portions of fluxes directly propor- unity. In writing Eq. (161) , use of Eq. (38) has been made in con- tional to the temperature gradients: junction with the definitions (37) and (39) for the pressure ratio ρD , k , ∂T P and the fuel compressibility factor Z , respectively. Utilization of T = − F O F T , F JF , n (157) D = D / D T ∂ nˆ Eqs. (32) and (161) in Eq. (155), along with F , O F , O F , OO as the binary diffusion coefficient normalized with its value D F , OO in the coflow stream, provides the definition of the nondimensional ρκF , F D F , O W F W O X F (1 − X F ) J T = F , n portion of the standard species flux involving barodiffusion trans- 0 R TW port, ∂T ∂ 2 ∂ × ( ρβ ) + ( ρβ ) . δ v YF KI v YF (158) T T J , ∂ nˆ ∂n 2 ∂ nˆ = F n . JF , n (162) ρF D F , OO PZ F Using this notation, and in the moving curvilinear frame de- T , scribed in Section 5.3 , the species conservation equation (46) for Using Eqs. (81) and (160) in Eq. (157), the remainder JF , n cor- the fuel near the interface becomes responding to the Soret effect, is nondimensionalized as ∂ ∂ ∂ δ J T YF YF T T T T F , n ρ + ρv n = − J , + J , + J , + J , J , = , (163) ∂ ∂ ∂ F n F n F n F n F n αρ D t nˆ nˆ F F , OO − + T + J + J T . KI JF , n JF , n F , n F , n (159) where α is a thermal-expansion ratio defined in Eq. (5) . In order to isolate the most important terms in Eq. (159) , con- The portion of the interfacial flux independent of the tem- J sider nondimensionalizing Eq. (159) with the same units as those perature gradient F , n is normalized by substituting Eqs. (32), employed to write the momentum Eq. (89) , namely Eqs. (32) and (81), (86), (32), (160) , the nondimensional molecular weight W = (81)–(86). Additional characteristic scales for the temperature and W /W O and the nondimensional partial molar volume v = ρF v /W F species fluxes are obtained as follows. into Eq. (156) , thereby yielding For 1, the temperature gradient near the interface is locally T δ J ∂ /∂ ∼ ( − ) /δ T T n, F a constant that can be approximated as T nˆ TO TF T . As a J , = . (164) F n ρ D P result, the variations of the temperature across the interface with F F , OO ZF respect to the interface temperature Te are of order T relative to J T The remainder F , n is normalized as the temperature difference between the free streams T O − T F . The dimensionless temperature is therefore defined as δ WJ T T T n, F J , = (165) F n α ρ D P T − T b¯ F F F , OO ZF T = e . (160) T (T O − T F ) 34 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

−1 −1 by using Eqs. (81), (86), (32), and (160) in Eq. (158), along whereas α/Le F = O (1) , PZ F /Le F = O (10 ) , N PZ F /Le F = O (10 ) , β = β /β (α P ) / ( W) = ( −3 ) with v v v F for the nondimensional volume expansivity, with and b¯ F ZF LeF O 10 are much smaller. Corre- β v F being the corresponding value in the fuel stream [see Eqs. spondingly, the first approximation to Eq. (168) is (C.22) for an expression of βv particularized for the equation of ∂ state (14) ]. In this formulation, b¯ F = T F β F is the ratio of β F to J , + J , = 0 . (174) v v ∂ F n F n n the ideal-gas volume expansivity in the fuel stream 1/TF , whereas The integration of Eq. (174) up to large distances compared with W = W F /W O is the ratio of molecular weights. The scalings in Eqs. (162) –(165) anticipate that the effect of the the interface thickness involves a constant that corresponds to the | + J | temperature gradients on the species transport fluxes near the in- sum of fluxes JF , n nˆ → + ∞ F , n nˆ → + ∞ . While the interfacial com- terface is small, ponent of that constant is zero for all practical purposes, the standard one J | may not be zero away from the interface, since T J T F , n nˆ → + ∞ J , α , αb¯ F n ∼ T 1 and F n ∼ T F 1 , (166) the coflow may mix with hot gas-like fuel transferred from the fuel P J W JF ,n ZF F ,n side. This large-scale mixing of propellants away from the interface is described by the species conservation equation (46) minus the (see Table 4 for typical values of α, , P, Z , b¯ and W), whereas T F F interfacial transport term, the dominant components of the standard and interfacial species fluxes are of the same order of magnitude ∂Y F ρ + ρv · ∇Y F = −∇ · J F , (175) ∂ / J = ( ) . t JF , n F , n O 1 (167) whose solution near the interface provides the aforementioned in- Eq. (167) suggests a balance between the two portions of the tegration constant J | → + ∞ . However, since Eq. (175) is driven by species fluxes that are independent of the temperature gradients. F , n nˆ gradients of composition over distances comparable to the thermal This aspect is further investigated below by examining the species mixing-layer thickness, the standard flux away from the interface conservation equation and is ratified by the numerical examples J | → + ∞ is anticipated to be of order compared to that near F , n nˆ T that will be provided in Sections 6 and 7. the interface J , and can therefore be neglected. As a result, the Upon substituting the variables (32), (81) –(86) , and (162) –(165) F , n integration of Eq. (174) leads to the transport equilibrium condition into Eq. (159) , the nondimensional species conservation equation + J = , JF , n F , n 0 (176) ∂Y ∂Y 1 ∂ ρ F + ρ F = − + J + N + J vn JF , n F , n T JF , n F , n ∂t ∂ nˆ E ∂ nˆ or equivalently s T T ∂ ∂ ∂ α ∂J α P ∂J ln fF XF XF vWF P F ,n T b¯ F ZF F ,n T + V¯ − − + T N J , − + T N J , 0 F F n F n ∂ ln X ∂ nˆ R T e ∂ nˆ Le F ∂ nˆ Le F W ∂ nˆ F P,T W 3 (168) κ , W W X (1 − X ) ∂ = F F F O F F (ρ ) , YF (177) 0 ∂ 3 is obtained, with R T e W nˆ

∂ ln f F ∂X F where use of Eqs. (155) and (156) has been made, and where J , = − ρ D , F n F O ∂ ∂ dimensional variables have been recovered in the notation. The ln XF , nˆ P T transport equilibrium condition (176) states that the Fickian and X F v W ∂P barodiffusion components of the standard species flux (153) bal- + V¯ − , (169) ( + α ) F ∂ τ T T W nˆ ance the portion of the interfacial species flux (148) independent of the temperature gradient, as sketched in Fig. 20 . ρ κ D ( − ) 3 2 F , F F , O XF 1 XF ∂ ∂ The interface temperature Te , which is generally a function of J , = (ρ Y ) + N (ρ Y ) , F n ∂ 3 F T ∂ 2 F time and the tangential coordinate sˆ , participates in Eq. (177) as (τ + αT T )W n n demanded by the leading-order asymptotic expansion of the fluxes (170) (169) and (170) for T 1. The temperature field around the inter- ρ D 2 , kF , T ∂T face evolves in time scales of order δ /D T , O , which are much larger J T = − F O , (171) T F , n δ (τ + αT T ) ∂ nˆ than the flow transit time across the interface I /UT . As a result, Te is a quantity that varies mostly quasi-steadily at interfacial scales. ρ κ D X (1 − X ) The boundary conditions required in the integration of Eq. T F , F F , O F F J , = F n (177) can be summarized as follows. First, the local thermody- W (τ + α T ) T namic pressure on the coflow side of the interface (see Fig. 17 ), 2 ∂T ∂ ∂ which can be approximated as the combustor pressure P ∞ at suf- × ( ρ β Y ) + N ( ρ β Y ) . (172) ∂ nˆ ∂n 2 v F T ∂ nˆ v F ficiently small Mach numbers, is recovered by the solution away from the interface, V¯ = ρ V¯ / In this formulation, F F F WF is the nondimensional fuel → ∼ → + ∞ . partial volume, τ = T e /T F is a temperature ratio, and Le F = P PO P∞ at nˆ (178) D , / D , is the fuel Lewis number evaluated in the coflow T O F OO Second, the fuel partial densities tend to their phase-equilibrium stream (see Table 4 for typical values of τ and LeF ). Additionally, values away from the interface, (∂ ln f F /∂ ln X F ) , denotes the partial derivative of the logarithm of P T ρ → ρ → −∞ , the fugacity with respect to the logarithm of the fuel molar frac- YF e YFe at nˆ (179) tion divided by PZ F , as suggested by Eq. (161) .

Following Table 4 , the largest factor premultiplying the differ- ρ → ρg g → + ∞ . YF e YFe at nˆ (180) ent terms in Eq. (168) is the one associated with the ﬁrst term on Concurrent with (180) and the equation of state (14) is the re- the right-hand side, which is inversely proportional to T and corresponds to the inverse of the transport-equilibrium parameter covery of the phase-equilibrium molar fraction on the coﬂow side of the interface, ρ 0 δ2 T LeF F DT , O R TF I −3 E = = = O (10 ) 1 , (173) g g s X → X = Y W /W at nˆ → + ∞ . (181) PZ F σ0 D F , OO W F δT F Fe Fe e F L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 35

→ ±∞ 0 In Eqs. (178)–(181), the limit nˆ denotes distances much αλ T αb¯ ρ D , R T PZ T ∼ O F , T ∼ F F F OO F F , δ δ qn Qn (189) larger than I but much smaller than T . Both phase-equilibrium δ δ W W g T T F ∞ values YFe and YFe are evaluated at P and Te. for the heat fluxes. These expressions indicate that qn and Qn are T T 5.6.2. The transport equilibrium condition in terms of a balance of of the same order of magnitude, whereas qn and Qn are negli- heat fluxes gible since Utilizing Eqs. (147) and (152) , it can be shown that the trans- T IG T port equilibrium condition (177) is equivalent to a balance of heat q αLe cˆ W γ Q αb¯ n ∼ T F O O n ∼ T F . 1 and 1 fluxes across the interface, q RPZ γ IG − Q W n F O 1 n + Q = , qn n 0 (182) (190) where qn is the sum of the Dufour effect and the interdiffusion of , heat by Jn, F IG In Eq. (190) , γ is the ideal-gas adiabatic coefficient defined in O 0 = / IG ρD F , O R T F W k F , T ∂ ln f F ∂X F Eq. (C.27), and cÔ cp , O cp , O is a ratio of real-to-ideal specific q = − n ( − ) ∂ ∂ heats at constant pressure, both parameters being evaluated in the WF WO XF 1 XF ln XF , nˆ P T γ IG coflow stream (see Table 4 for typical values of O and cÔ ). In X F vW F ∂P the vicinity of the interface, it can be also shown that convec- + V¯ − + (h − h ) J , (183) 0 F F O n F R T F W ∂ nˆ tion of enthalpy, viscous dissipation, pressure advection, curvature-

related terms, and enthalpy production by interfacial forces in Eq. and Q is the portion of the interfacial heat flux independent of n (185) are negligible compared to the normal derivative of the sum temperature gradients + Q qn n . 3 2 ∂ ∂ Q = ρκ , D , k , (ρY ) + K (ρY ) + (h − h ) J , . n F F F O F T ∂ nˆ 3 F I ∂ nˆ 2 F F O n F 5.6.3. Combination of mechanical and transport equilibrium (184) conditions in terms of chemical potentials Q Both qn and n participate in the enthalpy conservation equa- As discussed in Appendix E.1, the pressures within the inter- tion (51) when referred to the same curvilinear, moving coordinate face may become small enough to cross the mechanical spinodals system described above, when the combustor pressure decreases but it is still high rel- V¯ ∂ ∂ ∂ ∂ ∂ ative to the atmospheric value. In that case, singularities in F h h PGD PGD T T ρ + ρv n = + v n − q + q + Q + Q and ( ∂ ln f / ∂ ln X ) arise that can be regularized by combining ∂t ∂ nˆ ∂t ∂ nˆ ∂ nˆ n n n n F F P,T the Fickian and barodiffusion components into a single finite term ∂ T T vn − K q + q + Q + Q + K n,n , (185) given by the relation I n n n n ∂ nˆ where the viscous dissipation has been neglected by assuming ∂ W vW F ∂P ( μ − μ ) = V¯ − T F O ( − ) F small Mach numbers. In Eq. (185), the flux qn is the sum of ∂ nˆ W F W O 1 XF W ∂ nˆ Fourier conduction and interdiffusion of heat by the Soret effect, 0 R T ∂ ln f F ∂X F T T + q = −λ∇T + (h − h ) J , (186) (191) n F O F , n X ∂ ln X ∂ nˆ F F P,T Q T while n is the portion of the interfacial heat flux that depends on temperature gradients, obtained by evaluating Eq. (F.22) for N = 2 and approximating the constant-temperature gradients by ordinary ones because of the lo- 2 ∂T ∂ ∂ Q T = ρκ D ( ρβ ) + ( ρβ ) cally uniform temperature prevailing in the vicinity of the interface n F , F F , O kF , T v YF KI v YF ∂ nˆ ∂ nˆ 2 ∂ nˆ in the limit T 1. When expression (191) is used in Eqs. (155) and (183) , the scaling analysis performed above does not change in any ∂ ∂v n + κ , (ρY ) ρY + K v n fundamental manner and gives a transport equilibrium condition F F ∂ nˆ F F ∂ nˆ I valid over the entire high-pressure range that can be written as ∂ + T + J T + T + J T ∂ ∂ 3 JF , n F , n KI JF , n F , n ∂ nˆ ( μ − μ ) = κ , (ρY ) . (192) ∂ nˆ F O F F ∂ nˆ 3 F ∂ 2 ∂ T T − κ , J , + J , (ρY ) + K (ρY ) Eq. (192) is subject to the boundary conditions (113) - (114) away F F F n F n ∂ nˆ 2 F I ∂ nˆ F from the interface. ∂ 2 ∂ The transport equilibrium condition (192) can be combined T + T ( ρβv Y F ) + T K I ( ρβv Y F ) + (h F − h O ) J , . (187) ∂ nˆ 2 ∂ nˆ F n with the mechanical equilibrium condition (112) and integrated once giving the system of equations Similarly to the scaling analysis made for the species conservation Eq. (159) , it can be shown that the heat balance (182) is the ∂ 2 μ − μ = κ , (ρY ) , (193) result of integrating the first approximation to the enthalpy con- F Fe F F ∂ nˆ 2 F servation Eq. (185) ,

∂ ( + ) = . q Q 0 (188) μ − μ = 0 . (194) ∂ nˆ n n O Oe Expressions (183), (184), (186) and (187) , along with the rela- Eq. (193) is subject to the boundary conditions (109) –(110) , tion (161) and the dimensionless variables (32), (81), (82), (85), whereas Eq. (194) predicts that the chemical potential of the (86) , and (160) –(165) , motivate the characteristic scales coﬂow species remains constant across the interface, and equal 0 ρ D , R T PZ to its phase-equilibrium value, when only the fuel gradient-energy ∼ ∼ F F OO F F , qn Qn coeﬃcient is considered in the analysis. T δT W F 36 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

5.6.4. Generalizations to multicomponent systems for k = 1 , . . . N, which can be integrated once subject to the bound- The formulation can be easily generalized to multicomponent ary conditions (122) and (123) yielding systems. First, consider the generalized version of the species flux N ∂ 2 balance (176) , namely μ − μ = κ , (ρY ) (202) k ke k j 2 j ∂ nˆ + J = , = , . . . − . j=1 Ji,n i,n 0 i 1 N 1 (195) = , . . . In multicomponent systems, the species fluxes of the i -th com- for k 1 N. The system of Eq. (202) is subject to the boundary ponent, Eqs. (139) and (F.25) , depend on the Onsager coefficients conditions − of N 1 components, thereby making Eq. (195) of little interest for ρ → ρ → −∞ = , . . . , Yi e Y at nˆ for i 1 N (203) large N . A more useful version of Eq. (195) independent of the On- ie sager coefficients is derived in this section. ρ → ρg g → + ∞ = , . . . , Upon substituting the species fluxes (139) and (F.25) into Eq. Yi e Yie at nˆ for i 1 N (204) − (195) , the system of N 1 equations corresponding to the phase-equilibrium compositions away from N−1 the interface evaluated at P ∞ and T e . Additional approximations F = Li,k k 0 (196) made to arrive at Eq. (202) involve negligible curvatures, small k =1 Mach numbers, and negligible effects of viscous stresses at interface scales, as explained in Section 5.3.3 ; the consideration of these is obtained, with i = 1 , . . . N − 1 . In this formulation, F are com- k effects would require modification of Eq. (202) by using instead the ponents of an auxiliary vector defined as general form of the mechanical equilibrium condition (118) in the N ∂ 3 derivations above. F = (κ , − κ , ) (ρY ) k k j N j ∂ 3 j A relation exists between σ and the energy excess, or Lan- nˆ j=1 = ρ( − N μ ) dau potential, f k =1 Yk ke at the interface in condi- N−1 N−1 tions of mechanical and transport equilibrium. Upon multiplying X δ , ∂X j ∂ ln f − m + m k 0 m R Te Eq. (202) by ρY , summing the resulting expression from k = 1 to ∂ ∂ k = WNXN Wk = nˆ Xj , , m 1 j 1 P T Xr | r =1 , ...N −1 N , and using Eqs. (120) , (16) , and (C.7) and (C.8) , the equation ( r = j ) N N ∂ ∂ N−1 V¯ v X m V¯ m ∂P − e = κ , (ρY ) (ρY ) (205) − k − + , (197) GD k j ∂ ˆ k ∂ ˆ j = = n n W W N X N W N X N ∂ nˆ k 1 j 1 k m =1 is obtained. In Eq. (205) , = −P ∞ is the phase-equilibrium value e where use of Eqs. (56) and (140) has been made in order to re- of , as prescribed by Eq. (C.7), and GD is the gradient-dependent place the symbols ∇ , ψ , and ψ in Eq. (139) by their corre- T k N Landau potential sponding expressions under the approximation of locally uniform temperature. A similar substitution of the definition of heat fluxes N = ρ − μ . (138) and (F.24) into Eq. (182) yields GD fGD Yk ke (206) k =1 N−1 F = . Lq,k k 0 (198) The integration of Eq. (205) across the interface yields k =1 ∞ σ = ( − ) , GD e dnˆ (207) The concatenation of Eqs. (196) and (198) represents an over- −∞ determined system MF = 0 , with N equations for N − 1 un- knowns. The resulting matrix of coefficients M is the Onsager which provides a quantitative link between the interfacial excess matrix L given in Eq. (143) excluding the first column, with energy and the surface-tension coefficient. rank ( M ) ≤ N − 1 because L is positive semidefinite. In particular, rank ( M ) = N − 1 is achieved when L is positive definite, or equiv- 5.7. Remarks on transcritical interfaces in equilibrium and alently, when the entropy production (142) is strictly positive. In near-equilibrium conditions that case, the only possible solution to the over-determined sys- E E tem (196) and (198) is the trivial one, The dimensionless parameters m in Eq. (95) and s in Eq. (173) measure, respectively, the tendency of the interface to attain F = 0 (199) k mechanical and transport equilibrium. The smaller E m and E s , the E E for k = 1 , . . . N − 1 . more equilibrated the interface is. Both m and s are proportional Equation (199) represents the multicomponent version of the to the thermal Cahn number T , or dimensionally, to the tempera- transport equilibrium condition (177) and is also equivalent to the ture gradient across to the interface. simultaneous verification of the heat and species flux balances In isothermal systems, the thermal Cahn number is exactly = , E = E = (182) and (195) . Using Eq. (F.22) , the multicomponent transport zero, T 0 and therefore m s 0. The interface is in me- equilibrium condition (199) can be rewritten in terms of gradients chanical and transport equilibrium, in that the variations of the of chemical potentials as gradient-dependent pressure are exactly balanced with the variations of the interfacial stress normal to the interface, and the sum N ∂ ∂ 3 of the standard and interfacial species fluxes is exactly zero. The ( μ − μ ) = (κ , − κ , ) (ρY ) (200) ∂ nˆ k N k j N j ∂ nˆ 3 j structure of the interface can be described by the combination j=1 of the mechanical equilibrium condition (120) and the transport for k = 1 , . . . N − 1 . The combination of the multicomponent me- equilibrium condition (199) , or equivalently, by the system of Eq. chanical equilibrium condition (121) and the multicomponent (202) subject to the phase-equilibrium partial densities (203) and transport equilibrium condition (200) gives (204) in the far field. The system of Eqs. (202) has been utilized in early work addressing interfaces in multicomponent equi- N 3 ∂μ ∂ k = κ (ρ ) librium isothermal systems [17,37,100,109,118,121,131,132] , and can k, j Yj (201) ∂ nˆ ∂ nˆ 3 j=1 be alternatively derived by minimizing the volume integral of the L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 37 gradient-dependent Helmholtz free energy [116,117,120] . In addi- 6. Transcritical interfaces in isothermal bicomponent systems tion, the system (202) equivalently states that the generalized μ chemical potential k given by Eq. (58) remains constant through This section provides numerical results describing steady trans- the interface and equal to the phase-equilibrium chemical poten- critical interfaces in C12 H26 /N2 isothermal systems. The configura- μ tial ke . Correspondingly, the sources of entropy related to gra- tion analyzed here corresponds to the limit of infinitely thick ther- dients of temperature and generalized chemical potential in Eq. mal mixing layers, or equivalently, to the case of zero thermal Cahn (142) are exactly zero in mechanical and transport equilibrium. numbers, T = 0 , in which the interface is in mechanical and trans- For the binary mixtures studied here, the system of Eqs. port equilibrium. (202) simplifies to Eqs. (193) and (194) subject to the boundary conditions (109) and (110) . The solution to that problem 6.1. Formulation is a steady planar transcritical interface bearing surface tension and separating a liquid-like supercritical mixture rich in The equations integrated here are (193) and (194) , which are fuel from a gas-like supercritical mixture rich in coflow species, supplemented with the equation of state (14) and mixing rules with both mixtures being in phase equilibrium as discussed in (15) particularized for binary mixtures [see Eq. (212) introduced Section 6 . below], along with the expressions for the gradient-energy coeffi- None of the different forms of the transport equilibrium condi- cient (23) , chemical potential (B.23) , ideal-gas Gibbs free energy tion [i.e., Eqs. (176) , (177), (182), (192), (199), (200) , and (202) ] de- (C.46) , fugacity coefficient (C.47) , and the constants provided in pend on transport coefficients. Similarly, although high-order vis- Table C.1 and Appendix C . The boundary conditions away from cous effects can be retained in the derivation of the mechanical the interface are the phase-equilibrium partial densities (109) and equilibrium condition in Eq. (97) , the leading-order balance is in- (110) . The problem has a steady solution consisting of a thin inter- dependent of viscosity, as suggested by Eqs. (107), (112), and (120). face that bears surface tension and separates a liquid-like C12 H26 - As a result, the internal structure of a transcritical interface in rich supercritical mixture from a gas-like nitrogen-rich supercriti- mechanical and transport equilibrium is independent of viscos- cal mixture. A summary of the formulation is provided in Table 5 . ity, thermal conductivity, binary diffusion coefficient, and thermal- Details associated with the numerical integration of these equa- diffusion ratio. tions are discussed in Appendix G . In non-isothermal systems, the thermal Cahn number is larger The calculations focus on the four cases summarized in Table 6 . than zero, although in practical situations it remains small com- In all cases, the thermodynamic pressure away from the interface pared to unity, T 1, because the temperature gradients created P ∞ = 50 − 100 bar is larger than the critical pressures of the in- by the flow are never as large as the composition gradients across dividual components, while the temperature T e = 350 − 500 K is the interface. Even in the wake of the orifice ring separating both smaller than both the fuel critical temperature T c, F = 658 K and propellant streams in Fig. 1, where the temperature gradient is the the temperature of the diffusional critical point Tc ,diff at the cor- largest, the characteristic length for the temperature variations is responding pressure (i.e., see Table 3 ). As a result, the pseudo- the thickness of the orifice ring, which is always much larger than trajectories along the interface in thermodynamic space traverse the interface thickness. the coexistence region, as shown in Fig. 21 . Furthermore, while < For 0 T 1, the mechanical (95) and transport (173) equilib- case D only involves crossings of the diffusional spinodal, cases rium parameters attain small but non-zero values, 0 < {E m , E s } 1 . A, B, and C involve crossings of both diffusional and mechanical As a result, the interface is neither in mechanical equilibrium nor spinodals. It is therefore convenient to use the combined mechan- in transport equilibrium. However, the departures from equilibrium ical and transport equilibrium conditions written in terms of gra- are small, and the equilibrium conditions still describe the overall dients of chemical potentials (193) and (194) , since, as explained structure of the interface to a good approximation, as discussed in in Appendix E.1 , these forms are unaffected by singularities arising Section 7 . The interface temperature T e and the far-field bound- in the fuel partial molar volume and non-ideal diffusion prefac- ary conditions away from the interface may become functions of tor near the mechanical spinodals. It is also in cases A, B, and C space and time because of the large-scale evolution of the flow, where the thermodynamic pressure reaches negative values within but the characteristic length and time scales of these variations, the interface. This effect is inconsequential because the formula- δ = δ / δ2 / = / , T I T and T DT tI T are much larger, respectively, than tion in Table 5 is independent of the transport coefficients. δ the interface thickness I and the flow transit time across the in- δ terface tI (see Table 4 for typical values of I and tI ). As a result, 6.2. Results: the steady transcritical interface the structure of the interface evolves quasi-steadily in response to those flow variations, as sketched in Fig. 20 . The complete descrip- The profiles of density, fuel mass fraction, and thermodynamic tion of the evolution of the interface and the flow surrounding it pressure across the interface are shown in Fig. 22 (a–c). Density ra- requires the Navier-Stokes equations outlined in Section 4.7 sup- tios of order R ∼ 13 (for cases A and C at P ∞ = 50 bar) and R ∼ 7 plemented with the closures for interfacial terms derived (for cases B and D at P ∞ = 100 bar) are observed in the solu- above. tion that are spatially localized and accompanied by order-unity An intermediate case may exist in non-isothermal systems in variations of the fuel mass fraction in accordance with the phase- which the interface is much closer to mechanical equilibrium than equilibrium composition on each side of the interface. In particular, − < E E E / E = ( 3 ) to transport equilibrium, 0 m s 1 [e.g., m s O 10 on the fuel side of the interface, the mixtures are rich in C12 H26 in the conditions addressed in Table 4 ]. In this case, the species and have compressibility factors within the range Z ∼ 0.30–0.75, conservation equation (46) can be integrated simultaneously with which clearly depart from the ideal behavior corresponding to the mechanical equilibrium condition (120) . The latter is subject unity. The oscillation of the pressure across the coexistence region to boundary conditions corresponding to partial densities gener- is localized near the fuel side of the interface, where an under- ally away from phase equilibrium, and may involve pure com- pressure of order unity relative to P ∞ occurs. It will be shown that ponents in the propellant streams. This intermediate case is ad- this large underpressure leads to significant values of surface ten- dressed in Section 7 . The solution provides the spatiotemporal sion in accord with Eq. (108) . In contrast, on the other side of the evolution of the transcritical interface and the associated surface- interface, the mixture is rich in N2 and much less dense. The corre- tension coefficient downstream of the injection orifice, as pre- sponding compressibility factors are within the range Z ∼ 1.01–1.03, viewed in Figs. 2 and 3 . thereby indicating that the fluid approaches there the behavior of 38 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Table 5 Dimensional formulation for equilibrium transcritical ﬂat interfaces in isothermal bicomponent systems.

Conservation equations 2 μ − μ = κ d (ρ ) , Mechanical and transport equilibrium F Fe F , F YF d nˆ 2 μO − μOe = 0 .

Boundary conditions ρ → ρ → −∞ , Phase-equilibrium composition YF e YFe at nˆ ρ → ρg g → + ∞ . YF e YFe at nˆ Supplementary expressions (see Appendix B, Appendix C , and Appendix F for details)

Equation of state Eq. (14) , Coeﬃcients of the equation of state Eqs. (212) , and (C.10) - (C.12) , Gradient-energy coeﬃcient Eqs. (23) , (C.17) , and (C.18) , Thermodynamic relations Eqs. (213) , (214), (B.19), (B.20) , (C.43) , and (C.47) .

Fig. 21. Thermodynamic-space pseudo-trajectories across transcritical interfaces in C 12 H 26 /N 2 isothermal systems for cases A-D described in Table 6 . The ﬁgure includes isocomposition cross-sections of the phase-equilibrium surface colored by fuel mass fraction.

Table 6 Section 4.6 . In contrast, the barodiffusion component acts in the

Thermodynamic conditions and phase-equilibrium composition corresponding to opposite direction by transporting fuel along the positive pressure the pseudo-trajectories A-D in Fig. 21 for transcritical interfaces in C H /N 12 26 2 isothermal systems. gradient, albeit with less intensity than the Fickian component. As a result, the standard species ﬂux J is dominated by Fickian an- Phase-equilibrium composition F = nˆ → −∞ nˆ → + ∞ tidiffusion within the interface, with zero-ﬂux points JF 0 being (liquid-like supercritical) (gas-like supercritical) coincident with the diffusional-spinodal states given by the max-

ρ kg ρg kg g imum and minimum locations of the fuel chemical potential, as case Te [K] P∞ [bar] e Y e Y m 3 Fe m 3 Fe shown in Fig. 23 (b). A 350 50 642.1 0.983 48.1 0.002 No steady solution of the problem exists for κF , F = 0 (i.e., B 350 100 642.5 0.966 95.3 0.003 J = 0 ). Specifically, κ , = 0 would impede the balance shown in C 500 50 541.9 0.979 40.2 0.202 F F F J D 500 100 539.8 0.957 74.9 0.151 Fig. 23(a) between the interfacial species flux F and the stan- J dard one JF . The interfacial species flux F provides the necessary amount of positive transport of fuel, in the direction of decreasing fuel concentration, to yield a finite-thickness interface that persists an ideal gas despite the high pressures, the reason being that N2 indefinitely in time. is highly supercritical in temperature. The interface thickness can be computed as A point-wise cancellation between the standard and interfacial g g ρ Y − ρ Y fluxes of species is observed within the interface in Fig. 22(d). Pos- e Fe e Fe δI = . (208) itive values of the standard diffusion flux J (i.e., in the direction of ρ F d ( YF ) max decreasing fuel concentration) are observed on both flanks of the d nˆ interface, whereas negative values (i.e., in the direction of increasing fuel concentration) occur inside. Despite the high pressures considered here, Fig. 24 (a) indicates δ A detailed breakdown of the species fluxes is provided in that I calculated using Eq. (208) remains small compared to hy- δ = Fig. 23(a) for case D. In the diffusionally unstable region of the drodynamic scales of interest in practical systems (i.e., I 1.59– δ interface, the Fickian component of the species flux antidiffuses 2.98 nm). Specifically, the values of I observed here are compara- fuel in the direction of increasing concentration, as anticipated in ble to those arising in subcritical monocomponent systems of sep- L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 39

Fig. 23. (a) Standard species ﬂux J F (dark line), including the Fickian (dot-dashed line) and barodiffusion (dashed line) components, along with the interfacial species

flux J F (green line) arising from the present diffuse-interface theory. (b) Spe- cific fuel chemical potential highlighting the diffusional-spinodal states. The results shown in this figure correspond to case D in Table 6 for transcritical interfaces

in C 12 H 26 /N 2 isothermal systems. The normalizations in this ﬁgure have been carried out with the phase-equilibrium composition provided in Table 6 along with

D g = . · −7 the phase-equilibrium values for the binary diffusion coefficient F , Oe 2 1 10 2 = . , = m /s, compressibility factor Ze 0 62 molecular weight W e 139 g/mol, fuel specific chemical potential μFe = −3 . 63 MJ/kg, and pressure parameter P e = 0 . 64 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

arate N2 or C12 H26 close to their critical points (see Appendix D). However, because of the high pressures attainable in the coexistence region of hydrocarbon-fueled mixtures, the associated Knudsen numbers Kn I ∼ 0 . 02 − 0 . 15 are relatively smaller here [see Fig. 24 (b)], and can be made arbitrarily smaller by further increasing the pressure, or by increasing the temperature, with → KnI 0 as the pressure or temperature conditions approach those of the diffusional critical point. As a result, the continuum hypothesis underlying the present diffuse-interface theory may be much more appropriate in transcritical bicomponent systems than in subcritical monocomponent ones. The surface tension coefficient σ computed from Eq. (99) is shown in Fig. 25 (a). Despite the high pressures, the predicted values of σ are dynamically significant over a wide tempera- Fig. 22. (a) Density, (b) fuel mass fraction, (c) thermodynamic pressure, and (d) σ fuel species flux across transcritical interfaces in C 12 H 26 /N 2 isothermal systems at ture range. They amount to 30–40% of of water in air at at- T e = 350 K (left column) and T e = 500 K (right column) at the pressures indicated mospheric pressure. Discrepancies of 10–20% are observed be- J in panel (a). Panel (d) includes the interfacial species flux F (green lines), and tween these calculations and experimental measurements of σ in the standard species flux JF (dark lines). The spatial coordinate is normalized with diesel/nitrogen interfaces reported in Ref. [15] [e.g., σ (50 bar, 308, δ = 1 . 59 nm, 1.71 nm, 2.75 nm, and 2.98 nm for cases A, B, C, and D, respectively I ≈ σ ≈ (see Table 6 ). (For interpretation of the references to color in this figure legend, the K) 23.0 mN/m and (50 bar, 323 K) 21.0 mN/m in experiments reader is referred to the web version of this article.) [15] , versus σ (50 bar, 308 K ) = 28 . 1 mN/m and σ (50 bar, 323 K ) = 27 . 2 mN/m in the present work]. Similar discrepancies are observed in Fig. 26 between the present work and the experiments 40 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 25. Surface-tension coeﬃcient (a) as a function of temperature at different pressures, and (b) as a function of pressure for different temperatures. All cases Fig. 24. (a) Interface thickness and (b) Knudsen number as a function of temper- correspond to C 12 H 26 /N 2 isothermal systems. ature in C 12 H 26 /N 2 isothermal systems at two pressures [refer to legend in panel (a)].

in Ref. [16] when decane and pentane are used instead of dodecane. As P ∞ and T e increase, the amplitudes of the species fluxes decrease, the interface becomes thicker, the Knudsen number decreases, and the variations of density, pressure, and composition across the interface decrease, thereby causing a decrease in σ , as shown in Fig. 25 (a). In particular, the surface tension vanishes and δ I diverges to infinity when Te equals the temperature of the diffu- = sional critical point Tc ,diff at P P∞ . Steady transcritical interfaces > cannot exist for Te Tc ,diff in C12 H26 /N2 mixtures. In those conditions, the mixture is fully supercritical, there is no mechanism that opposes the diffusion of fuel in the direction of decreasing con- Fig. 26. Surface-tension coefficient as a function of pressure in isothermal systems centrations, and therefore the composition gradient decreases with at T e = 313 K composed of N 2 and decane (C 10 H 22 ) or pentane (C 5 H 12 ), including time. experiments (symbols from Ref. [16] ) and numerical results from the present work

The limiting value for the temperature above which transcrit- (solid and dashed lines). ical interfaces disappear may not necessarily be Tc ,diff in mixtures whose thermodynamic phase diagram is substantially different from the ones engendered by the hydrocarbon-fueled mix- 6.3. The generalized Cahn-Hilliard equation tures studied here. For instance, while C12 H26 /N2 mixtures are ࣃ In isothermal systems, the constant-temperature gradient of characterized by Tc ,diff Tc ,mix (see Table 3), other mixtures may chemical potentials becomes an ordinary gradient, and therefore have Tc ,mix Tc ,diff, in such a way that the maximum temperature the species conservation equation (46) can be written as of the diffusional spinodal surface could be intermediate to Tc ,diff and Tc ,mix . In that case, transcritical interfaces could still exist for − > N1 Te Tc ,diff, but would disappear if Te becomes larger than the max- ∂Y L , ρ i + ρ · ∇ = ∇ · i k ∇ μ − μ imum temperature of the diffusional spinodal surface. v Yi k N ∂t T e k =1 Transcritical interfaces persist even after large increments in Standard transport (Fickian+barodiffusion) pressure. As shown in Fig. 25(b), at ﬁxed temperatures of prac- = tical interest, Te 350–500 K, transcritical interfaces survive until − ∇ · ( ψ − ψ ) k N (209) pressures P ∞ ∼ 60 0–120 0 bar, above which the coexistence region disappears and the mixture becomes fully supercritical. Diffuse-interface correction L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 41

= , . . . , , for i 1 N where use of Eqs. (137) and (139) has been b = X F b F + (1 − X F ) b O , ψ made. In Eq. (209), the auxiliary vectors k are deﬁned in Eq. ( ) = ( − ϑ ) ( ) ( ), aF , O Te 1 F , O aF Te aO Te (212) (56), whereas the Onsager coeﬃcient Li , k are calculated using Eq.

(F.30) and the method described in Appendix F.4. with aF , aO , bF , and bO being functions of temperature defined in ϑ Equation (209) is a generalized multicomponent version of the Eqs. (C.10) and (C.11), whereas F,O is a binary-interaction parame- original Cahn-Hilliard equation derived in Ref. [109] for binary sys- ter provided in Appendix C . In Eqs. (211) and (212) , the molar and tems near the critical point. In contrast to Ref. [109] , no assump- mass fractions of fuel species are related as tion of a simplified predetermined form of the difference of chem- X F = Y F W /W F , (213) μ − μ ical potentials k N has been made here. Instead, the chemical potentials are described exactly by Eqs. (B.22) , (B.23) , and by where W is the mean molecular weight defined as Eq. (C.47) for the Peng-Robinson equation of state (14) . Setting the W = W F / [Y F + W(1 − Y F ) ]. (214) right-hand side of the generalized Cahn-Hilliard Eq. (209) to zero The substitution of Eqs. (C.46) and (212) –(214) into Eq. (211) , with gives the transport equilibrium condition (200) . That is an unde- P being computed from the equation of state (14) , makes μF − μO − termined system of N 1 equations that needs to be combined ρ a sole function of and YF in isothermal systems. with the mechanical equilibrium condition (121) and integrated A constant-pressure approximation P ≈ P ∞ in Eq. (211) becomes once to yield the determined system of N Eq. (202), as described appropriate only at sufficiently high pressures near the diffusional in Section 5.6.4. critical point. For C12 H26 /N2 mixtures at uniform temperature of

Numerical simulations employing diffuse-interface frameworks 500 K, the pressure at the diffusional critical point is 688 bar, as in CFD codes have been aimed at predicting complex two-phase shown in Fig. A.1 . On approach to the diffusional critical point at

flows at near-atmospheric pressures [145–148]. Those simulations that temperature, the pressure parameter P defined in Eq. (38) , employ ad-hoc versions of Eq. (209) to avoid numerical smearing corresponding to the ratio of the characteristic amplitude of the of the interface [e.g., see Eq. (57) in Ref. [147]]. Key assumptions pressure oscillation through the interface to the combustor pres- made in those formulations are: −3 −4 −4 −7 sure, is P = 4 · 10 , 8 · 10 , 1 · 10 , and 6 · 10 at P ∞ = 500 , (a) P and ρ are uniform through the interface. 600, 650, and 685 bar, respectively. As a result, the thermodynamic κ pressure becomes increasingly more uniform near the diffusional (b) Li , k and i , j are constants. (c) The standard component of the species flux is approximated by critical point. a polynomial expansion. The equation of state (14) with fixed P = P ∞ and T = T e makes (d) The order of the Cahn-Hilliard equation is dropped twice to the density an implicit function of the fuel mass fraction, ρ = ρ( ) make it second order, and therefore numerically more manage- YF . However, the density variations across the interface decrease

able. as the diffusional critical point is approached. For C12 H26 /N2 mixtures at uniform temperature of 500 K, the relative density varia-

It is shown in this section that approximations (a), (b), and (c) in- (ρ − ρg) /ρ = , tions across the interface are e e e 37% 24%, 15%, and 4% cur large errors in the predictions of this theory unless the condi- at P ∞ = 500 , 600, 650, and 685 bar, respectively. These variations tions are very close to the diffusional critical point. Approximation decrease with pressure at a much slower rate than the pressure

(d) does not bear any physical justiﬁcation. variations.

To examine approximations (a-c), consider Eq. (209) particular- κ The gradient-energy coeﬃcient F,F is a sole function of tem- ized for a bicomponent hydrocarbon-fueled system, perature, and therefore remains constant in the isothermal con- ∂ YF LF , F 2 ditions that lead to Eq. (210). In contrast, the Onsager coeﬃ- ρ + ρ · ∇ = ∇ · ∇ μ − μ − κ , ∇ (ρ ) , v YF F O F F YF (210) ∂t T e cient LF,F is a function of P, T, and XF , as indicated by Eq. (145).

κ For C12 H26 /N2 at uniform temperature of 500 K, the variations of with F,F and LF,F being deﬁned, respectively, in Eqs. (23) and (145).

LF,F induced by variations of composition across the interface are Upon combining Eqs. (B.19), (B.20), and (C.47), the difference be- g (L − L ) /L = 73% , 60%, 45%, and 15% at P ∞ = 50 0 , 60 0, 650, tween chemical potentials can be expressed in dimensionless form F , Fe F , Fe F , Fe as and 685 bar, respectively. Similarly to the approximation of constant density, the approximation of constant L involves signifi- 0 0 0 0 F,F μ − μ g¯ (P , T e ) g¯ (P , T e ) F O = F − W O cant errors of order 15% even at 99.5% of the diffusional critical 0 / 0 0 R T e WF R Te R Te pressure. b b P W As shown in Fig. 23 , the standard species flux engenders antid- + F − W O − 1 0 iffusive transport of fuel species within the diffusionally unstable b b ρR T e zone of the interface, and forward diffusion everywhere else. The ρ 0 ρ( − ) 0 XF R Te 1 XF R Te slope of the chemical-potential difference must therefore change + ln − W ln μ − μ P 0W (1 − ρb/W ) P 0W (1 − ρb/W ) sign twice along the YF axis. An approximate model for F O √ that fulfills this requirement is [149] 1 + 1 − 2 ρb/W a + √ ln √ Y g + Y 0 + + ρ / g Fe Fe 2 2bR Te 1 1 2 b W μ − μ ≈ μ − μ + Aˆ μ(Y − Y )(Y − Y ) Y − , F O Fe Oe F Fe F Fe F 2 ( + ( − ) ) × 2 XF aF 1 XF aF , O − bF (215)

a b where Aˆ μ = Aˆ μ(P ∞ , T e ) is a calibrated model coefficient. The model 2 ( X F a F , O + (1 − X F ) a O ) b O (215) intersects the ordinate μF − μO = μFe − μOe at the phase- −W − . (211) g , a b equilibrium fuel mass fractions Y and Y and at their average Fe Fe value (Y g + Y ) / 2 . Its oscillatory trend is such that the integral of 0 0 Fe Fe In this formulation, g¯ and g¯ are ideal-gas Gibbs free energies g F O μ − μ with respect to Y equals (μ − μ )(Y − Y ) in accor- defined in Eq. (C.46) . In addition, a and b are the Peng-Robinson F O F Fe Oe F F dance with Maxwell’s construction rule. coefficients obtained from the mixing rules (15) particularized for Comparisons between the model (215) and the exact difference binary mixtures, namely of chemical potentials (211) are provided in Fig. 27 (a). The coeffi-

= 2 ( ) + ( − ) 2 ( ) + ( − ) ( ) , ˆ a XF aF Te 1 XF aO Te 2XF 1 XF aF , O Te cient Aμ has been tuned such that the model (215) reproduces the 42 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

and density, the latter being approximately equal to the phase- ρ ≈ ρ equilibrium density on the liquid-like side, e . With these approximations, Eq. (216) becomes 2 ˆ dYF Aμ g 2 2 κ , ρ ≈ (Y − Y ) (Y − Y ) , (217) F F e d nˆ 4 F Fe F Fe

subject to Y → Y at nˆ → −∞ . The solution of Eq. (217) is F Fe g g + − YFe YFe YFe YFe 2nˆ Y ( nˆ ) = − tanh , (218) F 2 2 ˆ δI with g Y − Y ρ κ F F 8 e F , F δˆ = = (219) I g dY Y − Y Aμ max F F F d nˆ

being a modeled interface thickness. The mass fraction profile (218) can be substituted into Eq. (99) under the assumption of con- ρ ≈ ρ , stant density e which gives the modeled surface-tension coefficient ( − g ) 3 YFe YFe σˆ = (ρ ) 3κ , Aˆ μ. (220) 12 e F F The model σˆ leads to large errors with respect to the exact surface-tension σ coefficient because (a) the density is not constant across the interface, and (b) the discrepancies in the mass- fraction profiles in Fig. 27 (b) get amplified when they are differentiated, squared, and integrated to obtain σˆ . For instance, at P ∞ = 500 bar and T e = 500 K, the model predicts σˆ = 1 . 0 mN/m, while the exact calculation gives σ = 1 . 5 mN/m. Similar errors are observed for all tested pressures up to 685 bar. Additional shortfalls of the model expressions (218) –(220) are that their derivation does not involve the equation of state (14) nor the mechanical equilibrium condition (107) . Near the diffusional critical point, it is however plausible that Eq. (107) simplifies to P ≈ P ∞ with relative errors of order P 1 . In contrast, the equation of state (14) is ρ strictly incompatible with variations in YF once P, T, and are set ρ to their constant values P∞ , Te , and e . Fig. 27. Comparison between modeled (blue dashed lines) and exact (dark solid lines) profiles of (a) dimensionless chemical-potential difference in C H /N 12 26 2 7. Transcritical interfaces in non-isothermal bicomponent isothermal systems at T e = 500 K and different pressures, and (b) corresponding mass fractions as a function of distance normalized with the interface thick- systems ness computed using Eq. (208) based on the exact solution ( δI = 6 . 8 nm, 9.9 nm, 16.4 nm, and 62.3 nm for P ∞ = 500 , 600, 650, and 685 bar, respectively). To ob- This section provides numerical results describing unsteady ˆ ˆ = tain these profiles, the coefficient Aμ in Eq. (215) has been calibrated as Aμ transcritical interfaces in C12 H26 /N2 non-isothermal systems. This 391 . 8 kJ/kg, 344.4 kJ/kg, 323.5 kJ/kg, and 313.2 kJ/kg for P ∞ = 500 , 600, 650, and

685 bar, respectively. Similarly, the equilibrium densities on the liquid-like side are conﬁguration corresponds to small but non-zero thermal Cahn ρ = . 3 3 3 3 = , < e 520 6 kg/m , 509.7 kg/m , 499.0 kg/m , and 478.8 kg/m for P∞ 500 600, numbers, 0 T 1. In this limit, which is perhaps the one that 650, and 685 bar, respectively. (For interpretation of the references to color in this has the most practical interest, temperature gradients exist across

figure legend, the reader is referred to the web version of this article.) the interface that are however small compared to the composition gradients. maximum value of (211) evaluated at P = P ∞ . However, the amplitude match occurs at increasingly dissimilar fuel mass fractions as 7.1. Formulation the pressure decreases, since the exact profile of μF − μO becomes increasingly asymmetric. The temperature gradient across the interface changes funda- The good agreement between the model and the exact expres- mentally the solution with respect to that described in Section 6 . sion (211) at high pressures in Fig. 27 (a) translates into corre- Here, T e varies along the interface, and therefore the structure of spondingly good predictions of the spatial distribution of fuel, with the interface evolves with distance downstream of the injection discrepancies with the numerical results subsiding with increasing orifice. The spatial functionality of T e is determined by the rate of heat transfer across the interface. The solution provides the pressures, as observed in Fig. 27 (b). An expression for Y F ( nˆ ) can be derived from the model (215) by integrating twice the right-hand size of the transcritical region, within which the interface survives, σ side of Eq. (210) , subject to zero gradients and phase-equilibrium and where attains values of practical relevance before vanish- conditions away from the interface, giving ing at the interface edge. The transcritical interface is away from equilibrium, although the departures are small, as discussed in d 2 κ (ρ ) = μ − μ − (μ − μ ) . Section 5.7 . The problem must be solved using the Navier-Stokes F , F YF F O Fe Oe (216) d nˆ 2 equations in Section 4.7 , along with the closures for the interfa- Equation (216) can be integrated once more by using the identity cial stress K and interfacial fluxes J F and Q provided in Section 5 . 2 2 2 d Y F /d nˆ = (d/dY F )(dY F /d nˆ ) , and by assuming uniform pressure The numerical simulation of the problem sketched in Fig. 1 is not L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 43

exists in which the streawmise velocity remains equal to U everywhere. In this way, time and space can be used interchangeably in the equations of motion by means of the relation

t = x/U. (221)

The origins t = 0 and x = 0 are chosen such that the initial material surface coincides with the ﬂuid state of the propellants at the injection plane. Using Eq. (221) , the characteristic supercriticalization length (13) can be alternatively expressed as the time scale

t TR = LTR /U, (222)

which is used for normalization of the results. The formulation integrated in this section is summarized in Table 8 , where t plays the role of the streamwise marching coordinate x in accordance with Eq. (221) . Listed in Table 8 are the conservation equations (44), (46) and (47) , along with the species and heat fluxes (148), (147), (150) , and (152) . The mechanical equilibrium condition (107) is employed in place of the full momentum Eq. (45) . These equations are supplemented with Fig. 28. Schematics of the non-isothermal bicomponent model problem. the equation of state (14) and the mixing rules (212) , along with the expressions for the specific gradient-dependent internal energy (17) , gradient-energy coefficient (23) , total energy (48) , spe- straightforward, in that the large disparity between the interface cific entropy (151) , fuel molar fraction (213) , mean molecular δ thickness I and the orifice radius RF involves significant numerical weight (214), enthalpy (B.9), partial enthalpy (B.12), chemical po- stiffness, as discussed in Section 4.7 . A number of simplifications tential (B.19) and (B.20) , volume expansivity (C.22) , enthalpy de- outlined below are employed in order to facilitate the analysis. parture function (C.23) , ideal-gas partial enthalpies (C.33) , partial- The configuration studied in this section is summarized in enthalpy departure function (C.34) , partial molar volume (C.43) , Fig. 28 and corresponds to a simpler version of that depicted in ideal-gas Gibbs free energies (C.46) , fugacity coefficient (C.47) , and Fig. 1 . This configuration does however serve to illustrate the de- the associated constants and supplementary expressions provided velopment and disappearance of the transcritical interface in a in Appendix C . Effects of shear and interface roll-up require the manner qualitatively analogous to that sketched in Fig. 1 . The two consideration of streamwise transport and interface curvature. Al- propellant streams are injected at the same velocity U creating though these effects are not included in this section, they are ac- a slender, planar, laminar thermal mixing layer at a moderately counted for in the general formulation provided in Sections 4.7 and high Péclet number Pe R = UR F /D T , O 1 . The transcritical interface 5 . separating both propellant streams remains hydrodynamically thin The thermal conductivity, binary diffusion coefficient, and and slender until it vanishes downstream at an edge, where the thermal-diffusion ratio are calculated using Eqs. (F.13) , (F.36) , and transverse velocity, which is of the same order as the thermal- (F.43) , respectively, along with the supplementary relations in = , expansion velocity scale UT defined in Eq. (80), is much smaller Appendix F. Those expressions are evaluated at P P∞ as dis- than U by a factor of order 1 / (Pe R R ) 1 . Because of this slender- cussed in Appendix E.2 . In addition, the regularization (E.8) intro- ness, the transversal coordinate y and the coordinate normal to the duced in Appendix E.1 is employed here to maintain finite values interface nˆ are approximately equal, y nˆ. The Marangoni effect is of the difference of partial specific enthalpies h F − h O and of the neglected for simplicity. volume expansivity βv , since both would otherwise diverge across Table 7 summarizes the operating conditions of the configura- the mechanical spinodal surface. Specifically, Eq. (E.8) is used with = . = = tion. The fuel (C12 H26 ) and coflow (N2 ) injection conditions corre- m 0 01 from x 0 (or t 0) until a distance x (or time t) where spond, respectively, to liquid-like and gas-like supercritical fluids, T e becomes larger than the corresponding temperature threshold with injection temperatures T F = 450 K and T O = 10 0 0 K. Addition- provided in Fig. E.1 at the operating pressure (i.e., T e > 480 K at ally, the combustor pressure P ∞ = 100 bar is larger than the critical P ∞ = 100 bar). A sweep in the regularization constant m is per- pressures of the separate components. formed in Appendix E.1 that suggests negligible influences on the Under these simplifying assumptions, the boundary-layer ap- streamwise evolution of σ , as shown in Fig. E.4 . proximation, in which streamwise diffusion is neglected, may be The boundary condition far away from the axis ( nˆ → + ∞ ) cor- used for describing the dynamics of the interface along with the responds to the coflow temperature T = T O and composition Y F = 0 . mixing process. A solution of the streamwise momentum equation A symmetry boundary condition is used at the axis nˆ = 0 , where

Table 7

Operating conditions employed for the investigation of transcritical interfaces in C 12 H 26 /N 2 non-isothermal systems. The regularization constant is m = 0 . 01 in Eq. (E.8) , and the dimensional radius of the oriﬁce is assumed

to be R F = 50 μm. propellant streams fuel coﬂow liquid-like supercritical gas-like supercritical = = C12 H26 (YF 1) N2 (YF 0) parameters for initial conditions kg kg ρ ρ 0 = δ0 / 0 = δ0 /δ0 δ0 0 P∞ [bar] TF [K] F TO [K] O RF [nm] Te [K] m3 m 3 R I T I T I − 100 450 606 1000 33 5 · 10 −5 5 · 10 4 2.5 460 44 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Table 8

Dimensional formulation for transcritical interfaces in non-isothermal bicomponent systems. ( ) Using Eq. (221) , ∂ / ∂ x has been replaced by (1/ U ) ∂ / ∂ t .

Conservation equations ∂ρ ∂ Continuity + ( ρv ) = 0 , ∂t ∂ nˆ n 2 ∂ 2 ∂ − + ρ κ ρ − 1 κ ρ = , Mechanical equilibrium P∞ P YF F , F ( YF ) F , F ( YF ) 0 ∂ nˆ 2 2 ∂ nˆ ∂ ∂ ∂ Fuel species ( ρY ) + ( ρv Y ) = − ( J + J ), ∂t F ∂ nˆ n F ∂ nˆ F F ∂ ∂ ∂ ∂v n Total energy ( ρE ) + ( ρv E ) = − ( q + Q ) − P ∞ . ∂t ∂ nˆ n ∂ nˆ ∂ nˆ Transport ﬂuxes ρD , W W X (1 − X ) ∂ ∂T ρD , k , ∂T = − F O F O F F ( μ − μ ) + ( − ) − F O F T , Standard (fuel species) JF F O sF sO R 0T W ∂ nˆ ∂ nˆ T ∂ nˆ 2 2 ρκ , D , W W X (1 − X ) ∂ ∂ ∂T ∂ Interfacial (fuel species) J = F F F O F O F F (ρY ) + ( ρβ Y ) , F 2 F 2 v F R 0T W ∂ nˆ ∂ nˆ ∂ nˆ ∂ nˆ ∂T ∂ ∂T Standard (heat) q = −λ + (h F − h O ) J F − ρD F , O k F , T ( μF − μO ) + (s F − s O ) , ∂ nˆ ∂ nˆ ∂ nˆ

∂ ∂ 2 ∂ ∂ 2 ∂ ∂ ∂ ( + J ) Q = ρκ D (ρ ) + T ρβ + κ (ρ ) ρ vn + JF F Interfacial (heat) F , F F , O kF , T YF ( v YF ) F , F YF YF ∂ nˆ ∂ nˆ 2 ∂ nˆ ∂ nˆ 2 ∂ nˆ ∂ nˆ ∂ nˆ ∂ 2 ∂ 2 − κ + J (ρ ) + ρβ + ( − ) J F , F ( JF F ) YF T ( v YF ) hF hO F . ∂ nˆ 2 ∂ nˆ 2

Boundary conditions ∂ρ ∂Y ∂T Symmetry at the axis = F = = 0 and v = 0 at nˆ = 0 , ∂ nˆ ∂ nˆ ∂ nˆ n Coﬂow free stream Y F → 0 and T → T O at nˆ → + ∞ .

Initial conditions (t = 0 ) − ρF + ρO ρF − ρO 2 nˆ RF Density ρ( nˆ ) = − tanh for 0 ≤ nˆ < + ∞ , 2 2 δρ0 − 1 2 nˆ RF Fuel mass fraction Y ( nˆ ) = 1 − tanh for 0 ≤ nˆ < + ∞ , F 2 δ0 Y

T + T T − T 2 nˆ − R e Temperature T ( nˆ ) = F O − F O tanh for 0 ≤ nˆ < + ∞ , 2 2 δ0 T

Velocity v n = 0 for 0 ≤ nˆ < + ∞ .

Supplementary expressions (see Appendix B, Appendix E, Appendix C , and Appendix F for details)

Equation of state Eq. (14) . Coefficients of the equation of state Eqs. (212) , and (C.10) - (C.12) . Gradient-energy coefficient Eqs. (23) , (C.17) , and (C.18) . Transport coefficients Eqs. (F.13) , (F.36) , and (F.43) . Regularizations † Eqs. (E.8) and (E.9) . Thermodynamic relations Eqs. (17) , (48), (B.9), (B.12), (B.19), (B.20) , (C.22), (C.23), (C.33), (C.34), (C.43), (C.46) , and (C.47) .

the transversal velocity along with the gradients of temperature, tial value of the fuel partial density at the symmetry axis nˆ = 0 3 fuel mass fraction, and density are zero. (i.e., ρY F = ρF = 606 kg/m ) into the definition The presence of the interface complicates the characterization of the initial conditions, which may otherwise be calculated from ρ( nˆ = 0)Y ( nˆ = 0) δ = F . the self-similar solution of the conservation equations near the ori- I (223) ∂ ( ρ ) / ∼ , YF fice at x/RF 1 and nˆ RF 1 when the thickness of the separating max ∂ plate is also neglected. This procedure is not attempted here be- nˆ cause of the relatively unknown character of the formulation in Table 8 at present time, and is deferred to future work. Instead, The resulting value of the initial large-scale Cahn number is at t = 0 (or x = 0 ), the initial conditions for the density and fuel 0 = δ0 / = · −5 δ0 R I RF 5 10 . Similarly, the initial value T of the ther- mass fraction are assumed to be hyperbolic tangents centered at mal mixing-layer thickness can be obtained by substituting the = , nˆ RF whereas that of the temperature has a wider support and initial value of the temperature at the symmetry axis nˆ = 0 (i.e., = > , is centered at nˆ Re RF as formulated in Table 8. The origin of T = T F = 450 K) into the definition the temperature profile R e is chosen such that the initial value of 0 , the interface temperature, denoted by Te is initially imposed at T O − T ( nˆ = 0) = 0 0 < , δ = , nˆ RF . Specifically, the value of Te is selected such that Te Tc, diff T (224) ∂ T and is assumed to be closer to the fuel temperature than to the max coflow one because of the much smaller thermal diffusivity of the ∂ nˆ δ0 δ0 fuel stream. The parameters ρ and Y for the initial profiles of density and fuel mass fraction are such that the resulting initial which yields an initial thermal Cahn number of 0 = δ0 /δ0 = 5 · 0 T I T value of the interface thickness δ is equal to that calculated in −4 0 0 0 I 10 , with δ /R F = / = 0 . 1 . Further details of the configura- = 0 T R T the isothermal case at T Te using the formulation outlined in tion, including the numerical methods employed to integrate the Section 6 . In particular, δ0 can be obtained by substituting the ini- I formulation in Table 8 , are provided in Appendix G . L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 45

Q , and n as anticipated in Eq. (190). A similar result is observed in T J T Fig. 30(d) for the species fluxes JF , n and F , n [Eqs. (157) and (158)], which do not cancel each other and are much smaller than JF , n and J , F , n as anticipated in Eq. (166). These results ratify the analysis in Section 5.6 , in that most of the interfacial species flux is invested in counteracting the antidiffusion of fuel induced by the sum of Fickian and barodiffusion mechanisms, whereas most of the interfacial heat flux is invested in counteracting the Dufour effect. The small remainders of those cancellations, along with the components of the standard fluxes that are dependent on temperature gradients (Soret, Fourier, and interdiffusion), describe the transport of mass and energy across the interface in non-equilibrium conditions. The near-equilibrium behavior persists until the interface vanishes downstream at an edge. Because of the much smaller thermal diffusivity of the fuel, D T , F /D T , O = 0 . 03 , the cold fuel stream cools down the hot coflow more than what the hot coflow is capable of heating the fuel stream, as observed in the large-scale view of the temperature field in Fig. 31 . As a result, all isotherms, ࣃ including the supercriticalization one TTR Tc ,diff, are initially dis- placed outwards away from the axis into the coflow. This phenomenon is exacerbated by a mixing-induced augmentation of the constant-pressure specific heat c p , whose global maximum is attained within the interface for a significant portion of its length or lifetime, as shown in Fig. 33 . The line joining the global maxima of c p is the physical-space representation of the pseudo-boiling line discussed in Section 2.2 and Appendix B.2 . Similarly to the abrupt Fig. 29. Evolution with downstream distance (or time) of (a) large-scale ( ) and R termination of the pseudo-boiling line observed before intercept- thermal ( T ) Cahn numbers, and (b) interface temperature T e . In this figure, the

interface thickness δI and the thermal mixing-layer thickness δT are computed, re- ing the diffusional critical point in the C12 H26 /N2 phase diagram in

spectively, using Eqs. (223) and (224). The interface temperature Te is computed Fig. 10 (c), the global maximum of c p in physical space in Fig. 33 (a) as the temperature where the absolute value of the fuel partial-density gradient ceases to be within the interface upstream of the interface edge, ∂ (ρY ) /∂ nˆ attains its maximum value. F and switches abruptly thereafter to the jet axis. As the fuel is heated by the hot coﬂow, the supercriticalization isotherm eventually turns back toward the jet axis and crosses the transcritical interface. It is at that crossing point where the inter-

7.2. Results: The evolution of a transcritical interface downstream of face vanishes at an edge. The latter is located at a distance of or-

the injection oriﬁce der unity from the oriﬁce when normalized with the characteristic supercriticalization length (13) [or after a time of order unity The slowly growing values of R in Fig. 29(a) indicate that has passed when normalized with the characteristic supercritical-

the large composition gradients persist until the interface edge is ization time (222) ], as shown in Figs. 32 and 33 .

neared. In contrast, the temperature gradient across the interface A zoomed view of the structure of the interface edge is pro-

becomes increasingly smaller as the thermal mixing layer grows, vided in Fig. 2. For C12 H26 /N2 systems, the diffusional critical point thereby leading to decreasingly small values of for most of the T at the combustor pressure P ∞ = 100 bar plays a central role at the

length or lifetime of the interface. The interface temperature Te , interface edge because of the following reasons: shown in Fig. 29 (b), increases monotonically because the fuel is increasingly heated by the hot coflow as it flows downstream of (a) Whereas the thermodynamic pressure undergoes order-unity the injection orifice. Near the interface edge, where the interface oscillations across the interface in the near-equilibrium region, ࣃ the flow in the vicinity of the interface edge is characterized by approaches the supercriticalization temperature TTR Tc ,diff, the in- ࣃ terface thickness increases sharply, becoming more than 100 times being mostly isobaric at P P∞ . = . , δ0, (b) The isocontour YFc , diff 0 85 corresponding to the fuel mass thicker than its initial value I and causing a noticeable increase fraction of the diffusional critical point at 100 bar (see Table 3 ), in both R and T . The bathtub-like shape of the evolution of the thermal Cahn must necessarily emanate from within the transcritical interface since it is engendered in the coexistence region of the thermo- number T in Fig. 29(a) indicates that the interface is closest to transport equilibrium in the intermediate region away from the dynamic phase diagram, as shown in Fig. 10 (c).

orifice (where the temperature gradients are the largest) and away (c) The temperature of the diffusional critical point Tc ,diff is the from the interface edge (where the composition gradients are the maximum temperature of the diffusionally unstable region. smallest). The near-equilibrium behavior in that intermediate re- As discussed in Sections 3.2 and 6.2 , different mixtures may ex- gion is manifested in Figs. 30 (a,b) as almost-complete cancella- ist that may lead to a maximum value of the diffusional spinodal tions between the components of the standard (qn ) and interfacial surface that may be larger than the diffusional critical temperature. Q ( n ) heat fluxes independent of the temperature gradient [see Eqs. In those mixtures, the supercriticalization isotherm would corre- (183) and (184) ], and between the corresponding components of spond to the maximum temperature of the diffusional spinodal J the standard (JF , n ) and interfacial ( F , n ) species fluxes [Eqs. (155) surface, and the diffusional critical point would not be located at T and (156)]. In contrast, Fig. 30(c) shows that the standard (qn ) the interface edge, but at some distance upstream within the tran- T and interfacial (Qn ) components of the heat fluxes that depend scritical interface. on the temperature gradient [ Eqs. (186) and (187) ] do not can- Downstream of the interface edge, the temperature is every- cel each other, but are much smaller than their counterparts qn where larger than Tc ,diff, and consequently the system becomes dif- 46 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 30. Spatial distributions of heat and species fluxes across the interface at t/t TR = x/ L TR = 0 . 4 , where its thickness computed using Eq. (223) is δI = 5 . 96 nm. (a) Q Temperature-gradient-independent components of the standard heat flux qn [dark solid line, Eq. (183)] and interfacial heat flux n [green dashed line, Eq. (184)]. (b) J Temperature-gradient-independent components of the standard species flux JF , n [dark solid line, Eq. (155)] and interfacial species flux F , n [green dashed line, Eq. (156)]. T Q T (c) Temperature-gradient-dependent components of the standard heat flux qn [dark solid line, Eq. (186)] and interfacial heat flux n [green dashed line, Eq. (187)], along T J T with the net heat flux. (d) Temperature-gradient-dependent components of the standard heat flux JF , n [dark solid line, Eq. (157)] and interfacial species flux F , n [green dashed line, Eq. (158) ], along with the net species flux. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) fusionally stable. The interfacial fluxes of heat and species become the interface at the edge. The first stage of the decay is from negligible, and the standard diffusion flux of fuel reverses to the t/t TR = x/ L TR = 0 to 0.3, and is characterized by a rapid increase of usual forward direction along decreasing fuel concentrations, as the interface temperature near the orifice, as shown in Fig. 29 (b). shown in Fig. 34 . The transcritical interface morphs into a fully su- The decay of σ slows down from t/t TR = x/ L TR = 0 . 3 to 1.5, where percritical mixing layer that grows with distance downstream and near-equilibrium transport conditions are attained. Beyond t/t TR = is characterized by unimpeded mixing of the propellants by molec- x/ L TR = 1 . 5 , the decay rate increases on approach to the diffusional ular diffusion, as observed in Fig. 2 . In this zone, both propellants critical point, with σ plunging at the interface edge to dynamically resemble supercritical gas-like fluids. irrelevant values. The evolution of the pseudo-trajectories of the system with An accurate prediction of σ , given by the square symbols in distance (or time) are overlaid in Fig. 35 on a T − Y F thermody- Fig. 36 , can be obtained by solving an isothermal interface at ev- namic phase diagram evaluated at P = P ∞ . As the edge conditions ery time step using the formulation in Table 6 and the set-up are approached, Fig. 35 shows that the thermodynamic states of described Section 6 , where the local interface temperature T e is the mixture across the interface tend to intersect the coexistence obtained from Fig. 29 (b). This observation is consistent with the region at increasingly higher temperatures until the diffusional attainment of near-equilibrium conditions for most of the transcritical temperature is reached, above which intersections cannot critical region. Most importantly, it also suggests a potential route occur. for subgrid-scale modeling of transcritical flows based on pre- As indicated by the solid line in Fig. 36 , the surface-tension tabulation of equilibrium isothermal cases, with T e acting as a table coefficient σ , obtained from Eq. (99) , decays monotonically with input obtained from a coarse-grained calculation of the tempera- downstream distance (or time) and vanishes simultaneously with ture field. L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 47

Fig. 31. Large field of view of (a) solid contours of temperature (upper half) and fuel mass fraction (lower half), along with (b) corresponding profiles extracted at t/t TR = x/ L TR = 0 . 4 , 1.4, and 2.4. The figure includes isotherms corresponding to the fuel critical temperature (dark dashed line), critical mixing temperature (blue dashed line), and diffusional critical temperature (white dashed line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

8. Concluding remarks has been made on describing thermodynamically complex systems involving two components at high pressures. When the pressure Despite the remarkable progress made on augmenting the is supercritical with respect to both components, and when at thrust, range, and reliability of chemical propulsion technologies least one of the propellants is injected at subcritical temperature, over the last several decades, which has often relied on large in- transcritical conditions develop that appear to lead to atomization vestments in experimentation and testing, the fundamental fluid and spray phenomena according to existing experimental visual- mechanical processes participating in the injection, atomization, izations. vaporization, mixing, and combustion of propellants at high pres- The process of interface formation at high pressures cannot sures remain largely unknown. Specifically, the extreme pres- be explained using the thermodynamics of monocomponent sys- sure conditions involved in the transcritical flow of propellants tems, since only one thermodynamic state is possible in those into combustors make this problem quite formidable compared to when the temperature is fixed at supercritical pressures. In con- many other open questions in the technical discipline of general trast, in bicomponent systems, at least two states may be possible multiphase flows. at a given temperature for pressures above the critical pressures The present study has focused on a selected number of ba- of the separate components. Using previously well-established sic aspects of the problem of transcriticality. Particular emphasis high-pressure equations of state and mixing rules, it is shown 48 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 32. Zoomed version of Fig. 31 showing (a) solid contours of temperature (upper half) and fuel mass fraction (lower half), along with (b) corresponding profiles extracted at t/t TR = x/ L TR = 0 . 4 , 1.4, and 2.4. Panel (a) includes isotherms corresponding to the critical mixing temperature (blue dashed line), diffusional critical temperature (white dashed line), along with isocomposition lines corresponding to Y F = 0 . 82 (blue dotted line) and Y F = 0 . 87 (green dotted line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) that the coexistence region extends up to more than 100-fold The problem of translating by theoretical means the aforemen- higher pressures than the critical pressures of the separate com- tioned diffusional instability into an observable process in physi- ponents in systems of heavy-hydrocarbon fuels mixed with N2 , O2 , cal space is nothing short of laborious. Such translation requires a or air. tight connection between the thermodynamic space and the trans- At high pressures, the thermodynamic structure of the coexis- port of the propellants in physical space, and must therefore in- tence region is dominated by diffusional instabilities. Specifically, volve finite-rate processes of molecular diffusion that cannot be the components tend to separate in the coexistence region because described with the standard transport theory. This work has in- of an antidiffusion process that proves energetically favorable. In vestigated one possible way of addressing this question by cou- ordinary practice, systems like water brought into the coexistence pling an extension of the diffuse-interface theory of van der Waals region of their thermodynamic phase diagram at standard pres- flows with the Navier-Stokes conservation equations. The core of sure tend to separate into different phases, namely liquid and va- this theory is the folding of the coexistence region in thermo- por, across an interface. However, at high pressures, the phases are dynamic space into a thin interface in physical space. This can no longer vapor and liquid, but two supercritical fluids of distinct only be made possible by redefining the thermodynamic poten- character which, in the transcritical conditions addressed in this tials in the presence of strong composition gradients, and by re- study, behave at injection as a liquid-like supercritical fluid (for the visiting the constitutive laws participating in the Navier-Stokes heavy hydrocarbon fuel) and a gas-like supercritical fluid (for N2 , conservation equations. When these modifications are coupled

O2 , or air). with a high-pressure equation of state that reproduces a single L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 49

Fig. 33. (a) Solid contours of density (upper half) and constant-pressure specific heat (lower half), along with (b) corresponding profiles extracted at t/t TR = x/ L TR = 0 . 4 , 1.4, and 2.4. In panel (a), the thick green dashed line indicates the loci of points where the constant-pressure specific heat attains a global maximum; this line is not plotted for t/t TR = x/ L TR > 0 . 75 , when the global maximum switches from the interface to the jet axis. Additionally, panel (a) includes isotherms corresponding to the critical mixing temperature (blue dashed line), diffusional critical temperature (white dashed line), along with isocomposition lines corresponding to Y F = 0 . 82 (blue dotted line) and Y F = 0 . 87 (green dotted line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Maxwell loop within the coexistence region, the formulation can The theory of transcritical flows outlined in this work in- describe separation or mixing of phases or fluids in accordance corporates high-pressure models for thermophysical and trans- with their thermodynamic phase diagram. The ensuing interface port properties. Those are subject to large uncertainties, and has a non-zero thickness set by the nearly mutual cancellation their determination through analysis or experiments is a broad between an antidiffusion flux of matter provided by the stan- and active area of research. Furthermore, the formulation has in- dard transport theory, and an opposing interfacial flux arising as troduced additional parameters in the form of gradient-energy a regularization introduced by the diffuse-interface theory. The coefficients that are directly related to the thickness and sur- formulation is not exempt of approximations, among which the face tension of the interface. Most of the existing mod- most notable one is related to the continuum hypothesis across els for these coefficients rely on experimental calibration. Re- the interface. This is a comfortable approximation only at high cent improvements in experimental techniques for diagnostics pressures because the interface broadens relative to intercollision of high-pressure flow fields, along with new developments in distances. molecular-dynamics simulations for the computation of thermo- 50 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. 35. Thermodynamic-space pseudo-trajectories at t/t TR = x/ L TR = 0 , 0.4, 1.4, and 2.4, projected on a constant-pressure ( P = P ∞ = 100 bar) cross-section of the thermodynamic-phase diagram. The figure includes the diffusional critical point (purple diamond symbol), the VLE line (thick red dashed line), the diffusional spinodals (thin purple dashed lines), and the loci of the maximum of the molar heat / = / = . Fig. 34. Spatial distributions of (a) heat and (b) species fluxes at t tTR x LTR 2 4 at constant pressure (thick green dot-dashed line). (For interpretation of the refer- shortly downstream of the interface edge. ences to color in this figure legend, the reader is referred to the web version of this article.)

physical properties, transport coefficients, and possibly gradient- vents separation of the components thereafter. A fully supercritical energy coefficients, may assist in future work to reduce the mixing layer ensues from the interface edge, across which the pro- uncertainties. pellants mix by molecular diffusion as if they were two gas-like The utilization of this theory has discovered quantitative infor- fluids. mation about the dynamics of transcritical interfaces in coflow- A central motivation of this study has been to highlight that a ing configurations consisting of two propellant streams injected number of extensions of this theory could be made in order to re- at different temperatures. The interface separating both propellant lax the simplifying assumptions used in the examples above. These streams, along with the surface tension it engenders, survive from extensions could include the effects of turbulence, which is ex- the injection orifice until a supercriticalization zone downstream, pected to play an important role in the transport of heat in practi- where the interface vanishes at an edge upon reaching the temper- cal systems at high pressures, and consequently in the determina- ature of the diffusional critical point. The latter plays a prominent tion of the supercriticalization length. However, even in the sim- role at the interface edge in the C12 H26 /N2 systems tested here. ple cases that have been addressed in this work, the description Specifically, the interface edge is characterized by a fundamental of the coupling between the interface structure and the hydrody- transition in the transport characteristics of the mixture that pre- namic field is computationally expensive because of the resulting

Fig. 36. Surface-tension coefficient in a C 12 H 26 /N 2 non-isothermal system as a function of downstream distance (or time). The figure includes the solution obtained by a numerical integration of the formulation in Table 8 for non-isothermal bicomponent systems (solid line), along with solutions in mechanical and transport equilibrium (square symbols) obtained by a numerical integration of the formulation in Table 6 for isothermal systems assuming constant temperature equal to the local (or instantaneous) interface temperature T e read from Fig. 29 (b). L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 51 large disparity of scales. The reader should therefore be forewarned Fig. 1 , the discussion here will be precluded to bicomponent sys- that the formulation that has been presented here involves inter- tems ( N = 2 ), with brief references being made to monocompo- facial terms in the conservation equations whose spatiotemporal nent systems ( N = 1 ) for conceptual comparisons. The explana- resolution requirements are most likely untenable in CFD simulations are supplemented by the thermodynamic phase diagrams in tions of full engineering systems with current standard hardware Figs. 4 and A.1 . The reader is referred to Refs. [103–105] for gener- and numerical methods. alizations and full derivations of these conditions. Worthy extensions of this work that may decrease the com- Thermal stability is observed when the constant-volume molar putational cost could involve investigations of filtered versions heat of the mixture is positive, of the Navier-Stokes conservation equations augmented with the ∂ e interfacial terms derived here, where the filter width would be c v = > 0 , (A.1) ∂T much larger than the interface thickness but much smaller than v ,X F the hydrodynamic scales. This would necessarily lead to closure where X is the molar fraction of fuel. In particular, Eq. (A.1) en- problems in subgrid-scale interfacial terms, including the surface- F sures that the molar internal energy e increases with increasing tension force. The results presented here suggest that the interface temperatures at constant volume and composition. The condition is in mechanical and transport near-equilibrium for a significant (A.1) is satisfied by specific heats c calculated using the Peng- portion of its length, except near the orifice and near the interface v Robinson equation of state (14) in the conditions of interest for edge, where non-equilibrium considerations may become impor- the present study. Formulas for the calculation of c are outlined tant. It could therefore be of some interest to investigate the clo- v in Appendix C.7 , and a closer analysis of the resulting variations sure of subgrid-scale interfacial terms by reading them on-the-fly of specific heat with temperature and composition is provided in from pre-tabulated solutions of isothermal transcritical interfaces Appendix B.2 for C H /N mixtures. at a local interface temperature provided by the resolved flow field. 12 26 2 Mechanical stability is attained when the isothermal compress- β ibility of the mixture T is positive, Declaration of Competing Interest 1 ∂v β = − > 0 , (A.2) The authors declare that they have no known competing finan- T v ∂P T,X cial interests or personal relationships that could have appeared to F influence the work reported in this paper. which guarantees that the molar volume decreases as the pressure increases at constant temperature and composition. An expression for β particularized for the Peng-Robinson equation of Acknowledgments T state (14) is provided in Eq. (C.44) . It can be shown that a system that is mechanically stable is also thermally stable, but a system This work was funded by the US Department of Energy through thermally stable is not necessarily mechanically stable [104] . Most the National Nuclear Security Administration (NNSA) Grant # DE- importantly, the limit of mechanical stability is delineated by the NA0 0 02373 as part of the PSAAP-II Center at Stanford University. mechanical spinodal surface, whose equation is given by The authors are grateful to Dr. Mario Di Renzo for useful sugges- tions on the numerical integrations. ∂P = 0 , (A.3) ∂v T,X F Appendix A. Basic aspects of thermodynamic stability and β → ∞ phase equilibrium for transcritical systems or equivalently T . In monocomponent systems, the mechanical spinodals are easily visualized as lines intersecting all minima − The coupling between the diffuse-interface theory and the and maxima of the Maxwell loop in a P v diagram, as shown in − Navier-Stokes conservation equations depends fundamentally on Fig. 4 . Within the region of the P v diagram enclosed by the me- the diffusional stability of the mixture. In unstable conditions, a chanical spinodals, the isothermal compressibility is negative and transcritical interface is generated that separates mixtures near the system becomes mechanically unstable. In this unstable region, phase equilibrium. This appendix reviews basic concepts related the system tends to separate into two phases by means of a thin to thermodynamic stability and phase equilibrium that are ubiq- interface. uitously employed throughout the main text. A mechanical critical line exists on the mechanical spinodal surface where the pressure reaches an inflection point along the v axis at constant temperature and composition. The mechanical critical A1. Thermodynamic-stability considerations line is therefore defined as the locus of points in the thermodynamic space { P , T , X } where the condition (A.3) is satisfied simul- The Peng-Robinson equation of state (14) , and all others avail- F taneously with able, constrain all possible thermodynamic states of the system 2 on a hypersurface P = P (ρ, T , Y 1 , Y 2 . . . Y N−1 ) . However, not all ther- ∂ P = 0 . (A.4) modynamic states on that hypersurface are observable. Specifi- ∂ 2 v , cally, there exist regions on that hypersurface where the system T XF is thermodynamically unstable. There, a homogeneous mixture of A mechanical critical point is therefore defined as the in- the components is energetically less favorable than a configuration tersection of the mechanical critical line with constant-pressure, where different phases or different fluids are separated by inter- constant-temperature, or constant-composition planes. faces. The criteria that determine whether the system is thermo- In monocomponent systems, the mechanical critical line dynamically stable are independent of the equation of state. How- (A .3) and (A .4) collapses on a single critical point in the thermo- ever, the single Maxwell loop predicted by cubic equations of state dynamic phase diagram, as observed in Fig. 4 . This critical point such as (14) naturally accommodates the stability landscape. determines the critical pressure above which the two-phase re- In closed multicomponent systems subject to differential dis- gion ceases to exist, or equivalently, the pressure above which the turbances from equilibrium, thermodynamic stability occurs when monocomponent system is unconditionally stable. In contrast, me- three different conditions corresponding to thermal, mechanical, chanical stability is not a sufficient condition for thermodynamic and diffusional stability are simultaneously satisfied. Motivated by stability in multicomponent systems, since diffusional processes 52 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. A.1. Thermodynamic phase diagrams for C 12 H 26 /N 2 mixtures at uniform temperature T = 500 K in terms of (a) fuel fugacity f F as a function of the fuel molar fraction at different constant pressures, (b) fuel molar chemical potential μ¯ F as a function of the fuel molar fraction at different pressures [refer to legend in panel (a)], and (c) pressure as a function of the fuel molar fraction indicating different stability regions of the diagram.

can additionally drive separation of components at pressures above is satisﬁed, or alternatively the mechanical critical point. Diffusional stability requires that the molar chemical potential of one of the constituents, for instance ∂ ln f F = 0 , (A.9) that of the fuel μ¯ F , increases upon increasing the concentration of ∂ ln XF , the same species, namely P T as prescribed by Eq. (A.6) . ∂ μ¯ F > 0 . (A.5) The transcritical C H /N systems analyzed in this study are ∂X 12 26 2 F P,T diffusionally unstable but thermally and mechanically stable at pressures larger than approximately 200 bar, whereas they become The partial derivative of the fuel chemical potential μ¯ F with

= both mechanically and diffusionally unstable at pressures lower respect to XF can be rewritten in terms of the fuel fugacity fF than those (see Appendix E ). It is the breach of the diffusional sta- X F ϕ F P as bility constraint (A.5) what induces transcritical interfaces in the

∂ μ¯ R 0T ∂ ln f flow field, as discussed in Sections 4.6 and 5.4–5.6. Briefly, the dif- F = F , (A.6) fusional spinodals represent thermodynamic states where the Fick- ∂X F X F ∂ ln X F P,T P,T ian diffusion coefficient becomes zero. In particular, the quantity ϕ ( ∂ ln f / ∂ ln X ) fundamentally participates as a non-ideal prefac- where F is the corresponding fugacity coefficient defined in F F P,T Eq. (C.47) . The formal justification for Eq. (A.6) can be found in tor in the Fickian diffusion coefficient at high pressures defined in ∂ ∂ Appendix B.3 . In particular, Eq. (A.6) indicates that the diffusional Eq. (43), in such a way that changes in the sign of ( ln fF / ln XF )P , T stability criterion (A.5) can be alternatively stated as change the direction of the diffusion flux. Outside the region de- ∂ ∂ > limited by the diffusional spinodals [i.e., ( ln fF / ln XF )P , T 0], the ∂ ln f mixture is stable and the components tend to diffuse into flow re- F > 0 . (A.7) ∂ ln X gions in directions aligned with decreasing gradients of their con- F P,T centration. In contrast, within the region delimited by the diffu- ∂ ∂ < It can be shown that a diffusionally stable system is also sional spinodals [i.e., ( ln fF / ln XF )P , T 0], the chemical potential both thermally and mechanically stable, but a system both ther- decreases with increasing molar fraction. As a result, the mixture is mally and mechanically stable is not necessarily diffusionally stable unstable there and tends to separate by means of antidiffusion of [104] . The limit of diffusional stability is attained at the diffusional matter. This phenomenon is well characterized in physical chem- spinodal surfaces, where istry [136] , and cannot be described by the standard transport theory. ∂ μ ¯ F A relevant curve on the diffusional spinodal surfaces is the dif- = 0 (A.8) ∂X fusional critical line of the mixture, which is defined as the locus F P,T L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 53 of points where (A.8) is satisfied along with by the VLE lines, is provided in the phase diagrams presented in Fig. A.1 . ∂ 2μ¯ F = 0 , (A.10) As discussed in Appendix A.1 , the system is thermodynami- ∂X 2 F P,T cally stable along the coexistence envelope and everywhere out- or equivalently, where the chemical potential reaches an inflection side the volume enclosed by the coexistence envelope. The system is metastable in the concentric volume enclosed between the point along the XF axis at constant pressure and temperature. Al- ternatively, based on Eq. (A.6) , the diffusional critical line can be coexistence envelope and the diffusional spinodal surface, as indi- defined as the locus of points in the thermodynamic space { P , T , cated in Fig. A.1 (c). Thermodynamic instability occurs within the volume enclosed by the diffusional spinodal surface, where no ho- XF } where Eq. (A.9) is satisfied along with mogeneously mixed state consisting of phases and g can exist. 2 ∂ ln f F = 0 . (A.11) Under these unstable conditions, the system tends to become sep- ∂X 2 arated into the two equilibrium states and g by means of a thin F P,T interface. The diffusional and mechanical critical lines of the mixture As explained in Section 4.4 , the diffuse-interface theory maps are generally different. The intersection of the diffusional criti- the volume enclosed by the phase-equilibrium surface in thermo- cal line with constant-pressure, constant-temperature, or constant- dynamic space into a thin interface in physical space. In its sim- composition planes gives rise to a diffusional critical point. These plest representation corresponding to isothermal systems, the the- considerations are summarized in the thermodynamic phase dia- ory predicts that phase-equilibrium conditions are exactly satisfied grams shown in Fig. A.1 . on both sides of the interface. In most practical cases, however, The verification of the stability criteria (A .1), (A .2) , and (A .5) [or the propellant streams are injected in the combustor at different (A.7) ], does not prevent the system from becoming unstable temperatures, as depicted in Fig. 1 . In the presence of tempera- to finite-amplitude disturbances. Whereas unconditional stabil- ture gradients across the interface, the phase-equilibrium condi- ity is achieved along the phase-equilibrium surface, or coexis- tions (A .12) –(A .14) are only satisfied in an asymptotic sense up to tence envelope, and everywhere outside the volume enclosed an appropriate order of approximation provided in Sections 5.3 and by it, interstitial regions exist in the thermodynamic phase di- 5.6 . agrams in Fig. 4 and A.1 (c) that are enclosed between the phase-equilibrium and spinodal surfaces. The system is metastable Appendix B. Transcritical behavior of thermophysical there, in that finite-amplitude disturbances can render it unstable properties and molecular transport coefficients [103,104] . A complete description of the fluid motion requires specifica- A2. Phase equilibrium tion of the thermophysical properties and transport coefficients of the mixture. At high pressures, increasing departures from the Consider two thermodynamic states { P , T , X } and { P g , T g , X g} F F ideal-gas theory are observed, and consequently derivation of more in the same bicomponent system addressed above. These states complex expressions are necessary. are in phase equilibrium across flat interfaces when the following three conditions corresponding to equal pressures B1. Molar enthalpy g P = P = P e , (A.12) The specific gradient-dependent internal energy e can be ob- equal temperatures GD tained from the total energy using Eq. (47) . The molar enthalpy h = g = , T T Te (A.13) can be easily related to e GD by rewriting Eq. (52) on a molar basis as and equal chemical potentials = + , μ = μg = μ , μ = μg = μ , h eGD W PGD v (B.1) ¯ F ¯ F ¯ Fe ¯ O ¯ O ¯ Oe (A.14) are simultaneously satisfied [103,104] , with the subindex e being where PGD is given by Eq. (50). An expression for h as a function of employed here to indicate phase-equilibrium values. At low pres- the state variables at high pressures is provided in this section. sures, and g denote liquid and vapor phases, respectively. In the The utilization of molar rather than specific values is expedi- high-pressure systems studied here, and more particularly above ent since the resulting departure of the enthalpy from its ideal- the critical pressures of the separate components, and g denote gas counterpart involves integration of the equation of state (14), liquid-like and gas-like supercritical fluids, respectively. which is written in terms of the molar volume v and of the coef-

Based on Eq. (A.6) , the condition (A.14) is equivalent to the con- ficients a and b on a molar basis. Specific and molar values of the dition of equal fugacities, enthalpy are simply related through the mean molecular weight as h = h /W . Similarly, partial specific and partial molar values of the = g = , = g = . fF fF fFe fO fO fOe (A.15) enthalpy are related through the molecular weight of the particu- = / In monocomponent systems, the two conditions in (A.14) are lar component, hi hi Wi . Analogous relations apply to the entropy replaced by the single constraint and all other thermodynamic potentials, including their partial val-

ues. μ = μg = μ , ¯ ¯ ¯ e (A.16) The variation of enthalpy at high pressures can be calculated in where μ¯ is the molar Gibbs free energy. the following way. Consider the exact differential of the enthalpy At the equilibrium values of pressure P and temperature T , of the system H , e e the combination of Eqs. (A .12) , (A .13) and (A .15) , along with the N ∂ ∂ H H equation of state (14) and the fugacity (B.21), provides the equi- d H = d T + d P + hi d n i , (B.2) g ∂T ∂P librium molar volumes v and v , as well as the equilibrium mo- P,n T,n i =1 e e i | i =1 , ...N i | i =1 , ...N g lar fractions XFe and XFe . The subspace of states satisfying the = above phase-equilibrium constraints forms a surface or coexis- with ni being the number of moles of species i and hi (∂ H/∂ n ) , , | = , ... ;( = ) as the partial molar enthalpy. The en- tence envelope in the thermodynamic space {P, T, XF }. A constant- i P T n j j 1 N j i temperature cross-section of the coexistence envelope, indicated thalpy of the system can be expressed in additive form as H = 54 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

N , 0 = ni hi which can be differentiated and combined with Eq. the reference pressure P is chosen to be small compared to P (i.e., i 1 (B.2) yielding P 0/ P 1), which makes the change in enthalpy along the isobar of this particular integration path to occur under ideal-gas conditions. N ∂ h ∂ h The integration of Eq. (B.8) using this combined trajectory yields X i d h i = d T + d P. (B.3) ∂ ∂ = T , P , the expression i 1 P Xi | i =1 , ...N −1 T Xi | i =1 , ...N −1 IG = ( , , . . . ) + ( , , , . . . ) Further simpliﬁcations of Eq. (B.3) can be made by rewriting the h h T X1 XN−1 d h P T X1 XN−1 (B.9)

ﬁrst principle of thermodynamics (C.1) on a molar basis, namely for the molar enthalpy at the high-pressure state { P , T }. N−1 The right-hand side of Eq. (B.9) is composed of two different T d s = d h − v dP − ( μ¯ i − μ¯ N ) dX i , (B.4) terms corresponding to the two legs of the particular trajectory IG i =1 mentioned above. In particular, h is the molar enthalpy of an where the last term represents a chemical work done at constant ideal gas evaluated at the end of the isobaric portion of the tra- μ volume due to a variation in the composition, with ¯ i being the jectory, namely molar chemical potential of species i [deﬁned formally further be- T IG IG, 0 = + IG , low in Eq. (B.23)]. In particular, the partial differentiation of Eq. h h cp dT (B.10) 0 (B.4) with respect to pressure yields T IG, 0

∂ ∂ where h is the enthalpy of an ideal gas at the reference temper- h s = T + v . (B.5) 0 IG = N IG ∂ ∂ ature T , and cp i =1 Xi cp,i is the ideal-gas value of the constant- P , P , T Xi | i =1 , ...N −1 T Xi | i =1 , ...N −1 pressure molar heat of the mixture, which depends on composi- IG Substituting (B.5) into (B.3) and using the definition of the tion and temperature. A more practical way of computing h con- 0 = (∂ / ∂ ) sists in fixing the arbitrary reference temperature at T = 298 K constant-pressure molar heat cp h T P,X | i =1 , ...N −1 and the i and rewriting (B.10) as Maxwell relation (∂ s /∂P ) , | = , ... − = −(∂v /∂T ) , | = , ... − , the T Xi i 1 N 1 P Xi i 1 N 1 expression N T IG 298 K IG h = X h + c dT , (B.11) N i f,i p,i 298 K = + − β i =1 Xi dhi cp dT ( 1 T v )v dP (B.6) = 298 K i 1 where h f,i is the standard enthalpy of formation of element i. is obtained, where The bracketed quantity in (B.11) is the ideal-gas partial molar en- IG , 1 ∂v thalpy hi whose origin is chosen to be the formation value. By β = (B.7) v ∂ using the conservation equations of mass (44) and species (46), v T , P Xi | i =1 , ...N −1 it can be shown that the conservation equations for total energy (47) and enthalpy (51) are independent of this origin as long as is the volume expansivity. An expression for βv particularized for there are no chemical reactions in the problem. In chemically re- the Peng-Robinson equation of state (14) is given in Eq. (C.22) . acting systems, the first term on the right hand side of Eq. (B.11) , Because of its form amenable to isolation of non-ideal effects which involves the sum of the formation partial enthalpies multi- arising when T βv = 1 , the second term on the right-hand side plied by their corresponding molar fractions, is responsible for the of Eq. (B.6) is the basis for the computation of the pressure- changes in enthalpy due to chemical heat release without the need induced departure of the enthalpy with respect to that of an of adding any chemical sources on the right-hand sides of neither ideal gas [54] . Since that term cannot be straightforwardly ex- the total-energy Eq. (47) nor the enthalpy equation (51) . A chemi- pressed as a linear sum of contributions from the different com- cal source would however have to be added on the right-hand side ponents when the equation of state (14) and the mixing rules of the species conservation equation (46) . Despite the absence of (15) are employed, in multicomponent systems the standard ap-

chemical reactions in the present study, the choice is made here proach neglects enthalpy variations due to mixing, in such a way 298 K that the total variation of the enthalpy is assumed to be given by to use the formation enthalpy h f,i as reference value to facili- IG the linearized weighted sum of the variations of the partial en- tate the computation of hi for different species as a function of = N ( ) N thalpies, namely dh i =1 d Xi hi i =1 Xi dhi [150]. This proce- temperature from existing tables, since the latter are most promi- dure is equivalent to neglecting the last term on the right-hand nently found within the combustion-related literature [e.g., see Ref. side of Eq. (B.2) by freezing the chemical composition. In this way, [151] and Eq. (C.33) in Appendix C ]. Eq. (B.6) becomes The second term on the right-hand side of Eq. (B.9) corresponds to the enthalpy departure function d h c p d T + ( 1 − T βv )v d P. (B.8) IG P For ideal gases, βv = 1 /T and therefore the second term on the = − = − β , d h h h ( 1 T v )v dP (B.12) right-hand side of Eq. (B.8) vanishes, with h being only a function 0 of T and composition. In contrast, the second term on the right- which quantiﬁes the enthalpy variations of a real gas along the hand side of Eq. (B.8) is not zero at high pressures when real-gas isotherm at temperature T as the pressure changes from P 0 → 0 effects are considered, which makes h to additionally depend on to P . A closed expression for (B.12) can be obtained upon substi- = ( , , , . . . , ) pressure, h h P T X1 XN−1 . tuting the equation of state (14) into the integrand [e.g., see Refs. In coupling Eq. (B.8) with the Navier-Stokes equations, it is im- [92,152] and Eq. (C.23) in Appendix C ]. portant to note that the integration of Eq. (B.8) is independent A similar methodology can be employed for calculating partial of the thermodynamic trajectory. Consider an arbitrary reference molar enthalpies at high pressures using the expression 0 0 state {P , T } in which the gas is approximately ideal, with the en- IG IG, 0 ( , , , . . . ) = ( ) + ( , , , . . . ) . thalpy there being denoted as h . The change in molar enthalpy hi P T X1 XN−1 hi T d hi P T X1 XN−1 (B.13) IG, 0 = − 0 0 h h h between {P , T } and the ﬁnal state {P, T} can be Here, d hi is a partial-enthalpy departure function that can expressed as the sum of an isobaric change from { P 0 , T 0 } to { P 0 , be expressed in closed form by making use of the Peng-Robinson T }, and an isothermal change from { P 0 , T } to { P , T }. The value of equation [e.g., see Ref. [153] and Eq. (C.34) in Appendix C ]. L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 55

real-gas effects and is plotted in Fig. B.1(b) for C12 H26 /N2 mixtures at two pressures. Much larger values of c p are observed for dodecane than for nitrogen, with c p, F / c p, O = 10 − 15 under typical injection temperatures. However, as indicated in Eqs. (12) and (13) , the relevant ratio of free-stream heat capacities that influences the supercriticalization length is written in specific form and is therefore much smaller, with c p, F /c p, O = 2 at P ∞ = 100 bar, T F = 350 K and T O = 900 K. Figure B.1 (b) also provides the loci where local maximum values of c p are attained inside and outside the coexistence region. The resulting line is reminiscent of the pseudo-boiling, or Widom line, observed in monocomponent systems (i.e., see thick green dot-dashed line in Fig. 4 ), although important differences are worth highlighting. In monocomponent systems undergoing phase change, the temperature remains constant and therefore c p develops a singularity at boiling, where a finite change of enthalpy occurs under zero variations of the temperature. As the pressure is increased above the critical point, a finite-amplitude peak occurs in c p across the pseudo-boiling condition that is characteristic of a second-order phase transition of the monocomponent system and, as described in Section 2.1 , is often associated with transcriticality in the existing literature. In monocomponent systems, the pseudo- boiling line therefore emanates from the critical point and pene- trates in the supercritical region of the phase diagram to progressively fade away as the peak amplitude of c p vanishes with increasing pressures, as shown in Fig. 4 . In contrast, in multicomponent systems, the phase change necessarily requires a finite tempera-

Fig. B.1. Constant-pressure molar heats for C12 H26 /N2 mixtures as a function of ture change at all pressures because of the unfolding of the vapor- temperature and mass fraction of dodecane YF for (a) ideal gases, and (b) real gases equilibrium line into dew and boiling branches, thereby yielding at two pressures and different compositions (refer to the legend in the top panel).

The plots in (b) include the diffusional critical point of the mixture at the corre- ﬁnite-amplitude peaks in cp even inside the coexistence region, as sponding pressure (purple diamond symbol), the loci of the maximum of the molar shown in Figs. 10 and B.1 (b). The resulting line that joins all lo- heat at constant pressure (thick green dot-dashed line), along with the VLE line cal maxima of c p does not necessarily emanate from the critical (thick red dashed line). (For interpretation of the references to color in this ﬁgure point, and, depending on the pressure and mixture components, legend, the reader is referred to the web version of this article.) may not even enter the fully supercritical region of the thermodynamic phase diagram. While the peak in c p for pure dodecane is visible in Fig. B.1 (b) at 50 bar and 730 K, the corresponding peak in c p for nitrogen oc- B2. Constant-pressure molar heat curs at much lower temperatures of order 130 K that are of no interest for the problem analyzed in Fig. 1 . When both compo- Similar departures to those observed above in the enthalpy oc- nents are mixed at not too high pressures such that the peak in cur in the molar heat as a result of the high pressures. To account c p for each individual component is still discernible, a line can be for these, the constant-pressure molar heat of the mixture can be traced that joins all peaks of the mixture c p , traverses the coexis- derived by differentiating Eq. (B.9) with respect to temperature at tence region, crosses the vapor-liquid equilibrium line at the dif- constant pressure and composition, which gives fusional critical point, and terminates within the single-phase su-

( , , , . . . ) = IG ( , , . . . ) percritical region at the peak of cp for dodecane, as shown in the cp P T X1 XN−1 cp T X1 XN−1 left panel of Fig. B.1 (b). However, at higher pressures, as in the ∂( ) d h right panel in Fig. B.1 (b), where the peak of c p for dodecane is + (P, T , X 1 , . . . X N−1 ) . (B.14) ∂T no longer discernible, the line of maximum c p ends short within P,X | = , ... − i i 1 N 1 the coexistence region and never crosses the diffusional critical The right-hand side of Eq. (B.14) is the sum of two terms corre- point. Additional lines of maximum c p are observed in molecular- sponding to ideal and non-ideal components. The ideal component dynamics simulations that tend to occur at much lower tempera- is the ideal-gas constant-pressure molar heat that can be obtained tures and may be of interest for cryogenic propellants, but those from weighted sums of the ideal-gas molar heats of the individ- may not be computable using the Peng-Robinson equation of state IG = N IG , IG ual components as cp i =1 Xi cp,i with cp,i being a temperature- [154]. dependent function typically obtained from tabulated data from different species [e.g., see Ref. [151] and Eq. (C.25) in Appendix C ].

IG The overall dependence of cp on temperature and composition for B3. Chemical potentials

C12 H26 /N2 mixtures of ideal gases is illustrated in Fig. B.1(a), which indicates that the variations are smooth and monotonic within the As shown in Section 5 , the diffuse-interface theory for mul- range of operating conditions studied here. ticomponent flows makes ubiquitous use of chemical potentials. The non-ideal component in Eq. (B.14) corresponds to the par- Specifically, the energy excess inherent to the interface structure, tial derivative of the enthalpy departure function with respect along with the different chemical composition of the propellant to temperature. The resulting combined expression (B.14) for the streams, results in gradients of chemical potentials that are respon- constant-pressure molar heat of the mixture c p , which can be sible for molecular diffusion. μ = readily calculated by making use of the Peng-Robinson equation The molar chemical potential of species i, ¯ i μ ( , , , . . . − ) , of state [see Ref. [92] and Eq. (C.28) in Appendix C], accounts for ¯ i P T X1 XN 1 is defined as the partial molar Gibbs 56 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 free energy, ∂G μ¯ = , (B.15) i ∂ ni , , P T n j | j =1 , ...N ( j = i ) where G = n μ¯ is the Gibbs free energy of the system, μ¯ is the molar chemical potential of the mixture, and n the total number of μ moles. A more useful definition of ¯ i can be derived by applying the fundamental reciprocity relation to G , ⎡ ⎤ ∂ ⎢ ∂G ⎥ ⎣ ⎦ ∂ ∂ P ni , , P T n j | j =1 , ...N ( j = i ) , T n j | j =1 , ...N ∂ ∂G = , (B.16) ∂n ∂P i T,n i | i =1 , ...N , , P T n j | j =1 , ...N ( j = i ) or equivalently, ∂ μ¯ i = V¯ , (B.17) ∂ i P , T X j | j =1 , ...N −1 where use of Eq. (B.15) has been made along with the definitions of the system volume V = (∂ G/∂ P ) , | = , ... and the partial molar T ni i 1 N volume ∂V V¯ = . (B.18) i ∂ ni , , P T n j | j =1 , ...N ( j = i )

V A closed expression for the partial molar volume ¯i is provided in Eq. (C.43) for the case of the Peng-Robinson equation of state

(14). Fig. B.2. Constant-pressure distributions of the logarithm of fugacity coeﬃcients for

Direct integration of Eq. (B.17) yields the expression for the the individual components ( ϕF for C 12 H 26 and ϕO for N 2 ) of a C 12 H 26 /N 2 mixture as = chemical potential a function of temperature and mass fraction of dodecane YF at (a) P 50 bar and (b) P = 100 bar. The plots include the zero-fugacity line corresponding to an ideal

μ ( , , , . . . ) = μIG ( , , ) ¯ i P T X1 XN−1 ¯ i P T Xi gas (thick blue dot-dashed line), the diffusional critical point of the mixture at the corresponding pressure (purple diamond symbol), along with the VLE line (thick + 0 ϕ ( , , , . . . ) , R T ln i P T X1 XN−1 (B.19) red dashed line). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) μIG where ¯ i is the chemical potential of an ideal gas given by

μIG = 0 + 0 ( / 0 ) . ¯ i g¯i R T ln Xi P P (B.20) In the notation, P 0 is an arbitrary reference value for the pressure that is chosen to be sufficiently low for ideal-gas conditions to pre- The partial molar volume and the molar volume are the same IG IG 0 0 = 0 ( 0 , ) for an ideal gas, V¯ = v = R T /P, as shown by substituting V = vail, and g¯ g¯ P T is the partial molar Gibbs free energy of i i i 0 0 / = N the pure component at that pressure. In practice, P = 1 bar and nR T P along with n i =1 ni into Eq. (B.18). Correspondingly, the ϕ 0 fugacity coefficient i tends to unity and the fugacity fi tends to tables are accessed to evaluate the temperature dependence of g¯i ϕ the partial pressure in thermodynamic conditions where the ideal- [e.g., see Ref. [151] and Eq. (C.46)]. Additionally, i is a fugacity coefficient whose logarithm is defined as gas approximation is appropriate. Such conditions typically consist of high temperatures in the pressure range studied here, as P 0 R T 0 shown in Fig. B.2 for C12 H26 /N2 mixtures. Departures from ideality R T ln ϕ i = V¯ i − dP. (B.21) 0 P in chemical potentials, which are caused by the second term on the right-hand side of Eq. (B.19) , are observed to be the largest at The fugacity coefficient ϕ is related to the fugacity f by the ex- i i low temperatures, and become increasingly important as the pres- pression V , ϕ sure increases. Away from ideality, ¯i departs from v i attains = ϕ . fi Xi i P (B.22) values which are far from unity, and fi ceases to be the partial

ϕ pressure. A closed expression for i is provided in Eq. (C.47) by substituting the Peng-Robinson equation of state (14) into Eq. (B.21) . The combination of Eqs. (B.19) , (B.20) and (B.22) yields the alternative B4. Molecular transport coeﬃcients deﬁnition of the chemical potential

0 0 0 The high pressures involved in the multicomponent systems μ¯ = g¯ + R T ln (f / P ) (B.23) i i i studied in this work prevent the use of simple relations for η directly in terms of the fugacity fi . The differentiation of Eq. the calculation of the dynamic viscosity , the thermal conduc- λ D , (B.23) with respect to Xi readily leads to Eq. (A.6) anticipated in tivity , the binary diffusion coeﬃcients i, j and the thermal- η λ Appendix A.1. diffusion ratio ki , T . Standard methods for computing and are L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 57

Fig. B.3. Constant-pressure distributions of (a) dynamic viscosity, (b) thermal conductivity, (c) binary diffusion coeﬃcient, and (d) thermal-diffusion ratio for C 12 H 26 /N 2 mixtures at high pressures as a function of temperature and mass fraction of dodecane Y F [refer to the legend in the right panel in (a)]. The plots include the diffusional critical point of the mixture at the corresponding pressure (diamond symbol), as well as projections of the loci of the maximum of the molar heat at constant pressure

(thick dot-dashed line), and the VLE line (thick dashed line) on the planes T − η, T − λ, T − D F , O and T − k F , T . In interpreting these plots, it is important to note that the entire coexistence region is not exactly enclosed within the space bounded by the VLE line in panels (a), (b) and (d) since the partial variation of the transport coeﬃcient with respect to Y F attains negative values there.

detailed in Appendix F, and follow the modeling approaches in Linear superposition of the high-pressure model of kF,T in Refs. [155,156] . Those models have been used in previous stud- Ref. [167] with the ideal-gas counterpart from kinetic theory ies on direct numerical simulations (DNS) of mixing layers [157] , [140] would likely yield more realistic values at high temperatures, large-eddy simulations (LES) of diesel jets [158] and thermoacous- as suggested by the comparisons between different models in tic instabilities [159] , and have been successfully utilized to nu- Ref. [168] . ≥ merically predict experiments of N2 jets into N2 environments The positive values kF,T 0 observed in all cases indicate that at high pressures [28,160,161]. The formulation developed in Refs. C12 H26 tends to migrate toward cold flow zones, thereby sharpen- D [162,163] is employed here to compute i, j and ki , T , and is also ing the transcritical interface. Despite the fact that kF,T is an order- explained in Appendix F . That methodology accounts for real-gas unity parameter at high pressures, the contribution of the Soret ef- effects and has been satisfactorily compared against experimen- fect to the species diffusion flux is only fractional compared to the tal data involving liquid and gas mixtures of alkanes, CO2 , and N2 contribution of the composition gradients because of the compar- [164–166] . atively smaller temperature gradients across the interface, as dis- The temperature and composition dependence of η, λ, D F , O , cussed in Section 5.6 . η λ and kF,T for C12 H26 /N2 mixtures at high pressures is shown in A significant change in the temperature dependence of , , η λ D D Fig. B.3. In N2 -rich mixtures, , , and F , O increase monotonically and F , O is observed in C12 H26 -rich mixtures near ambient tem- with temperature in a manner that is reminiscent of that encoun- peratures. In particular, it is found that μ and λ increase as T tered in gases, since the range of temperatures studied here is well decreases, while D F , O attains values that are mostly independent above the critical temperature of nitrogen. A similar behavior is of temperature. The aforementioned trends are typical of liquids, also observed in C12 H26 -rich mixtures at sufficiently high tempera- and suggest that C12 H26 -rich mixtures behave in a liquid-like man- tures, and more particularly, at temperatures much larger than the ner at high pressures and ambient temperatures. Those corre- critical temperature of C12 H26 . spond to typical conditions in the fuel injection port. This shift

By deﬁnition, the thermal-diffusion ratio of C12 H26 , kF,T , be- in the trend of the transport coeﬃcients is accompanied by a = = comes zero in the limits YF 0 and YF 1 corresponding to maximum in kF,T at intermediate concentrations, and occurs at a pure components. At intermediate concentrations and high tem- pressure-dependent temperature that is well correlated with the peratures, kF,T decays to small values as a result of the par- pseudo-boiling one corresponding to the maximum molar heat ticular model employed here, in which kF,T is proportional to capacity. the energy-departure function (see Ref. [167] and Appendix F ). 58 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Appendix C. Supplementary thermodynamic expressions C2. Coeﬃcients of the Peng-Robinson equation of state applicable to multicomponent systems at high pressures

The individual coefficients ai and bi in the mixing rules (15) for Despite the gradient corrections used in the thermodynamic the Peng-Robinson equation of state are given by [92] potentials (16) –(18) , the classic theory of thermodynamics must 2 0 2 still be locally satisfied. A number of standard thermodynamic re- R Tc,i = . + − / , ai 0 457 1 ci 1 T Tc,i (C.10) lations are listed in this appendix for use in the analysis. In addi- P c,i tion, the formulation of thermodynamic systems at high pressures requires complex equations of state. Quantities including thermo- 0 R T c,i dynamic potentials, specific heats, volume expansivity and others b = 0 . 078 , (C.11) i are involved in the diffuse-interface formulation described above Pc,i and depend on the particular form of the equation of state. This where Pc , i and Tc , i are, respectively, critical pressures and temper- appendix provides a summary of expressions for those quantities. atures obtained from Table 1. Additionally, the coefficients ci are A number of relations are particularized for the Peng-Robinson defined as equation of state (14) , whose coefficients are also listed here, but general definitions are given as well that allow for utilization of . + . ω − . ω 2 + . ω 3 ω > . , 0 380 1 485 i 0 164 i 0 017 i if i 0 49 = other equations of state by direct substitution. Closure param- ci 2 0 . 375 + 1 . 542 ω − 0 . 270 ω otherwise, eters for the gradient-energy coefficient (23) are also supplied i i here. (C.12)

ω ω C1. Local thermodynamic relations with i as the acentric factor. The values of i used in this study are [40] The ﬁrst principle of thermodynamics ω = . , ω = . , ω = . , N 2 0 037 O 2 0 022 CO 2 0 224 N−1 = + ( /ρ) − (μ − μ ) , ω = 0 . 011 , ω = 0 . 349 , ω = 0 . 574 . (C.13) Tds de Pd 1 i N dYi (C.1) C H4 C 7 H16 C 12 H26 = i 1 ϑ The binary-interaction parameters i , j utilized in Eq. (15) are or its alternative form [75,157] N−1 dP = − − (μ − μ ) , ϑ = . Tds dh ρ i N dYi (C.2) F , O 0 156 (C.14) i =1

for mixtures of alkanes with N2 or O2 , and are both satisfied locally, with h denoting the specific enthalpy ϑ = − . h = e + P/ρ, (C.3) F , O 0 017 (C.15) which will be shown to be an exclusively local quantity in this for- for mixtures of CO2 with N2 or O2 . Additionally, the binary- mulation under the approximation (24) . In writing Eqs. (C.1) and interaction parameter of an element with itself is assumed to be − = − N 1 zero, (C.2), the mass conservation constraint dYN i =1 dYi has been used. Equations (C.1) and (C.2) can be recast in terms of the local ϑ , = ϑ , = 0 . (C.16) specific Helmholtz free energy f as F F O O N−1 C3. Parameters of the model for the gradient-energy coefficient df = −sd T − P d (1 /ρ) + (μi − μN ) dY i , (C.4) i =1 The dimensionless coefficients A and B in the model expres- or in terms of the local specific Gibbs free energy, or local specific i i sion for the gradient-energy coefficient (23) are given by [120] chemical potential of the mixture μ, as −16 2 / 3 N−1 A = −10 N / ( 1 . 233 + 1 . 376 ω ), (C.17) dP i A i μ = − + + (μ − μ ) , d sdT ρ i N dYi (C.5) = i 1 −16 2 / 3 B = 10 N / ( 0 . 905 + 1 . 541 ω ), (C.18) with i A i ω μ = h − T s. (C.6) where the acentric factors i for dodecane and nitrogen are given by Eq. (C.13) . The combination of Eqs. (19) , (C.3) and (C.6) leads to μ − = /ρ. f P (C.7) C4. Molar Helmholtz free energy and chemical potential In addition, an alternative definition of μ can be given in terms μ of the weighted sum of partial specific chemical potentials i as The molar Helmholtz free energy and chemical potential corre-

N sponding to the Peng-Robinson equation of state (14) are √ μ = Y i μi , (C.8) + − = P (v − b) a v 1 2 b i 1 ¯ = ¯ IG − 0 + √ √ f f R T ln ln (C.19) μ μ R 0T + + with i being related to its corresponding partial molar value ¯ i as 2 2b v 1 2 b μ = μ / i ¯ i Wi . Equation (C.8) can be substituted into Eq. (C.5) yielding the Gibbs-Duhem relation and N ( − ) dP IG 0 P v b v Y d μ = −sd T + . (C.9) μ¯ = f¯ + P v = μ¯ − R T 1 + ln − i i ρ R 0 T v − b i =1 √ √ Interfacial effects are incorporated in the above relations in a v + 1 − 2 b 2 2b v + √ √ − , Section 5 leading to important modiﬁcations of the constitutive ln (C.20) 2 2b v + 1 + 2 b v 2 + 2 v b − b2 laws in the Navier-Stokes equations. L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 59

Table C.1 Coeﬃcients for the evaluation of the ideal-gas speciﬁc heats, entropies, enthalpies and Gibbs free energies for selected species [151] .

−1 −2 −3 −4 Species r 1, i [-] r 2, i [K ] r 3, i [K ] r 4, i [K ] r 5, i [K ] r 6, i [K] r 7, i [-]

1 −2 −4 −7 −10 4 1 C 12 H 26 2.133 × 10 −3 . 864 × 10 3 . 995 × 10 −5 . 067 × 10 2 . 007 × 10 −4 . 225 × 10 −4 . 858 × 10 −4 −7 −9 −12 3 N 2 3.531 −1 . 237 × 10 −5 . 030 × 10 2 . 435 × 10 −1 . 409 × 10 −1 . 047 × 10 2.967

respectively. In the notation, f¯ IG = μ¯ IG − R 0 T is the molar The ideal-gas adiabatic coeﬃcient of the mixture is deﬁned as Helmholtz free energy of an ideal-gas mixture, where

γ IG = IG/ IG, N cp cv (C.27) IG 0 0 0 0 μ = ¯ ( , ) + ( / ) ¯ Xi gi P T R T ln XiP P (C.21) IG N IG IG N IG with c = X c and c = X c being the ideal-gas spe- i =1 p i =1 i p,i v i =1 i v ,i ciﬁc heats of the mixture. is the corresponding chemical potential and g¯0 is the ideal-gas i The real-gas constant-pressure molar heat of the mixture for molar Gibbs free energy of species i deﬁned in Appendix Ap- the Peng-Robinson equation of state (14) is obtained by substitut- 0 pendix C at a reference pressure P . ing Eq. (C.23) into Eq. (B.9) , which gives [103] ∂ ∂ C5. Volume expansivity h IG 0 0 Z c¯ = = c + R (Z − 1) + R T p ∂T p ∂T P,X P,X The volume expansivity (B.7) can be expressed as i | i =1 , ... N−1 i | i =1 , ... N−1 ⎧⎡ ⎤ ⎫ −1 ∂a ⎪ − ⎪ 1 ∂v 1 ∂T ⎪ a T √ ⎪ β = = ⎨ ⎢ ∂T ⎥ ⎬ v ∂ ⎢ X ⎥ + 1 − 2 b v ∂T v ∂v i | i =1 , ... N−1 v P,X P,X + ⎢ √ ⎥ ln √ . i | i =1 , ...N −1 i | i =1 , ...N −1 ∂ ⎪⎣ ⎦ + + ⎪ T ⎪ 2 2b v 1 2 b ⎪ −1 ⎩⎪ ⎭⎪ 0 ( 2 − + 2 ) R a v 2bv b , = P − (C.22) P Xi | i =1 , ... N−1 ( 2 + − 2 ) 2 v v 2bv b (C.28) for the Peng-Robinson equation of state (14) . In this formulation, Z = P v / R 0 T is the compressibility factor. Its derivative with respect to temperature is given by C6. Enthalpy departure function ∂ − Z 2 2 1 = 3 Z + 2(B − 1) Z + (A − 2 B − 3 B ) The enthalpy departure function (B.12) particularized for the ∂ T , Peng-Robinson equation of state is [92] P Xi | i =1 , ...N −1 IG = − = − 0 d h h h Pv R T P ∂a 2 a √ × − (B − Z) ( 0 ) 2 ∂ + − R T T T 1 ∂a v 1 2 b Xi | i =1 , ...N −1 + √ a − T ln √ , ∂T + + 2 2b X v 1 2 b bP i | i =1 , ...N −1 − + − 2 − + − 2 , 6BZ 2Z 3B 2B A Z (C.23) R 0T 2 where (C.29) N N = / ( 0 ) 2 = / ( 0 ) ∂a 1 X i X j a i, j da j da where A aP R T and B bP R T . The partial derivative = + i , ai a j (∂ a/∂ T ) in Eq. (C.29) can be calculated by combining ∂T 2 a i a j dT dT Xi | i =1 , ...N −1 X | = , ... − i =1 j=1 i i 1 N 1 Eqs. (C.10) and (C.24) .

(C.24) An expression for the real-gas constant-volume molar heat of while the coefficients a and b are obtained by combining Eqs. the mixture for the Peng-Robinson equation of state (14) can be (15) and (C.10) –(C.12) . For a given specific volume and composi- derived by substituting Eqs. (C.28) and (C.22) , along with the β tion field, P and T are related in Eq. (C.23) through the equation of isothermal compressibility T introduced below in Eq. (C.44), into state (14) . the fundamental equation for the heat capacities

β2 C7. Molar heats, adiabatic coefficient, and isentropic compressibility Tv v c v = c p − , (C.30) βT The ideal-gas constant-pressure molar heat of species i is given which is obtained by substituting the identity by [151] (∂ P/∂ T ) ,X = ( c p − c v ) / (T v βv ) into Eq. (E.5) . Corre- v i | i =1 , ...N −1 IG ∂ spondingly, the adiabatic coefficient for a real gas, defined here as IG hi c , = p i ∂ the ratio of molar heats, can be calculated as T , P X j | j =1 , ...N −1 ( = ) c p β j i γ = = T , (C.31)

0 2 3 4 c v βs = R r 1 ,i + r 2 ,i T + r 3 ,i T + r 4 ,i T + r 5 ,i T , (C.25) . . . with where the coeﬃcients r1, i , r2 ,i r5 ,i are listed in Table C.1 209for 2 C12 H26 and N2 . The corresponding value of the constant-volume 1 ∂v T v β β = − = β − v molar heat for an ideal gas can be easily derived by substituting s T (C.32) ∂ v P , cp Eq. (C.25) into the Mayer’s relation s Xi | i =1 , ...N −1

IG = IG − 0. cv ,i cp,i R (C.26) being the isentropic compressibility. 60 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

IG C8. Partial molar enthalpies and partial-enthalpy departure functions where h is the ideal-gas enthalpy deﬁned in Eq. (B.10) . Corre- spondingly, the departure function for the molar internal energy IG

The partial molar enthalpy hi of an ideal gas referred to the is standard reference state can be expressed as the polynomial ex- = − IG = + 0 ( − ) , pansion d e e e d h R T 1 Z (C.39) 2 3 4 with h being provided in Eq. (C.23) for the Peng-Robinson equa- IG T T T T r , d = 0 + + + + + 6 i , h R T r1 ,i r2 ,i r3 ,i r4 ,i r5 ,i (C.33) tion of state (14) . i 2 3 4 5 T

. . . where the coeﬃcients r1, i , r2 ,i r6 ,i are listed in Table C.1 for C10. Partial molar internal energies

C12 H26 and N2 [151]. Note that Eq. (C.33) is equivalent to substituting the polynomial expansion of the specific heat (C.25) into IG The partial molar internal energy of an ideal gas ei referred to IG 298 K the definition h = h + T c IG dT . The coefficient R 0 r = the standard reference state can be expressed as i f,i 298 K p,i 6 ,i 298 K 298 K IG IG h , − c , dT is a constant employed to refer the enthalpy IG = − 0 , f i 0 p i ei hi R T (C.40) IG 298 K to the formation value h = h , . IG i f i where h is the partial molar enthalpy obtained from the poly- The departure function of the partial molar enthalpy of a real i nomial expansion (C.33) . By definition, the partial molar internal gas described by the Peng-Robinson equation of state (14) is given energy of a real gas is related to the partial molar enthalpy as by [153] e = h − P , (C.41) ∂( ) i i ivi n d h 0 h = = P V¯ − R T d i ∂ i = = / ni with Pi Xi P and vi v Xi . The combination of Eqs. (C.40) and P,n | = , ... j j 1 N ( j = i ) (C.41) yields the departure function of the partial molar internal energy ∂ V¯ − / a i vbi b + a − T = − IG = + 0 ( − ) , ∂ 2 + − 2 d ei ei ei d hi R T 1 Z (C.42) T v 2bv b Xi | i =1 , ...N −1 ⎧ where d hi is the departure function of the partial molar enthalpy √ ⎪ ⎨ provided in Eq. (C.34) for the Peng-Robinson equation of state (14) . 1 v + 1 − 2 b ∂a + √ ln √ ⎪ ∂ 2 2b v + 1 + 2 b ⎩ Xi , T X j | j =1 , ...N −1 C11. Partial molar volume and isothermal compressibility ( j = i ) ⎫ ⎬⎪ The partial molar volume is given by [92] 2 ∂ a ∂a b − − − i . T a T 0 0 ∂ ∂ ∂ ∂V R T R T b Xi T T b ⎭⎪ i X | = , ... − X | = , ... − V¯ = = v β + j j 1 N 1 i i 1 N 1 i ∂ T − 2 ( j = i ) ni , , v b ( v − b ) P T n j | j =1 , ...N (C.34) ( j = i ) (∂ /∂ ) N In Eq. (C.34), the quantity a T X can be obtained using 2 a ( v − b )b 2 j=1 Xj ai, j i | i =1 , ...N −1 + i − (C.43) 2 2 + − 2 Eq. (C.24) , whereas the remaining partial derivatives are calculated v 2 + 2 bv − b2 v 2bv b as β N for the Peng-Robinson equation of state (14). In Eq. (C.43), T is an ∂ a = 2 X a , , (C.35) isothermal compressibility that can be calculated as ∂ j i j Xi T,X | = , ... − j=1 −1 j j 1 N 1 ∂ ∂ ( j = i ) 1 v 1 P β = − = − T ∂ ∂ v P , v v , T Xi | i =1 , ...N −1 T Xi | i =1 , ...N −1 2 N ∂ a a , da da −1 i j j i 0 = X j a i + a j . (C.36) 1 R TBT ∂ ∂ = , (C.44) Xi T = aia j dT dT 2 X j | j =1 , ...N −1 j 1 v (v − b) ( j = i ) with B being a dimensionless variable defined as The partial-enthalpy departure function (C.34) is a sole function T of P when the values of the specific volume and molar fractions 2 a B = 1 − . (C.45) T 2 are fixed, since pressure and temperature are related through the ( v + b )R 0 T [v / ( v − b ) + b/ ( v + b )] equation of state (14) .

C12. Ideal-gas Gibbs free energies C9. Internal-energy departure function The molar Gibbs free energy of an ideal gas at the reference Using the deﬁnition pressure P 0 = 1 atm is [151] e = h − P v , (C.37) 0 2 3 4 g¯i T T T T r6 ,i = r , ( 1 − ln T ) − r , − r , − r , − r , + − r , , the internal molar energy e can be computed at high pressures af- R 0 T 1 i 2 i 2 3 i 6 4 i 12 5 i 20 T 7 i ter obtaining h from Eq. (B.9) . Similarly, its ideal-gas counterpart (C.46) e IG can be expressed as . . . where the coeﬃcients r1, i , r2 ,i r7 ,i are listed in Table C.1 for IG IG = − 0 , e h R T (C.38) C12 H26 and N2 . L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 61

C13. Fugacity coeﬃcients Table D.1 Dimensional formulation for high-pressure subcritical interfaces in isothermal monocomponent systems. ϕ The logarithm of the fugacity coeﬃcient i is [103]

Conservation equation 2 bi Pv P 2 ρ ρ ln ϕ = − 1 − ln ( v − b ) d 1 d i 0 0 Mechanical equilibrium P ∞ − P + ρκ − κ = 0 . b R T R T d nˆ 2 2 d nˆ √ N + − a 2 j=1 Xj ai, j b i v 1 2 b Boundary conditions + √ − √ ln Phase-equilibrium densities ρ → ρ at nˆ → −∞ , 0 a b + + e 2 2bR T v 1 2 b ρ → ρg → + ∞ . e at nˆ

(C.47) Supplementary expressions for the Peng-Robinson equation of state (14) . Equation of state Eq. (14), Coeﬃcients of the equation of state Eqs. (C.10) –(C.12) , Gradient-energy coeﬃcient Eqs. (23) , (C.17) , and (C.18) . C14. Speed of sound

The speed of sound c of a real gas is given by ∂P − / = = (ρβ ) 1 2, c s (C.48) 1 1 ∂ρ = − g + − = , f fe P∞ 0 (D.4) s,X | = , ... − e ρ ρg i i 1 N 1 e e where β is the isothermal compressibility defined in Eq. (C.32) . s g l with fe and fe being the phase-equilibrium Helmholtz free energies. Equation (D.4) indicates that the oscillation of the pressure Appendix D. High-pressure interfaces in subcritical encloses zero net area along the axis. In contrast, Eq. (108) states monocomponent systems v that the net area under the oscillation of the pressure in physical space is non-zero and is proportional to the surface tension. It can This appendix focuses on interfaces arising in monocompo- be shown that the mechanical equilibrium condition (D.1) is con- nent systems. In this case, interfaces can only emerge at pres- sistent with Maxwell’s construction rule (D.4) by substituting the sures and temperatures below the critical point, where two former into the latter and using the identity thermodynamic states are possible at the same pressure. The resulting system is therefore subcritical everywhere. Nonethe- nˆ → + ∞ ∞ 2 2 + d 2 ρ 1 dρ 1 dρ 1 dρ less, only relatively high pressures not too far from the crit- ρ − = = , dnˆ 0 ical point can be considered because of the continuum as- −∞ d nˆ 2 2 d nˆ ρ2 d nˆ 2 ρ d nˆ nˆ →−∞ sumption, as emphasized in Section 4.5 . Further details on the (D.5) utilization of diffuse-interface concepts in monocomponent systems can be found in a number of previous investigations as prescribed by the vanishing gradients of the density away from [53,111,14 9,16 8,169] and in van der Waals’ original contribution the interface. Analogous oscillatory profiles of P are observed in [108] . bicomponent systems, as shown in Section 6 . The origin of Eq. (31) anticipated in Section 4.4 , which relates D1. Formulation the chemical potential with the density gradients through the interface, is the mechanical equilibrium condition for monocompo- Similar arguments to the ones made in Section 4.4 for concep- nent flows (D.1) evaluated at uniform temperature. In this partic- tualizing the diffuse-interface formalism as a thermodynamic-to- ular case, Eqs. (31) and (D.1) are equivalent, as easily shown by physical space mapping of the oscillations of the chemical poten- evaluating Eq. (121) for N = 1 , which gives tial also apply to the pressure variations within the interface. To dμ ∂ 3 ρ illustrate this, consider the simplified form of the mechanical equi- = κ . (D.6) ∂ 3 librium condition (120) for N = 1 , d nˆ nˆ 2 Eq. (D.6) can be integrated once subject to the phase- d 2 ρ 1 dρ − + ρκ − κ = , equilibrium boundary condition μ = μe at nˆ → + ∞ , thus leading P∞ P 0 (D.1) d nˆ 2 2 d nˆ to Eq. (31) . subject to the phase-equilibrium densities The equations integrated in this appendix are summarized in Table D.1 and consist of the mechanical equilibrium condition for ρ → ρ → −∞ , e at nˆ (D.2) monocomponent systems (D.1) supplemented with the equation of state (14) and the model (23) for the gradient-energy coefficient.

ρ → ρg → + ∞ . e at nˆ (D.3) The numerical results are obtained by solving the system of equations on uniform 1D meshes with approximately 50–75 grid points Similarly to Eq. (31) for the chemical potential in systems in across the interface. The boundary conditions away from the in- phase equilibrium, Eq. (D.1) maps the pressure oscillations from terface correspond to phase-equilibrium densities (D.2) and (D.3) . thermodynamic space into a thin interface in physical space. The The problem has a solution consisting of a steady high-pressure oscillation of the pressure in thermodynamic space is induced subcritical interface that separates a liquid from its vapor in phase by the equation of state (14) evaluated in conditions where two equilibrium. phases exist, as sketched in Fig. 18 (c) for the simplest case corre- The isothermal cases summarized in Table D.2 for C H and sponding to a ﬂat interface in phase equilibrium. In that case, a 12 26 N are considered here. In all cases, the temperature T and the Maxwell’s construction rule for P can be obtained by integrating 2 e thermodynamic pressure away from the interface P ∞ are smaller Eq. (28) and using Eq. (C.7) subject to the phase-equilibrium con- than their corresponding critical values T and P . As a result, the dition (A.16) , which gives c c pseudo-trajectories along the interface in thermodynamic space g ρ 1 ve e dρ are subcritical and traverse the coexistence region, as shown in ( − ) = ( − ) P P∞ dv P P∞ ρg ρ2 Fig. D.1. W ve e 62 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. D.1. Thermodynamic pseudo-trajectories of cases A-H listed in Table D.2 for (a) dodecane and (b) nitrogen subcritical isothermal systems. The portion of the pseudo- trajectories involving negative pressures is not plotted in these diagrams.

Table D.2 ﬂat. This limiting behavior is accompanied by (a) an increase of the

Thermodynamic conditions corresponding to cases A-H in Fig. D.1 for subcrit- δ interface thickness I [deﬁned in Eq. (208)] and a decrease in the ical interfaces in monocomponent systems. Knudsen number [ Eq. (40) ], as shown in Fig. D.3 , along with (b)

C12 H26 system a decrease in the surface-tension coeﬃcient [ Eq. (22) ], as shown

ρ kg ρg kg P∞ Te in Fig. D.4 . The Knudsen number attains acceptable values for the case P∞ [bar] Te [K] e e m 3 m 3 P T c c continuum range only when conditions are close to the critical A 14.6 641.6 302.0 92.0 0.816 0.975 point. The interface disappears and the surface tension vanishes B 11.7 625.2 353.8 63.2 0.652 0.950 when the critical point is reached. C 9.3 608.7 393.1 46.4 0.516 0.925 Figure D.4 shows comparisons between the surface-tension D 5.5 575.8 453.0 25.4 0.310 0.875 coeﬃcient calculated by utilizing the numerical solution in Eq. N2 system

kg g kg P∞ Te (22) and the standard correlation [170] case P ∞ [bar] T [K] ρ ρ e e 3 e 3 m m P c T c

E 29.2 123.0 446.9 170.9 0.861 0.975 / / 1 . 86 + 1 . 18 ω F 25.1 119.9 514.8 129.6 0.738 0.950 σ = 2 3 1 3 Pc Tc G 21.3 116.7 567.7 102.1 0.628 0.925 19 . 05 / / H 15.1 110.4 651.6 65.9 0.444 0.875 3 . 75 − 0 . 91 ω 2 3 T 11 9 × 1 − [ mN/m ] (D.7) 0 . 291 − 0 . 08 ω T c

D2. Results [see also Eq. ( 12 -3.7) in Ref. [54] ], with ω being the acentric factor provided in Appendix Appendix C . Whereas good agreement is

The proﬁles of density and pressure obtained from integrating observed in the case of nitrogen over the entire range of tested the formulation in Table D.1 are provided in Fig. D.2. Density ratios conditions, the comparison reveals disagreements of order 20%- of order 10 are observed in cases D and H, which correspond to the 30% for dodecane. These discrepancies are caused by the relatively farthest conditions from the critical point. As the critical point is large accentric factor of dodecane, for which the standard correla- approached, the density and pressure proﬁles become increasingly tion is not designed to operate accurately [54] . In contrast, the nu- L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 63

Fig. D.4. Surface-tension coeﬃcient as a function of normalized temperature in monocomponent isothermal systems consisting of nitrogen or dodecane, including numerical results from the present work (solid lines), the standard correlation (D.7) (dashed lines), and experimental measurements for nitrogen (triangles, Ref. [72] ) and dodecane (asterisks, Ref. [73] ; squares, Ref. [74] ).

Appendix E. Transcritical bicomponent systems approaching mechanically unstable conditions

Relevant aspects of the formulation outlined above lead to intrinsic singularities at conditions approaching the limit of mechanical stability (A.2) . The analysis in this section builds on considerations made by Gaillard et al. [38,39] , and provides thresholds in thermodynamic conditions leading to this behavior along with a

Fig. D.2. Profiles of (a) density and (b) pressure across the interface in isothermal palliating regularization. systems composed of dodecane (left panels) and nitrogen (right panels). In dodecane systems, the interface thicknesses used for normalizing the spatial coordinate E1. Singularities of the theory at the limit of mechanical stability are δI = 4 . 98 nm, 3.59 nm, 2.91 nm, and 2.18 nm for cases A, B, C, and D, respectively. In nitrogen systems, the corresponding values are δ = 4 . 55 nm, 3.07 nm, I 2.42 nm, and 1.77 nm for cases E, F, G, and H, respectively. As discussed in Secs. 1, 2.2, and 3.2, transcritical conditions in systems fueled by heavy hydrocarbons involve combustor pressures P ∞ ࣡ 34 bar (for pure nitrogen coflows), P ∞ ࣡ 36 bar (for air coflows), and P ∞ ࣡ 50 bar (for pure oxygen coflows). These combustor pressures lead to fully supercritical pressures with re- merical solution agrees well with experimental data in Refs. [72– spect to both fuel and coflow propellants (see Table 1 ). Within 74] . Remarkably, the agreements between experiments and numer- the practical range of propellant injection temperatures ( 300 − ical results occur even at conditions far away from the critical 10 0 0 K), those combustor pressures are also larger than the maxi- point, where the continuum hypothesis across the interface be- mum mechanical critical pressures of mixtures of heavy hydrocar- comes hardly justifiable. bons with N2 , O2 , or air (15–25 bar), and smaller than the maximum diffusional critical pressures (50 0–10 0 0 bar). However, it is shown in Sections 4.4 and 5.3.3 that the thermodynamic pressure undergoes order-unity oscillations within the interface. These oscillations are the physical-space counterparts of the Maxwell loops in the coexistence region of the thermodynamic phase diagram, and are predicted by the equation of state (14) , or any other cubic equation of state available in the literature. Conse- quently, when P ∞ is not sufficiently high, the Maxwell loops may generate underpressures within the interface smaller than the mechanical critical pressure, including negative thermodynamic pressures. The Stefan-Maxwell forms of the standard diffusion fluxes of heat and species, given by Eqs. (152) and (153) , are of limited practical use in conditions where the mechanical stability criterion (A.2) is not satisfied. This is because both the fuel partial molar V ∂ ∂ volume ¯F and the non-ideal diffusion prefactor ( ln fF / ln XF )P , T participating in the Fickian, barodiffusion, and Dufour components of those fluxes diverge at the mechanical spinodal surface defined

Fig. D.3. Variations of (a) the interface thickness and (b) Knudsen number as a in Eq. (A.3) . To understand this, note that the partial molar vol- function of the temperature nondimensionalized with the critical temperature of the corresponding component. ume deﬁned in Eq. (B.18) can be equivalently expressed for the fuel 64 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Fig. E.1. Constant-pressure distributions of (a) isothermal compressibility, (b) volume expansivity, (c) fuel partial molar volume, (d) non-ideal diffusion prefactor, and partial speciﬁc enthalpies of (e) C 12 H 26 and (f) N 2 as a function of the C 12 H 26 molar fraction at mechanically supercritical pressure ( P = 50 bar, solid lines) and mechanically subcritical pressure ( P = 15 bar, dashed lines). All panels correspond to C 12 H 26 /N 2 mixtures at uniform temperature T = 500 K.

species as served in Fig. E.1(c) and prescribed by Eq. (A.6). The singularity in the non-ideal diffusion prefactor can be traced back to the partial ∂P

V¯ F = v βT , (E.1) molar volume as a consequence of a non-integrable singularity of ∂n F , , the latter in pressure, T V nO where β is the isothermal compressibility defined in Eq. (A.2) and T ∂ V¯ F ∂ ∂ μ¯ F = , (E.3) particularized for the Peng-Robinson equation of state in Eq. ∂ ∂ ∂ XF , P XF , (C.44) in Appendix C . The coflow partial volume satisfies the simi- P T P T T,X F lar expression as indicated by differentiating Eq. (B.17) and utilizing the funda- ∂P mental reciprocity relation of thermodynamics. V¯ = v β . (E.2) V ∂ ∂ O T ∂ The singularities in ¯F and ( ln fF / ln XF )P , T arising at the me- nO , , T V nF chanical spinodals cancel out each other upon summation in Eq. Whereas the partial derivatives of the pressure with respect to (F.22) , in such a way that the constant-temperature gradient of the the number of moles of fuel or coflow species in Eqs. (E.1) and chemical-potential difference (E.2) are generally finite and different from zero on approach to ∇ ( μ − μ ) = [∇ ( μ − μ ) + (s − s ) ∇T ] the mechanical spinodals, the isothermal compressibility diverges T F O F O F O there as demanded by Eq. (A.3) and illustrated in Fig. E.1 (a). As W vW R 0T ∂ ln f = V¯ − F ∇ + F ∇ F P XF a consequence, the partial molar volumes diverge, as shown in W W (1 − X ) X ∂ ln X F O F W F F P,T Fig. E.1 (b), although the molar volume v = X V¯ + (1 − X ) V¯ re- F F F O (E.4) mains finite. It is also at the mechanical spinodals where the fuel molar remains finite. Note that other singularities emerging when chemical potential develops two turning points along the com- ∇ ( μF − μO ) in Eq. (E.4) is evaluated in the pure components (i.e., position axis, as shown in Fig. A.1 (b). As a result, its derivative X F = 0 and X F = 1 ) disappear upon multiplying Eq. (E.4) by the On- (∂ μ¯ F /∂X F ) P,T becomes infinite there, and correspondingly the non- sager coefficients (145) and (146) to construct the standard diffu- ∂ ∂ ideal diffusion prefactor ( ln fF / ln XF )P , T also diverges there, as ob- sion fluxes of heat and species (149) and (150). L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 65

The considerations above indicate that, in configurations where the temperature gradients are exactly zero, such as those addressed in Section 6 , the standard species diffusion flux can be computed across the mechanical spinodals by direct spatial differentiation of the chemical potentials as in Eq. (150) , instead of using the Stefan-Maxwell form (153) . Similarly, the Dufour effect in the standard heat diffusion flux can also be safely computed across the mechanical spinodals using Eq. (149) instead of the corresponding Stefan-Maxwell form (152) . Additional singularities need to be tackled in configurations such as that addressed in Section 7 , in which there is a temperature gradient across the interface. In particular, singularities arising in the volume expansivity βv and in the partial specific enthalpies

hF and hO at the mechanical spinodals play a role in transport mechanisms associated with temperature gradients. Note that βv participates in the portion of the interfacial species flux (148) that depends on the temperature gradient. It is also required for the calculation of the terms in the interfacial heat flux (147) that depend on the temperature gradient and on the heat transport by Fig. E.2. Regime map for C 12 H 26 /N 2 mixtures showing the range of operating com- + J , the net species flux JF F the latter being small but non-zero in bustor pressures leading to crossings of the mechanical spinodals (region hatched practice because of the small but non-zero thermal Cahn numbers with red solid lines) and to negative underpressures (region hatched with blue

resulting from moderate temperature gradients. The difference of dashed lines) as a function of the interface temperature. The plot includes the critical point of dodecane (square symbol), the VLE line of dodecane (dotted line), and partial specific enthalpies h − h participates in the interdiffusion F O the maximum mechanical critical pressure (brown dot-dashed line). (For interpre- heat flux in Eq. (149) , in the last term of the interfacial heat flux tation of the references to color in this figure legend, the reader is referred to the (147) , and implicitly in the calculation of the constant-temperature web version of this article.) gradient of chemical potentials in Eq. (E.4) through the difference of partial entropies in Eq. (151) . β ∇T = 0 and J + J = 0 . In this way, the local pressure and com- Since T diverges at the mechanical spinodals by definition, and F F β β position can be monitored in the calculations to ascertain whether T is related to v by the identity the mixture has entered the mechanically unstable region of the ∂P thermodynamic phase diagram, and to evaluate the two problem- βv = βT , (E.5) ∂T atic quantities βv and h F − h O participating in the formulation of v ,X F non-isothermal systems. then βv also diverges there, as shown in Fig. E.1 (d). In Eq. (E.5) , As indicated in Fig. E.2 , the minimum combustor pressure

the partial derivative of P with respect to temperature generally P ∞ required to initiate crossings of the mechanical spinodals in

remains ﬁnite and different from zero. C12 H26 /N2 mixtures increases with decreasing values of the inter-

The singularities of the partial enthalpies at the mechanical face temperature T e . No regularizations are needed at suﬃciently

spinodals, shown in Fig. E.1(e and f), can be understood by con- high combustor pressures P ∞ ࣡ 200 bar over the entire range of sidering the identities relevant injection temperatures, since the system is always mechanically stable. ∂H ∂H ∂H h = = + V¯ , (E.6) In the conﬁguration sketched in Fig. 1 , T e is much closer to F ∂ ∂ F ∂ nF , , nF , , V , , P T nO V T nO T nF nO TF than to TO near the injection oriﬁce. As a consequence, Te in

Fig. E.2 can be approximated as TF plus a small temperature dif-

∂H ∂H ∂H ferential that depends on the details of the wake ﬂow created h = = + V¯ , (E.7) O ∂ ∂ O ∂ by the oriﬁce ring. In practice, the boundary of the mechanically nO , , nO , , V , , P T nF V T nF T nF nO unstable region in Fig. E.2 indicated by the red thick solid line

which can be combined with Eqs. (E.1) and (E.2) for the partial provides approximately the minimum TF necessary for warranting mechanical stability for a given P ∞ . For instance, at the pressure molar volumes, thereby suggesting that the singular behavior of h F β P ∞ = 100 bar considered in the results presented in Section 7 for and hO is closely related to that of T , since the remaining partial

derivatives generally remain finite and different from zero. Despite non-isothermal C12 H26 /N2 systems, the mechanical spinodal sur- ࣠ the singularities arising in the partial molar enthalpies, the molar face is crossed if TF 480 K. Downstream of the injector, however, the interface temperature increases because of the heat re- enthalpy of the mixture h = X F h F + (1 − X F ) h O remains finite across > the mechanical spinodals. ceived from the hot coflow. When T e 480 K, the system ceases β , to cross the mechanical spinodal surface and regularizations of β The need to regularize v hF , and hO occurs at conditions v − where temperature gradients exist across the interface and the and hF hO become unnecessary. These aspects are further ana-

combustor pressure P ∞ is not suﬃciently high. The threshold for lyzed in Section 7. Since the unboundedness of β on the mechanical spinodal sur- P ∞ depends on the interface temperature T e , since both T e and T β , the local pressure P determine the values of the fuel molar frac- face dominates the divergent behavior of v hF , and hO , a simple β tion on the mechanical spinodal surface. A quantiﬁcation of the way of limiting the value of T is by using the regularization

range of operating conditions {P∞ , Te } within which the mechani- (v − b) 2 ≈ ( , ) , cal spinodal surface is crossed in C12 H26 /N2 mixtures is provided max BT m (E.8) R 0T v β in Fig. E.2 as the region hatched with red solid lines. Speciﬁ- T

cally, the results shown in Fig. E.2 pertain to 1D interfaces arising where BT is a dimensionless variable deﬁned in Eq. (C.45), and m in a phase-equilibrium mixture at pressure P ∞ and uniform tem- is a positive dimensionless regularization constant whose value is perature equal to T e . In this case, whose details are provided in much smaller than unity. Figure E.3 shows the resulting regular- Section 6 , the singularities in βv , h F , and h O have no effect since ized proﬁles of βv and h F − h O . 66 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

mixtures are associated with near or full transgression of the mechanical-stability limit, and are denoted in Fig. E.2 as the region hatched with dashed lines. It should be emphasized that negative pressures do not lead to any transcendental behavior in the conservation equations, neither do they induce any singularities in the thermodynamic variables discussed in Appendix E.1 . They do, however, influence the calculation of transport coefficients in the following manner. The dynamic viscosity μ, thermal conductivity λ, binary diffu- D , sion coefficient F , O and thermal-diffusion ratio kF,T are generally functions of the thermodynamic pressure and participate in the conservation equations (44) –(47) through the viscous stress tensor λ D , and the fluxes of heat and species. In particular, , F , O and kF,T are required for the calculation of the Onsager coefficients (144) – (146) . Whereas models for these coefficients are well-established in the literature for ideal and real gases, including fully supercritical fluids, relatively much less is known about their behavior within the mechanically unstable region of the thermodynamic phase diagram, since homogeneous mixtures of phases or fluids of fundamentally different nature are untenable there. The standard μ λ D , models for , , F , O and kF,T in fluids at high pressures, employed in Appendices B.4 and F , cease to have physical significance at negative pressures where, for instance, D F , O becomes undefined because of Eq. (F.41) . However, note that the mechanical and transport equilibrium conditions (107) and (177) [or their equivalent forms (193) and (194) in terms of chemical potentials] are inde-

Fig. E.3. Regularized profiles of (a) volume expansivity and (b) difference of partial pendent of viscosity and of the Onsager coefficients, and there- specific enthalpies across transcritical interfaces in C 12 H 26 /N 2 isothermal systems at fore are not influenced by the locally erroneous behavior of any = = = . P∞ 50 bar and Te 500 K. The plot includes regularized profiles with m 0 01 of the transport coefficients under conditions leading to negative = . (dot-dashed line) and m 0 1 (dashed line), along with the non-regularized case pressures. m = 0 (solid line). The considerations above suggest that, in isothermal systems such as the ones treated in Section 6 , the structure of the inter- λ The differences in the regularized profiles of βv and h F − h O in face is completely insensitive to the specific manner in which , D , Fig. E.3 have a negligible impact on the overall solution of the non- F , O and kF,T are modeled. In contrast, in non-isothermal systems isothermal system addressed in Section 7 . This is shown in Fig. E.4 , such as the one in Fig. 1 , the mechanical and transport equilib- which shows that distribution of σ over the transcritical region is rium conditions are only satisfied asymptotically to leading order largely independent of the regularization constant m used in Eq. in the limit of small thermal Cahn numbers, T 1. Second-order (E.8) . The reason for the lack of sensitivity of the solution to m is effects lead to departures from equilibrium and require consider- that the terms multiplying βv and h F − h O in the fluxes of heat and ation of transport induced by temperature gradients. This involves λ D , species become increasingly smaller as mechanical and transport evaluation of , F , O and kF,T across the interface, and gives rise equilibrium are approached, a condition that is always satisfied for to small fluxes of heat and species. In absence of improved models a long portion of the interface, as discussed in Sections 5.7 and 7.2 . for transport coefficients at mechanically unstable conditions, the The singularities observed above at the limit of mechanical sta- approximations bility are not unique to the Peng-Robinson equation of state (14) . λ(P, X F , T ) ≈ λ(P ∞ , X F , T ) , They also occur when employing other cubic equations such as the D ( , , ) ≈ D ( , , ) , van der Waals [51] or Redlich-Kwong [96,97] equations of state. F , O P XF T F , O P∞ XF T k F , T (P, X F , T ) ≈ k F , T (P ∞ , X F , T ) , (E.9) E2. Negative thermodynamic pressures are used in this study, where the transport coefficients are evaluated at the combustor pressure P ∞ , the latter being representative The thermodynamic pressure P is always a positive quantity in of the thermodynamic pressure everywhere away from the inter- ideal gases, in which the intermolecular forces are negligible. In face in flows at small Mach numbers. contrast, P can attain negative values in liquids since the molecules are closer and therefore exert a non-negligible attractive force on Appendix F. Supplementary expressions for molecular each other. Negative pressures in liquids are well-documented in transport coefficients at high pressures the literature and are always associated with fully tensile mechanical states such as those participating in cavitation, nucleation, or This appendix reviews models for the calculation of the vis- capillary flows [104,127,128] . These processes are thermodynami- cosity, thermal conductivity, diffusion coefficient, and thermal- cally metastable or unstable, thereby suggesting that negative pres- diffusion ratio at high pressures. sures often correspond to transient states in which the fluid is transitioning to a different phase. Negative pressures may also F1. Dynamic viscosity arise in fluids at high pressures, when the density is large and cohesive forces become important, as represented by the second The dynamic viscosity η of the mixture can be modeled as term on the right-hand side of Eq. (14) , or by equivalent terms in [155,156] other equations of state [99]. 1 / 2 . The combustor pressure P ∞ and interface temperature T lead- − ∗ 36 344 W Tc e η = 7η · , 10 / [Pa s] (F.1) ing to locally negative pressures within the interface in C12 H26 /N2 2 3 vc L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 67

Fig. E.4. Surface-tension coeﬃcient in the C 12 H 26 /N 2 non-isothermal system analyzed in Section 7 as a function of downstream distance (or time), including the baseline case with regularization constant m = 0 . 01 (solid line), along with two supplementary calculations with m = 0 . 1 (dotted line) and m = 0 . 001 (dot-dashed line), all other parameters being the same.

3 Table F.1 where T c [K], v c [cm /mol], and W [g/mol] are, respectively, representative values of the critical temperature, critical molar volume, Coeﬃcients for Ek in Eq. (F.11) [155]. and molecular weight of the mixture given by the expressions k w1 ,k w2 ,k w3 ,k w4 ,k − . 2 1 6.324 50.412 51 680 1189.0 N N −3 −3 −3 / 2W i W j 2 1 . 210 × 10 −1 . 154 × 10 −6 . 257 × 10 0.03728 = σ 2 1 2 εσ 2 , = , W Xi Xj i, j i, j Wi, j Wi, j 3 5.283 254.209 −16 8 . 4 8 3898.0 W i + W j i =1 j=1 4 6.623 38.096 −8 . 464 31.42 5 19.745 7.630 −14 . 354 31.53 = . ε, = (σ / . ) 3. Tc 1 2593 vc 0 809 (F.2) 6 −1 . 900 −12 . 537 4.985 −18 . 15 7 24.275 3.450 −11 . 291 69.35 In Eq. (F.2) , ε [K] is the minimum of the pair-potential energy 8 0.7972 1.117 0.01235 −4 . 117 σ ˚ divided by the Boltzmann constant, and [A] is the hard-sphere 9 −0 . 2382 0.06770 −0 . 8163 4.025 diameter. Both of these quantities are obtained by the empirical 10 0.06863 0.3479 0.5926 −0 . 727 mixing rules

N N N N

ε = ε σ 3 /σ 3, σ 3 = σ 3 , Xi Xj i, j i, j Xi Xj i, j (F.3) −E y E y i =1 j=1 i =1 j=1 E 1 1 − e 4 /y + E 2 G 1 e 5 + E 3 G 1 G 2 = , (F.9) σ E E + E + E where i , j and i , j are obtained from 1 4 2 3 1 / 2 1 / 2 ∗∗ ∗ − ∗ − ε , = ε , ε , , σ , = ξ , σ σ , 2 1 2 i j i i j j i j i j i j (F.4) η = E 7 y G 2 exp E 8 + E 9 ( T ) + E 10 ( T ) , (F.10) with the corresponding quantities for the individual components / ε = / . σ = . 1 3 being given by i,i Tc,i 1 2593 and i,i 0 809v , . Using the 4 c i E = w , + w , ω + w , M + w , k , k = 1 , 2 , . . . 10 , (F.11) Peng-Robinson equation of state (14) , the critical molar volume k 1 k 2 k 3 k r 4 k a = 0 / , = , . . . of the species i can be computed as vc,i Zc R Tc,i Pc,i with Zc where the coefficients w1 ,k w4 ,k are provided in Table F.1. In this . = . 3 ω 0 307 [92], thereby yielding vc, F 932 5 cm /mol for C12 H26 , and formulation, is an acentric factor of the mixture defined as = . 3 vc, O 94 5 cm /mol for N2 . Note, however, that these values do N N 3 not match the experimentally measured ones , = 750 . 4 cm /mol ω = ω σ 3 /σ 3, vc F Xi Xj i, j i, j (F.12) 3 and v c, O = 89 . 4 cm /mol [40] , since the Peng-Robinson equation of i =1 j=1 state has only two coefficients, and therefore cannot be forced to with ω = ω + ω / 2 . The reduced dipole moment M and the simultaneously satisfy measured critical values of pressure, tem- i, j i j r association factor of the mixture k can be obtained from tables perature and molar volume. a ∗ [54] and are set to zero for C H /N mixtures. In Eq. (F.1) the prefactor η is given by 12 26 2 ∗ 1 / 2 ∗ (T ) F c 1 ∗∗ F2. Thermal conductivity η = + E 6 y + η , (F.5) v G 2 λ ∗ In this study, the thermal conductivity of the mixture is mod- where T = T /ε and y = / (6 ) . The remaining parameters are vc v eled as [155,156] computed using the relations . × 3 η0 ∗ ∗ 31 2 10 − ∗ −0 . 149 −0 . 773T −2 . 438T λ = 1 + = 1 . 161 ( T ) + 0 . 525e + 2 . 162e , (F.6) G B6 y v W 3

2 1 / 2 W + qB y ( T /T c ) G , (F.13) 7 3 · 4 m K = − . ω + . M + , Fc 1 0 276 0 059 r ka (F.7) where W and T c are given in (F.2) . The rest of the parameters in Eq. (F.13) are 3 − 1 / 2 / G 1 = (1 − 0 . 5 y ) / (1 − y ) , (F.8) η0 = . × 7 / ( 2 3 ) , 40 785 10 Fc W T vc v 68 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Table F.2 is obtained, which connects the constant-temperature gradients of

Coeﬃcients for Bk in Eq. (F.17) [155]. the speciﬁc chemical potential with the gradients of pressure and δ = , . . . − k z1, k z1, k z1, k z1, k composition, where m , k is the Kronecker delta and k 1 N 1.

1 2.417 0.748 −0 . 919 121.720 In writing Eq. (F.22), use has been made of Eq. (B.17) to replace the 2 −0 . 509 −1 . 509 −49 . 991 69.983 partial derivative of the chemical potential with respect to pressure 3 6.611 5.621 64.760 27.039 by the partial molar volume, and of Eq. (B.23) to express the par- 4 14.543 −8 . 914 −5 . 638 74.344 tial derivative of the chemical potential with respect to the molar 5 0.793 0.820 −0 . 694 6.317 fraction as 6 −5 . 863 12.801 9.589 65.529 7 91.089 128.110 −54 . 217 523.810 ∂ μ¯ ∂ ln f m = R 0T m , (F.23) ∂ ∂ Xj , , Xj , , P T Xi | i =1 , ...N −1 P T Xi | i =1 , ...N −1 ( i = j ) ( i = j ) − / 1 / 2 / = . × 3 2 / / 2 3, q 3 586 10 Tc W vc (F.14) where f m is the fugacity of species m deﬁned by the combination of Eqs. (B.22) and (C.47) . α( 0 . 215 + 0 . 283 α − 1 . 061 β + 0 . 267 γ ) The utilization of Eq. (F.22) in Eqs. (136) - (137) for replacing the = 1 + , (F.15) 0 . 637 + βγ + 1 . 061 αβ constant-temperature gradients of chemical potentials by gradients

of composition and pressure leads to the Stefan-Maxwell forms − − B4 y + B5 y + N−1 B1 1 e B2 G1 e B3 G1 L , G = , (F.16) = − q q ∇ + ( − ) 3 q T hi hN Ji y B 1 B 4 + B 2 + B 3 2 T = i 1 Fourier conduction

4 interdiffusion B = z , + z , ω + z , M + z , k a , k = 1 , . . . 7 , (F.17) k 1 k 2 k 3 k r 4 k N−1 N−1 N−1 X δ , ∂ ln f − 0 m + m k m ∇ where Lq,k R Xj W N X N W ∂X = = m =1 k j P,T,X j 1 k 1 r | r =1 , ... N−1 ( r = j ) α = IG/ 0 − − . , β = . − . ω + . ω 2, c¯p R 1 1 5 0 786 0 711 1 317 Dufour effect 2 γ = 2 + 10 . 5 ( T /T ) , (F.18) − c N−1 N1 Lq,k V¯ v X V¯ − k − + m m ∇P (F.24) ω with Tc and given in (F.2) and (F.12), respectively. The parameters T W W N X N W N X N = k m =1 , ... k 1 z1 ,k z4 ,k are calculated based on Table F.2. Dufour effect (cont.) F3. Stefan-Maxwell forms of the standard fluxes of heat and species and − N1 The expressions (136) and (137) for the standard diffusion L q,i J = − ρD , ∇X − ∇T fluxes of heat and species can be recast in more physically in- i i j j 2 T j=1 sightful forms that involve gradients of temperature, pressure and Soret effect composition. Because of the functional dependence of the mo- Fickian diffusion (thermal diffusion) lar chemical potential on pressure, temperature and composition, N−1 N−1 V V μ = μ ( , , , . . . ) , Li,k ¯k v Xm ¯m ¯ k ¯ k P T X1 XN−1 given explicitly by Eqs. (B.22), (B.23), − − + ∇P (F.25) and (C.47) , the differential form T Wk WN XN WN XN k =1 m =1 N−1 ∂ μ¯ ∂ μ¯ barodiffusion ∇ μ = k ∇ + k ∇ T ¯ k P Xj ∂P ∂X j for the standard diffusion fluxes of heat and species, respectively. T,X | = , ... − j=1 P,T,X | = , ... − j j 1 N 1 i i 1 N 1 ( i = j ) In Eq. (F.25), Di , j is the Fickian diffusion coefficient. (F.19) F4. Fickian diffusion coefficients is satisfied. In addition, the Gibbs-Duhem relation (C.9) can be rewritten on a molar basis at constant temperature as The Fickian portion of the standard diffusion flux (F.25) can be N expressed in matrix form as X i d μ¯ i = v dP, (F.20) = −ρ , i =1 J DX (F.26) = = ∇ − which can be spatially differentiated yielding where J [Ji ] and X [ Xj ] are vectors with N 1 components, = ( − ) × ( − ) N−1 and D Di, j is a N 1 N 1 matrix composed of the Fick- 1 ian diffusion coefficients. The latter are related to the Onsager co- ∇ T μ¯ N = v ∇P − X j ∇ T μ¯ j . (F.21) X N j=1 efficients Li , k through Eq. (F.30), which, in matrix form, can be expressed as Upon dividing Eq. (F.19) by W and subtracting it from Eq. k = L W . (F.21) divided by W N , the relation D (F.27) N−1 In Eq. (F.27) , L = [ L ] is the (N − 1) × (N − 1) subset of the V¯ v X V¯ i,k ∇ μ − μ = k − + m m ∇ T ( k N ) P Onsager matrix L denoted by dot-dashed lines in Eq. (143) . This W W N X N W N X N k m =1 subset contains only mass-transfer-related coefficients. In addition, W = W ( − ) × ( − ) N−1 N−1 [ k,m ] is a N 1 N 1 matrix whose elements depend X δ , ∂ ln f m m k 0 m + + R T ∇X j on the local composition and are given by ∂ = WNXN Wk = Xj , , m 1 j 1 P T Xi | i =1 , ...N −1 0 δ ( i = j ) R Xm m,k W , = + . (F.28) k m ρ (F.22) Xm WN XN Wk L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 69

( − ) × ( − ) = Lastly, the N 1 N 1 elements of the matrix [ m, j ] represent high-pressure effects and are given by C = C , 1 10 2 (F.39) ∂ ln f m m, j = X m . (F.29) ∂X j P,T,X r | r =1 , ...N −1 ( r = j ) C = C − ω + ω + ω ω , 2 3 1 i 10 j 10 i j (F.40) At low pressures, the fugacity is equal to the partial pressure, and therefore becomes the identity matrix. Upon substituting 3 C C 10 C −C T , P , Eqs. (F.29) and (F.28) into (F.27) , the expression 5 5 6 6 r j r i C = C P − 6 P + 6 T + C T + C , (F.41) 3 4 r, j r,i r, j 7 r,i 2 Tr,i Pr, j N−1 N−1 0 δ ∂ Li,k R Xm m,k ln fm C = − . , C = . , C = − . , C = − . , C = D i, j = + with 1 0 0472 2 0 0103 3 0 0147 4 0 0053 5 ρ W X W ∂X 0 = = N N k j , , −0 . 3370 , C = −0 . 1852 , and C = −0 . 1914 . The product ( cD ) k 1 m 1 P T Xr | r =1 , ...N −1 6 7 ( r = j ) [mol/(m · s)] is calculated using [173] (F.30) 1 / 2 / + / 2 − / 1 Wi 1 Wj m is obtained for the effective Fickian diffusion coefficient of species ( cD )0 = 1 . 01 × 10 2T 3 4 , / / 2 i into j . 0 ( ) 1 3 + ( )1 3 s R ν i ν j By definition, the binary diffusion coefficients D satisfy the i,m (F.42) relation

−1 where ( ν)i is the diffusion-volume increment of component i J = −ρB X , (F.31) ( ) = . 3 ( ) = . 3 given by ν N 18 50 cm and ν C H 250 86 cm for ni- = ( − ) × ( − ) 2 12 26 where B [Bi , m ] is a N 1 N 1 matrix given by [137,162] trogen and dodecane, respectively [54] .

N Xi Xk F6. Thermal-diffusion ratios B , = + for i = m and i = 1 , . . . N − 1 , i m D D i , N i , k k =1 ( = ) k i The calculation of the Onsager coefficients Lq , i , which partici- (F.32) pate in the Soret and Dufour effects in the standard diffusion fluxes (F.24) - (F.25) , is connected to the coefficients L and the thermal- i,i and diffusion ratio ki ,T of species i by the expression [163] 1 1 − 0 N1 ∗ ∗ B , = X − for i = m and i , m = 1 , . . . N − 1 . W R T L , Q Q i m i D D = i i = k − N , i , N i , m Lq,i ki, T Li,k (F.43) W W X X W W i N i N = k N (F.33) k 1 where Q ∗ is the amount of energy that must be absorbed per mole Upon combining Eqs. (F.26) , (F.27) and (F.31) , the relations k of component k while diffusing out that component in order to − − L = B 1W 1 (F.34) maintain the temperature and pressure of the mixture locally constant. and Following the modeling approach described in Ref. [163] , the −1 D = B (F.35) expression are obtained. In particular, Eqs. (F.34) and (F.35) enable the calcu- N V lation of the Onsager coefficients L [from Eq. (F.34) ] and the Fick- ∗ d e¯k Xj d e¯ j ¯k i,k Q = − + for k = 1 , . . . N, k N ian diffusion coefficients D [from Eq. (F.35) ] by using the binary ι ι j V¯ i,j k j=1 j=1 Xj j D diffusion coefficients i, j described in Appendix F.5. (F.44)

F5. Binary diffusion coefficients is employed here, with d e¯k the departure function of the partial molar internal energy of component k defined in Eq. (C.42) . D The binary diffusion coefficients i, j can be modeled using the Additionally, ι is a model parameter corresponding to the ratio expression ι = . = of vaporization and viscous flow energies. The value k 4 5 (k N 1 , . . . N) is employed here, which is slightly larger than the stan- ∞ ∞ ∞ ∞ / D = ( ) Xj ( ) Xi ( ) Xk 2 2/ , ι = . i, j Di, j D j,i Di,k D j,k [m s] (F.36) dard calibrated value k 4 0 used in Ref. [167]. This increase in k =1 ι is made to warrant a positive semidefinite Onsager matrix when k = i, j

ki ,T is employed in conjunction with the aforementioned models λ D which follows the formulation described in Refs[162,171,172]. for for and i, j in the binary mixtures considered in this study (see non-ideal and nonpolar multicomponent mixtures. In Eq. (F.36) , Fig. 19 and discussion in Section 5.5 ). ∞ ∞ Di, j (D j,i ) corresponds to the molecular diffusion coefficient of component i ( j ) infinitely diluted in component j ( i ). These aux- Appendix G. Numerical methods for the calculation of iliary coefficients are given by transcritical interfaces ∞ C C ( ω ,ω )+ C ( T ,P ) cD 1 η [ 2 i j 3 r r ] i, j Tr, j Pr,i 2 The solver employed to integrate the formulation presented in = C [ m / s] . (F.37) 0 0 η0 ( cD ) Tr,i Pr, j Table 8 is based on: (a) a second-order central finite-difference method for spatial discretization on one-dimensional non-uniform In this formulation, c [mol/m 3 ] is the molar density, T = T /T r,i c,i grids (e.g., see Ref. [174] and Ch. 2 in Ref. [175] for details); and and P = P/P are the reduced temperature and pressure, η/ η0 is r,i c,i (b) a fully-implicit variable-step second-order backward differenti- the ratio between high- and low-pressure viscosities, and C are k ation formula (BDF2) especially suited for time integration of stiff model constants given by equations [176] . The resulting nonlinear system of equations is C = C , 0 exp 1 (F.38) solved through an iterative globally-convergent strategy based on 70 L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Table G.1 Dimensional formulation integrated numerically by the solver for transcritical interfaces in isothermal bicomponent systems. Pseudo-time variations are included in the solver to relax the solution to the equilibrium steady state that satisﬁes the formulation in Table 5 . ( ‡ ) The steady solution of the problem is independent of the binary diffusion coeﬃcient.

Conservation equations 2 ∂ 2 ∂ − + ρ κ ρ − 1 κ ρ = , Mechanical equilibrium P∞ P YF F , F ( YF ) F , F ( YF ) 0 ∂ nˆ 2 2 ∂ nˆ ∂ ∂ Fuel species ( ρY ) = − ( J + J ). ∂t F ∂ nˆ F F Transport ﬂuxes ρD F , O W F W O X F (1 − X F ) ∂ Standard (species) J F = − ( μF − μO ) , 0 ∂ R T e W nˆ 3 ρκF , F D F , O W F W O X F (1 − X F ) ∂ Interfacial (species) J F = (ρY F ) . 0 ∂ 3 R T e W nˆ Boundary conditions ρ → ρ → −∞ , Phase-equilibrium composition YF e YFe at nˆ ρ → ρg g → + ∞ . YF e YFe at nˆ Supplementary expressions (see Appendix B, Appendix C , and Appendix F for details)

Equation of state Eq. (14) , Coefficients of the equation of state Eqs. (212) , and (C.10) - (C.12) , Gradient-energy coefficient Eqs. (23) , (C.17) , and (C.18) , Binary diffusion coefficient ‡ Eq. (F.36) , Thermodynamic relations Eqs. (213) , (214), (B.19), (B.20), (C.43) , and (C.47) .

the Newton-Raphson method [177] combined with line searches close enough to the minimum where the quadratic convergence along the Newton direction that are guaranteed to decrease the behavior of the Newton method is achieved. The utilization of J , residual by incorporating a backtracking routine [178] . Details are which is constructed from local derivatives, only guarantees that r provided in this appendix along with specific considerations for decreases at the starting point of the Newton step, but not nec- isothermal and non-isothermal cases. essarily all the way. Therefore, to ensure that the Newton iteration reduces the residual, the updating step is premultiplied by a G Y = Y + G Y , < G ≤ G1. Spatial and temporal discretizations factor { k } as { k +1 } { k } { k } { k } with 0 { k } 1. Given Y that { k } is a local direction of descent, the full Newton step, G = , Consider the conservation equations expressed as the coupled i.e., { k } 1 is tested first; this will lead to quadratic convergence Y differential system y˙ = F (t, y ) subject to y (t 0 ) = y 0 , where y de- when is sufficiently close to the minimum. However, if the resid- ( Y ) > ( Y ) , Y notes the vector of independent variables, F (t, y ) represents the ual increases, i.e., r { k +1 } r { k } { k } is backtracked along G convection and diffusion fluxes, and y 0 is the initial value of y . The the Newton direction testing smaller values of { k } until a local variable-step BDF2 scheme advances the solution in time as minimum is found. The details of this methodology, which is guaranteed to efficiently decrease the residual norm, are fully described + 2 1 2 n n in Ch. 6 of Ref. [178] . y + − ( 1 + n )y n + y − + n 1 + n 1 1 n 1 n An important piece of the solution methodology is the efficient and accurate calculation of J at each sub-iteration of the Newton- = t n F (t n +1 , y n +1 ) , (F.1) Raphson solver. In an ideal scenario, the best option would be to where the subscript n indicates time level, and analytically derive the expressions for all the terms in the Jaco- bian matrix such that only analytical evaluations of functions are n = t n / t n −1 (F.2) needed to compute J at runtime. This strategy would typically is the ratio of time steps. provide the most accurate and computationally efficient calcula- The nonlinear implicit system of algebraic Eqs. (G.1) is iteration of the Jacobian, which is important since the complexity of tively solved by means of the Newton-Raphson method at each evaluating J scales with its size as M 3 . However, such approach is time level. Specifically, defining M as the number of indepen- not optimal for transcritical flows due to the complex relations be- dent variables multiplied by the number of spatial grid points, tween the independent variables and the residual expressions in-

Eq. (G.1) can be recast into R ( Y ) = 0, with R ∈ R M the vector volving standard and diffuse-interface fluxes, and real-gas thermo- of residuals and Y ∈ R M the discrete representation of y . To ob- dynamic functions. Instead, a practical solution is to approximate tain the distribution of Y at time n + 1 , minimization approaches the terms of J with finite differences. Similarly to most Jacobian- based on Newton’s method iteratively find an updated solution based solvers [177,178] , this work utilizes first-order forward finite Y = Y + Y Y { k +1 } { k } { k } from the previous state { k } by calculat- differences [175] to approximate the terms of the Jacobian matrix ing the Newton step Y = −J −1 · R , where the subscript { k } as it provides a good balance between accuracy and computational { k } { k } { k } refers to a sub-implicit iteration within the time-level advance- cost, whereas the inverse of the Jacobian is calculated by means of J = ∂ R /∂ Y a lower-upper (LU) decomposition. ment, and i, j i j is the Jacobian matrix. In this way, the norm of the residuals, r = R · R T , is reduced in every iteration. At Y the beginning of each time step, { k } is initiated with the solu- G2. Numerical considerations for cases involving equilibrium tion of the previous time level Y n to efficiently start the iterative isothermal bicomponent systems process that provides Y n +1 . Y Taking the full Newton step { k } may not decrease the resid- In isothermal cases, the solver used in this study integrates an ual. This is the case, for example, when the solution state is not unsteady formulation that simplifies to that in Table 5 in steady L. Jofre and J. Urzay / Progress in Energy and Combustion Science 82 (2021) 100877 71 state. In particular, the code integrates the mechanical equilibrium [2] Rosner DE . Liquid droplet vaporization and combustion. In: Harrje DT, Rear- condition (107) and relaxes a convectionless species conservation don FH, editors. Liquid propellant rocket combustion instability.; 1972 . NASA Tech. Rep. #NASA-SP-194. R equation (46) to the steady state. Correspondingly, corresponds [3] Sánchez-Tarifa C , Crespo A , Fraga E . A theoretical model for the combus- to the residual versions of these two equations. The boundary tion of droplets in super-critical conditions and gas pockets. Astronaut Acta conditions away from interfaces correspond to phase-equilibrium 1972; 17 :685–92 .

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Urzay / Progress in Energy and Combustion Science 82 (2021) 100877

Dr. Lluís Jofre , Postdoctoral Fellow at the Center for Tur- Dr. Javier Urzay , Senior Research Engineer at the Cen- bulence Research, Stanford University (USA). He obtained ter for Turbulence Research, Stanford University (USA). He his Mechanical Engineering Degree in 2008 from the obtained his B.Sc./M.Sc. degree in Mechanical Engineering Technical University of Catalonia BarcelonaTech (Spain), in 2005 from the Carlos III University of Madrid (Spain), jointly with the KTH Royal Institute of Technology, Swe- and his M.Sc. and Ph.D. degrees in Aerospace Engineer- den. In 2014, he obtained a Ph.D. in Fluid Mechanics and ing in 2006 and 2010 from the University of California Thermal Engineering also from the Technical University San Diego (USA) working on theoretical aspects of com- of Catalonia BarcelonaTech, Spain. His main research ar- bustion physics and fluid mechanics. His research inter- eas include high-pressure transcritical flows, two-phase ests include chemically reacting flows, multiphase turbu- flows, modeling and computational studies of turbulence lent flows, chemical rockets, hypersonic aerothermody- in multiphysics environments, uncertainty quantification, namics, supersonic combustion, and their engineering ap- and data science in Fluid Mechanics. plications to aeronautics and astronautics. He currently serves in the U.S. Air Force Reserves at Travis Air Force Base, California.