Sajad Jafari
email: sajad.jafari@ifpen.fr

Hesham Gaballa

Chaouki Habchi
email: chaouki.habchi@ifpen.fr

Jean-Charles De Hemptinne

Pascal Mougin

Exploring the interaction between phase separation and turbulent fluid dynamics in multi-species supercritical jets using a tabulated real-fluid model

Keywords: Real fluid model, Vapor-liquid equilibrium, Tabulation, Multicomponent, Largeeddy simulation, Condensation, Droplet formation process

Today, injection of liquid fuels at supercritical pressures is a frequently used technique to improve the efficiency of energy systems and address environmental constraints. This paper focuses on the analysis of the coupling between the hydrodynamics and thermodynamics of multi-species supercritical jets. Various phase transition phenomena, such as droplet formation process by condensation, which have been shown experimentally to significantly affect the flow and mixing dynamics of the jet, are studied. For this purpose, a tabulated multicomponent real fluid model assuming vapor-liquid equilibrium is proposed for the simulation of turbulent n-hexane jets injected with different inflow temperatures (480 K, 560 K, 600 K) into supercritical nitrogen at 5 MPa and 293 K. Numerical results are compared with available experimental data but also with published numerical studies, showing a good agreement. In addition, comparisons between different turbulence models, including the LES Sigma, Smagorinsky and RANS 𝐾 -𝜖 models have been performed, showing the relevance of the LES Sigma model for these very complex two-phase flows.

Introduction

In modern high-performance engines, the operating pressures now approach or exceed the critical pressures of the working fluids to increase the engine's efficiency and reduce 𝐶𝑂 2 emissions. However, the injection is such that the propellant may enter the chamber at subcritical (cryogenic temperature) or supercritical temperature, corresponding to liquid-like (LL) or gas-like (GL) states, respectively [START_REF] Crua | On the transcritical mixing of fuels at diesel engine conditions[END_REF][START_REF] Jofre | Transcritical diffuse-interface hydrodynamics of propellants in high-pressure combustors of chemical propulsion systems[END_REF][START_REF] Banuti | Crossing the Widom-line-Supercritical pseudo-boiling[END_REF]. The properties in these states significantly deviate from the ideal gas as they introduce some thermodynamics' non-idealities and transports anomalies. Two-phase flows covering (LL) and (GL) conditions are known in the literature as the transcritical regime [START_REF] Crua | On the transcritical mixing of fuels at diesel engine conditions[END_REF][START_REF] Jofre | Transcritical diffuse-interface hydrodynamics of propellants in high-pressure combustors of chemical propulsion systems[END_REF][START_REF] Banuti | Crossing the Widom-line-Supercritical pseudo-boiling[END_REF][START_REF] Lapenna | Characterization of pseudo-boiling in a transcritical nitrogen jet[END_REF][START_REF] Schmitt | Large-eddy simulation of transcritical flows[END_REF][START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF][START_REF] Gerber | Enhanced Rayleigh scattering in supercritical fluid injection across the Widom line[END_REF][START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Ma | On the numerical behavior of diffuse-interface methods for transcritical realfluids simulations[END_REF].

At subcritical conditions, the phase transition phenomena are either classified as evaporation or condensation. Generally, it has been accepted that the subcritical liquid-to-gas transition is characterized by a structural change from an ordered (liquid) to a disordered (gas) state. In this case, density can be chosen as the appropriate order parameter, which exhibits a sharp discontinuity when phase transition occurs [START_REF] Simeoni | The Widom line as the crossover between liquid-like and gas-likebehaviour in supercritical fluids[END_REF][START_REF] Gorelli | Liquid like behavior of supercritical fluids[END_REF][START_REF] Gorelli | Dynamics and Thermodynamics beyond the critical point[END_REF]. Besides, pseudo phase transition (i.e. pseudo-boiling or pseudo-condensation) may occur in the supercritical state at the crossover between LL and GL fluids, where the density changes sharply but continuously. The locus where this sharp transition from LL and GL fluids is currently not well understood. The Widom line [START_REF] Banuti | Crossing the Widom-line-Supercritical pseudo-boiling[END_REF][START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Simeoni | The Widom line as the crossover between liquid-like and gas-likebehaviour in supercritical fluids[END_REF][START_REF] Gorelli | Liquid like behavior of supercritical fluids[END_REF], defined as the heat capacity maximum, is usually used to separate the LL and GL supercritical fluids. However, the Widom line cannot be clearly defined at high pressure and high temperature (see Figure 1(b)), where the minima of the thermal diffusivity coefficients are found to be more appropriate by Gorelli et al. [START_REF] Gorelli | Dynamics and Thermodynamics beyond the critical point[END_REF] for tracking the transition between the supercritical LL and GL fluids. In the pseudo-boiling transition, no structural change has been observed in macroscopic light scattering experiments [START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF][START_REF] Gerber | Enhanced Rayleigh scattering in supercritical fluid injection across the Widom line[END_REF]. As such, these phenomena share a nonlinear behavior similar to that of the subcritical condition when the coexistence line is crossed, as illustrated in Figure 1. For pure n-Hexane (𝐶 6 𝐻 14 ), this figure shows the phase transition in subcritical states (i.e., evaporation and condensation processes) when the coexistence line is crossed. Also, the gradual phase-change processes in the transcritical regime (i.e., pseudo-boiling and pseudo-condensation) as soon as it meets the Widom line. This illustrates that phase transition, including bubble or droplet formation process, evaporation, and condensation, are mainly driven by changes in pressure or temperature for a single-component fluid.

As a matter of fact, researcher's interest in the behavior of supercritical fluids dates back to at least 1869 [START_REF] Andrews | The Bakerian lecture-On the continuity of the gaseous and liquid states of matter[END_REF]. However, despite recent theoretical, experimental, and computational efforts, these phenomena are still not fully understood especially for multicomponent systems. In the past decades, extensive studies have been performed for understanding the behavior of a single-component phase transition [START_REF] Banuti | Crossing the Widom-line-Supercritical pseudo-boiling[END_REF][START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Ma | On the numerical behavior of diffuse-interface methods for transcritical realfluids simulations[END_REF][START_REF] Gorelli | Dynamics and Thermodynamics beyond the critical point[END_REF][START_REF] Delplanque | Transcritical Vaporization of Liquid Fuels and Propellants[END_REF][START_REF] Chehroudi | Understanding injection into high pressure supercritical environments[END_REF][START_REF] Bellan | Supercritical (and subcritical) fluid behavior and modeling: Drops, streams, shear and mixing layers, jets and sprays[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Romei | Computational fluid-dynamic modelling of two-phase compressible flows of carbon dioxide in supercritical conditions[END_REF]. One of the pioneering and remarkable studies was conducted by Chehroudi et al. [START_REF] Chehroudi | Understanding injection into high pressure supercritical environments[END_REF]. They experimentally studied the jet behavior of pure liquid nitrogen injected into gaseous nitrogen at subcritical and supercritical pressures. They observed that in subcritical pressures, due to the significant surface tension, the interface between the liquid and the gas plays an important role, and it leads to the appearance of the primary and secondary breakup, known as classical atomization. However, as the pressure increases, the magnitude of the surface tension decreases and vanishes at the critical point, which suppresses the formation of ligaments and droplets for supercritical pressures. However, liquid-like ligaments (also called finger-like) could be seen in their shadowgraphy experimental results despite the absence of surface tension, especially for the transcritical injection regime. Therefore, injection above the critical pressure of a single-component propellant can be considered as a diffuse interface mixing process rather than a classical atomization of the interface. However, the same is not valid for multicomponent systems for which the local critical pressure of the mixture could dramatically exceed the value of the pure components. Indeed, various studies have been carried out for multicomponent systems, and their interfacial phase transition has been revealed to be more complex than for a single-component system, mainly due to the variation of the mixture critical point with the local composition of the fluid [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Mayer | Injection and mixing processes in high-pressure liquid oxygen/gaseous hydrogen rocket combustors[END_REF][START_REF] Roy | Disintegrating supercritical jets in a subcritical environment[END_REF][START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF][START_REF] Qiu | Simulation of supercritical fuel injection with condensation[END_REF][START_REF] Tudisco | Numerical investigations of phase-separation during multicomponent mixing at super-critical conditions[END_REF][START_REF] Matheis | multicomponent vapor-liquid equilibrium model for LES of high-pressure fuel injection and application to ecn spray a[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF]. Furthermore, this difference in phase transition mechanisms between single and multicomponent systems leads to various interface structures that influence the jet behavior [START_REF] Crua | On the transcritical mixing of fuels at diesel engine conditions[END_REF][START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF][START_REF] Qiu | Simulation of supercritical fuel injection with condensation[END_REF]. Crua et al. [START_REF] Crua | On the transcritical mixing of fuels at diesel engine conditions[END_REF] studied the injection of liquid (n-heptane, n-dodecane, n-hexadecane) into supercritical nitrogen at different temperatures and pressures. They investigated the transition from a two-phase to a single-phase mixing regime and established a criterion for switching between the two regimes as a function of the critical temperature and pressure of the pure fuels. However, due to the limited accuracy of the experimental setup, reliable data are still lacking for the precise identification of the transition between the two-phase and single-phase mixing regime that could be used for validating numerical models. In addition, there are still multiple doubts and questions about the conditions and mechanisms behind the interface disappearance for multicomponent systems.

Other interesting experimental studies have recently done by Gerber et al. [START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF][START_REF] Gerber | Enhanced Rayleigh scattering in supercritical fluid injection across the Widom line[END_REF]. These studies focused in particular, on under-relaxed jets for which a shock waves train was observed in the near nozzle region. They concluded that high-pressure jets have the same scattering footprint and morphological properties as low-pressure jets. However, enhanced scattering is only observed upon multiple crossings of the Widom region, e.g., due to the series of rapid recompressions caused by the shock train. This enhanced scattering is caused by the enhancement in compressibility and density fluctuation when the fluid switches between LL and GL. Therefore, the experimental findings of Gerber al. [START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF][START_REF] Gerber | Enhanced Rayleigh scattering in supercritical fluid injection across the Widom line[END_REF] show that density fluctuations are not negligible across the Widom line. They can be detected both at the macroscopic (continuum) and molecular scale. Indeed, these findings were confirmed at molecular scales by the small-angle X-ray scattering experiments in [START_REF] Nishikawa | Density fluctuation of supercritical fluids obtained from small-angle X-ray scattering experiment and thermodynamic calculation[END_REF]. This completely changes the physical meaning of pseudo-boiling/pseudo-condensation transition from the subcritical counterpart (evaporation/condensation).

Another interesting study was recently carried out by Traxinger et al. [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]. They experimentally and numerically studied a multicomponent system consisting of a high-pressure n-hexane jet at three different temperatures (480𝐾, 560𝐾, 600𝐾), injected into a chamber filled with supercritical nitrogen at (𝑃 = 5𝑀 𝑃𝑎 and 𝑇 = 293𝐾). Specifically, the focus was on the phase separation of an initially supercritical fluid due to the interaction with its surrounding. Simultaneous shadowgraphy and elastic light scattering (ELS) experiments were conducted by Traxinger et al [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF] to capture both the flow structure as well as the phase separation. Remarkable results have been shown experimentally for the case of supercritical n-hexane with T=560 K, exceeding its critical temperature and pressure, injected into supercritical nitrogen at 𝑇 = 293𝐾 and 𝑃 = 5𝑀 𝑃𝑎.

Indeed, the condensation of n-hexane has been observed in these conditions, as expected from their priory thermodynamic study. The phase transition is initiated in the mixing layer some distance downstream of the nozzle and eventually mixes with the jet core at large distances. In addition, at a higher injection temperature of n-hexane (𝑇 = 600𝐾), this phenomenon of phase transition does not occur, and the gaseous fuel injected into the chamber gradually mixes with the ambient gas. Finally, a classical subcritical atomizing n-hexane jet was imaged at the lowest n-hexane injection temperature (𝑇 = 480𝐾). The various experimental and numerical results of Traxinger et al. [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF] will be used in the present work to validate our Computational Fluid Dynamics (CFD) tabulated Real-Fluid Model (RFM) presented in Section 3.

Developing a CFD model capable of handling subcritical, transcritical, and supercritical conditions, including possible phase transition, is crucial to enable a more accurate analysis of multicomponent systems. To study these complex behaviors, it is therefore essential to use a real fluid model capable of automatically switching from a single-phase flow to a two-phase flow, and vice versa [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF][START_REF] Matheis | multicomponent vapor-liquid equilibrium model for LES of high-pressure fuel injection and application to ecn spray a[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF]. However, real fluid simulations have been proven to be computationally time-consuming. Indeed, it has been found that the complexity of cubic EoS and solving the Vapor-Liquid-Equilibrium (VLE) iteratively using the isoenergetic-isochoric (UVn)-flash set of equations are computationally expensive [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Matheis | multicomponent vapor-liquid equilibrium model for LES of high-pressure fuel injection and application to ecn spray a[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF][START_REF] Michelsen | The isothermal flash problem. Part I. Stability[END_REF][START_REF] Michelsen | The isothermal flash problem. Part II. Phase-split calculation[END_REF][START_REF] Saha | The isoenergetic-isochoric flash[END_REF][START_REF] Koukouvinis | High pressure/high temperature multiphase simulations of dodecane injection to nitrogen: application on ECN Spray a[END_REF]. Therefore, an accurate, robust, and most importantly an efficient real-fluid model is required, which is one of the main objectives of the presented work. A remedy for this problem is to set up a table before the start of the simulation and store the values of the required thermodynamic properties in it. Then, rapid searching, as well as an interpolation, are carried out in the table during the simulation. This process of searching and interpolation is usually called "a look-up table" in the literature. More details about tabulation, interpolation, and look-up of data can be found in [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Lorenzo | Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations[END_REF][START_REF] Brown | Thermodynamic interpolation for the simulation of two-phase flow of non-ideal mixtures[END_REF][START_REF] Ar. Azimian | Generation of steam tables using artificial neural networks[END_REF][START_REF] Fang | An Accurate and Efficient Look-up Table Equation of State for Two-phase Compressible Flow Simulations of Carbon Dioxide[END_REF][START_REF] Foll | On the use of tabulated equations of state for multi-phase simulations in the homogeneous equilibrium limit[END_REF][START_REF] Praneeth | Uncertainty quantification of tabulated supercritical thermodynamics for compressible Navier-Stokes solvers[END_REF].

In summary, in this study, a fully compressible, multicomponent real fluid model accounting for VLE is proposed using a generalized three-dimensional (3D) tabulation method [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Koukouvinis | High pressure/high temperature multiphase simulations of dodecane injection to nitrogen: application on ECN Spray a[END_REF][START_REF] Lorenzo | Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations[END_REF].

The tabulated thermodynamic properties of the binary mixture are calculated as a function of temperature (T), pressure (P), and mass fraction of the first species in the mixture (Y). In this model, an in-house thermodynamic library IFPEN-Carnot is used to generate a 3D table (with T-P-Y as the axis for a binary mixture) based on various real fluid EoS. This IFPEN-Carnot thermodynamic library includes different EoSs, such as Peng-Robinson (PR-EoS) [START_REF] Peng | A new two-constant equation of state[END_REF],

Soave-Redlich-Kwong (SRK-EoS) [START_REF] Soave | Equilibrium constants from a modified Redlich-Kwong equation of state[END_REF], and Cubic Plus Association (CPA-EoS) [START_REF] Kontogeorgis | An Equation of State for Associating Fluids[END_REF], which also represents one of its main advantages [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Gaballa | A tabulated real fluid modeling approach applied to renewable dual-fuel evaporation and mixing[END_REF]. For computational speed and robustness, a uniform look-up table is generated based on an isothermal-isobaric TPn-flash [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF][START_REF] Saha | The isoenergetic-isochoric flash[END_REF]. The RFM model was implemented together with the generalized 3D tabulation method in CONVERGE solver [START_REF] Senecal | CONVERGE 3.0, Convergent Science[END_REF]. This paper is organized as follows. Section 3 is dedicated to the explanation of the numerical and theoretical methods, including the governing equations, thermodynamic tabulation, and look-up methods. Section 4 is devoted to understanding the condensation phenomenon during the supercritical injection of hexane into nitrogen using the proposed RFM model. In addition, comparisons between various turbulence models, including large-eddy simulations (LES Sigma [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF] and Smagorinsky [START_REF] Sagaut | Large-Eddy Simulation for Incompressible Flows, 3rd edn. Scientific Computation[END_REF]) models as well as a Reynolds Averaged Navier-Stokes (RANS 𝐾-𝜖) [START_REF] Han | Turbulence Modeling of Internal Combustion Engines Using RNG k-e Models[END_REF] model will be carried out and discussed. Subsequently, the numerical results in sections 4 and 5 are compared with available experimental data, but also with published numerical studies.

Finally, this paper is summarized and the conclusions are presented in section 6.

The real-fluid RFM model

The real-fluid RFM model [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Gaballa | A tabulated real fluid modeling approach applied to renewable dual-fuel evaporation and mixing[END_REF][START_REF] Yi | Numerical analysis of subcritical evaporation and transcritical mixing of droplet using a tabulated multicomponent vapor-liquid equilibrium model[END_REF] used in the current work is presented in this section. It is a fully compressible model developed under the homogeneous equilibrium modeling (HEM) assumption, considering multicomponent in both phases. Indeed, the RFM model has been developed assuming the two-phase system to be under the assumptions of thermodynamic equilibrium (i.e., equilibrium of the velocity, pressure, temperature, and chemical potential at liquid-vapor interfaces). This assumption guarantees the required hyperbolic mathematical formulation for the RFM model. The set of governing equations as employed in the CONVERGE CFD solver [START_REF] Senecal | CONVERGE 3.0, Convergent Science[END_REF] has been used as a base framework for the development of the tabulated RFM model presented below. More details about this modelling approach could be found in Jafari et al. [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF].

Governing equations of the flow solver

The set of governing equations are presented through equations (1 -4) including the density transport equation, species transport equation, mixture momentum equation and mixture specific internal energy equation, respectively. (𝜌, 𝑢 𝑖 , 𝑃, 𝑇, 𝑒) are the mixture's density, velocity, pressure, temperature, and specific internal energy, respectively. (𝑌 𝑚 , ℎ 𝑚 ) are mass fraction and enthalpy of species (𝑚) in the mixture, respectively. 𝑁 𝑠 is the total number of species.

𝜕 𝜌 𝜕𝑡

where the viscous stress tensor (𝜏 𝑖 𝑗 ) is given by

𝜏 𝑖 𝑗 = 𝜇( 𝜕𝑢 𝑖 𝜕𝑥 𝑗 + 𝜕𝑢 𝑗 𝜕𝑥 𝑖 ) -2/3𝜇( 𝜕𝑢 𝑘 𝜕𝑥 𝑘 )𝛿𝑖 𝑗 (5)
It is worth mention that the calculations of the real fluid thermodynamic properties are performed based on the residual approach [START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] De Hemptinne | Select Thermodynamic Models for Process Simulation -A Practical Guide using a Three Steps Methodology[END_REF]. In this approach, any thermodynamic function is computed from the sum of an ideal gas part and a residual part. The ideal gas part is determined from a specific polynomial equation [START_REF] Passut | Correlation of Ideal Gas Enthalpy, Heat Capacity and Entropy[END_REF]. While the residual part, which represents the deviation from the ideal gas behavior is deduced from the employed equation of state (EoS). More details can be found in our previous studies [START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF]. Besides, the thermal conductivity (𝜆) and the dynamic viscosity (𝜇) cover laminar and turbulent contributions, as cumulative transport properties of the fluid. The laminar contribution of (𝜆, 𝜇) is computed by Chung et al. [START_REF] Chung | Generalized multi-parameter correlation for non-polar and polar fluid transport properties[END_REF] correlations. The turbulent conductivity is calculated using a given turbulent Prandtl number 𝑃𝑟 𝑡 , as

(𝜆 𝑡 = 𝐶 𝑃 𝜇 𝑡 𝑃𝑟 𝑡 ),
where (𝐶 𝑝 ) is the isobaric heat capacity and the turbulent viscosity (𝜇 𝑡 ) is computed by the adopted turbulence model. Also, the laminar, and turbulent diffusion coefficients are estimated using given laminar and turbulent Schmidt numbers, (𝑆𝑐) for 𝐷 = 𝜇/(𝜌𝑆𝑐), and (𝑆𝑐 𝑡 ) for 𝐷 𝑡 = 𝜇 𝑡 /(𝜌𝑆𝑐 𝑡 ), respectively. In this regard, various turbulence models including large-eddy simulations (LES Sigma [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF] and Smagorinsky [START_REF] Sagaut | Large-Eddy Simulation for Incompressible Flows, 3rd edn. Scientific Computation[END_REF]) models as well as a Reynolds Averaged Navier-Stokes (RANS 𝐾-𝜖) [START_REF] Han | Turbulence Modeling of Internal Combustion Engines Using RNG k-e Models[END_REF] model have been considered in this study (See Section 4) to investigate the effect of the turbulence modeling on the numerical results.

Numerical schemes

The above set of governing equations (1 -4) of the two-phase RFM model is solved numerically using a diffuse interface modeling approach (DIM). In the CONVERGE CFD solver, all the dependent parameters are collocated at the center of the computational cell. To prevent checkerboarding, the Rhie-Chow algorithm is employed [START_REF] Rhie | Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation[END_REF]. Besides, a second-order-accurate spatial discretization scheme, plus a first-order Euler temporal scheme, are used for the governing conservation equations discretization. The transport equations are solved using a density-based solver, where an extended Pressure Implicit with Splitting of Operators (PISO) algorithm, method of Issa [START_REF] Issa | The Computation of Compressible and Incompressible Recirculating Flows by a Non-Iterative Implicit Scheme[END_REF], has been adapted for real fluid equations of state. More details on the modified PISO algorithm can be found Jafari et al. [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF].

Equilibrium thermodynamic closure of the flow solver, and the tabulation look-up

For the supercritical jet with (𝐶 6 𝐻 14 , 𝑁 2 ) binary mixture, the real-fluid Peng-Robinson (PR) [START_REF] Peng | A new two-constant equation of state[END_REF] equation of state (EOS) has been selected for the thermodynamic closure of the RFM model transport equations (1 -4). A general form of this cubic EoS can be written as equations [START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF][START_REF] Gerber | Enhanced Rayleigh scattering in supercritical fluid injection across the Widom line[END_REF], along with the various parameters that are listed in Table 1 [START_REF] Peng | A new two-constant equation of state[END_REF], where index (𝑐) stands for

𝑃 = 𝑅𝑇 𝜈 -𝑏 - 𝑎(𝑇) (𝜈 + 𝛿 1 𝑏) (𝜈 + 𝛿 2 𝑏) (6)
𝑎 𝑖 (𝑇) = (𝑎 𝑐 ) 𝑖 [1 + 𝑚 𝑖 (1 - √ 𝑇 𝑇 𝑐 )] 2 (7)
Besides, when the PR-EoS is used for mixtures, van der Waals mixing rules are applied as equation [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF], where (𝑘 𝑖 𝑗 ) is the binary interaction parameter (BIP) that can be fitted to experimental data to well represent the phase diagram of a binary system, 𝑥 𝑖 is the molar fraction, and 𝑎 𝑖 (𝑇) is the parameter calculated using Equation ( 7) for component i. In this study, 𝑘 𝑖 𝑗 is set to zero.

𝑎(𝑇) = ∑ 𝑖 ∑ 𝑗 𝑥 𝑖 𝑥 𝑗 √ 𝑎 𝑖 (𝑇).𝑎 𝑗 (𝑇) (1 -𝑘 𝑖 𝑗 ) 𝑏 = ∑ 𝑖 𝑥 𝑖 .𝑏 𝑖 (8)
Generally, it is well known that cubic EoSs with van der Waals (vdW) mixing rule are not really well suited for computing densities [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF]. However, our achievement in this study demonstrate that the PR EoS [START_REF] Peng | A new two-constant equation of state[END_REF] can acceptably predict the density and specific heat capacity for n-hexane (a relatively light component) at 𝑃 = 5𝑀 𝑃𝑎, which are in an excellent agreement with the reference data taken from NIST [START_REF] Linstrom | The NIST Chemistry WebBook: A Chemical Data Resource on the Internet[END_REF], as shown in Figure 2. The current work adopts a tabulation approach, where the thermal and transport properties, as well as the phase states and compositions, are tabulated before the CFD simulation. During the simulation, the various tabulated properties, such as density, specific internal energy, isobaric heat capacity, isochoric heat capacity, viscosity, conductivity, gas volume fraction, and species liquid mass fraction in each phase, are linearly interpolated using three inputs, which are temperature (T), pressure (P), and species mass fraction in the mixture (𝑌 𝑚 , m = 1 : 𝑁 𝑠 -1), where 𝑁 𝑠 is the total number of species. As we saw above for the binary systems (𝑁 𝑠 = 2), 𝑌 1 is subsequently denoted 𝑌 , as the third axis of the 3D-Table . In this study, this table is generated based on the IFPEN-Carnot thermodynamic library, which uses a robust isothermal-isobaric (TPn)-flash algorithm coupled with the PR-EoS. This tabulation approach offers the advantage of avoiding the direct evaluation of the non-linear cubic EoS along with the VLE calculation during the simulation, which has been proven to be computationally demanding and costly [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF][START_REF] Koukouvinis | High pressure/high temperature multiphase simulations of dodecane injection to nitrogen: application on ECN Spray a[END_REF][START_REF] Lorenzo | Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations[END_REF].

Moreover, the tabulation approach based on the IFPEN-Carnot thermodynamic library allows simulating different fuels and surrogates without the need of hard coding a (TPn) flash for each EoS of interest.

With respect to interpolating the data in the table during runtime, the simplistic inverse distance weighting (IDW) is implemented [START_REF] Ware | Memory intensive statistical algorithms for multibeam bathymetric data[END_REF]. This critical step is mainly carried out during the simulation for two main tasks:

A Table look-up function:

Compute the thermal, transport properties as well as the phase state using the input parameters (𝑇, 𝑃, 𝑌 𝑚 ) with 𝑚 = 1 : 𝑁 𝑠 -1.

A Reverse look-up function:

Compute/Update the temperature using the inputs (𝑒, 𝑌 𝑚 , 𝑃) once the mixture mass fraction (𝑌 𝑚 ) and the energy (𝑒) transport equations are solved for each pressure (𝑃) in the PISO loop.

A general form of finding an interpolated value of 𝑢 𝑖 at a given point 𝑋 based on samples 𝑢 𝑖 = 𝑢(𝑋 𝑖 ), 𝑖 = 1, ...𝑁 using IDW can be expressed as equation [START_REF] Ma | On the numerical behavior of diffuse-interface methods for transcritical realfluids simulations[END_REF], where 𝜔 𝑖 (𝑋) =

𝑢(𝑋) =        𝑁 𝑖=1 𝜔 𝑖 (𝑋)𝑢 𝑖 𝑁 𝑖=1 𝜔 𝑖 (𝑋)
𝑖 𝑓 𝑑 (𝑋, 𝑋 𝑖 ) ≠ 0 for all i 𝑢 𝑖 𝑖 𝑓 𝑑 (𝑋, 𝑋 𝑖 ) = 0 for some i (9)

Validation of the VLE solver

As explained in the introduction, the state properties deviate considerably from those of the ideal gas, as they introduce some thermodynamic non-linearity and a transport anomaly at supercritical pressures, as shown in Figure 2. Indeed, as it can be seen in Figure 2, compared to the density and isobaric heat capacity from NIST [START_REF] Linstrom | The NIST Chemistry WebBook: A Chemical Data Resource on the Internet[END_REF] for n-hexane at 𝑃 = 5𝑀 𝑃𝑎, ideal gas EoS cannot predict the transcritical behavior of such properties, ideal gas EoS is outside the range. However, more accurate and complex EoS, as PR EoS, can predict the transcritical behavior with a good agreement, especially for the isobaric heat capacity.

To validate the VLE thermodynamic solver, the VLE for a binary mixture of 𝐶 6 𝐻 14 -𝑁 2 using PR-EoS has been compared with the available experimental data [START_REF] Eliosa-Jimenez | High-pressure vapor-liquid equilibria in the nitrogen + n-hexane system[END_REF], as shown in Figure 3.

It can be seen that the calculation results of the VLE solver are in good agreement with the experimental data over the entire ranges of pressure for the different temperatures. Furthermore, Figure 4(a,b) shows the variation of the mixture density and heat capacity, respectively, for a binary mixture of 𝐶 6 𝐻 14 -𝑁 2 as a function of temperature and 𝑁 2 mole fraction at P=5 MPa.

These results have been obtained using a uniform table with Δ𝑇 = 1𝐾, Δ𝑌 𝑁 2 = 0.005 in grid sizes on the axis. Due to the different non-linearity of the properties shown in Figure 4(a,b), a grid independence for the thermodynamic table is mandatory, which should correctly capture various interesting phenomena such as the heat capacity peak (𝐶 𝑝,𝑚𝑖𝑥 ) around the Widom line for certain compositions of this multi-species system, see Figure 4(b). It is worth mentioning that the values of properties in two-phase region, in the vapor liquid equilibrium dome, are computed from the mixing of the liquid and vapor phase properties using the vapor fraction, more can bee found here [START_REF] Jafari | Towards understanding the structure of subcritical and transcritical liquid-gas interfaces using a tabulated real fluid modeling approach[END_REF][START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF].

Analytical study of condensing supercritical jet

This section focuses on the analysis of the thermodynamic states of a condensing supercritical jet.

As reported in the introduction, condensation of n-hexane has been recently shown experimentally by Traxinger et al. [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF] when it is injected in supercritical state at 𝑇 𝑖𝑛 𝑗 = 560𝐾 into a supercritical nitrogen at 𝑇 𝑐ℎ = 293𝐾 and 𝑃𝑐ℎ = 5𝑀 𝑃𝑎. They observed that condensation is initiated in the mixing layer and eventually mixes in the core of the jet forming a spray of droplets at great distance from the nozzle. In such conditions, the mixture temperature distribution in the chamber is usually varying in the range between 𝑇 𝑖𝑛 𝑗 = 560𝐾 and 𝑇 𝑐ℎ = 293𝐾. Indeed, assuming an adiabatic process neglecting heat exchange, the enthalpy of the mixture at equilibrium 𝐻 𝑚𝑖𝑥 is equal to the weighted molar average of the enthalpy of the pure components, 𝐻 𝑁 2 and 𝐻 𝐶 6 𝐻 14 :

𝐻 𝑚𝑖 𝑥 (𝑇 𝐴𝑀 , 𝑃 𝑐ℎ , 𝑧 𝑁 2 ) = 𝑧 𝑁 2 𝐻 𝑁 2 (𝑇 𝑐ℎ , 𝑃 𝑐ℎ ) + 𝑧 𝐶 6 𝐻 14 𝐻 𝐶 6 𝐻 14 (𝑇 𝑖𝑛 𝑗 , 𝑃 𝑐ℎ ).
The adiabatic mixture temperature, denoted (𝑇 𝐴𝑀 ) is the solution of this equation, as the enthalpy values are calculated using PR-EoS using our in-house thermodynamic library. 𝑧 𝑁 2 is the mole fraction of 𝑁 2 (with 𝑧 𝑁 2 + 𝑧 𝐶 6 𝐻 14 = 1). 𝑇 𝑖𝑛 𝑗 , 𝑇 𝑐ℎ , 𝑃 𝑐ℎ denote the temperature of the injected n-hexane at the exit of nozzle, the initial temperature of 𝑁 2 in the chamber, and the initial chamber pressure (here constant and equal to 5𝑀 𝑃𝑎), respectively. More details on AMT can be found in [START_REF] Qiu | An investigation of thermodynamic states during high-pressure fuel injection using equilibrium thermodynamics[END_REF]. shows that by increasing the inflow temperature to 595𝐾, the AMT no longer crosses the phase boundary. Therefore, this theoretical analysis has demonstrated that due to the slope (dT/dx) near 𝑥 𝑁 2 = 1 in the phase diagram, the periphery of a supercritical jet may or may not nucleate and condense. This proves that jet structure in a multicomponent system does not follow the same behavior as in a single-component jet. In summary, for supercritical jet of 𝐶 6 𝐻 14 injected into 𝑁 2 colder ambient, droplet formation process by condensation are expected to be triggered at the jet's periphery when the AMT crosses the two-phase region. Otherwise, the periphery of the supercritical jet is going to mix with the ambient gas remaining in the supercritical regime without any droplet formation process. This supercritical droplet formation process/condensation phenomenon, which has been the subject of some recent studies [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF][START_REF] Qiu | Simulation of supercritical fuel injection with condensation[END_REF], will be investigated in more detail in the next section based on the tabulated RFM model applied in the LES numerical framework.

Results and discussion

Test cases definition

This section is devoted to the numerical study of the turbulent jet mixing process of n-hexane

(

Configuration setup

Figure 6(a,b) present the boundary conditions, the computational domain, as well a the mesh structure, respectively, employed in this study. The nozzle orifice diameter (𝐷 𝑖 ) is 236𝜇𝑚, as can be seen in this figure on the left side of the hexahedral computational domain. The length and width of the grid are 90𝐷 𝑖 and 40𝐷 𝑖 , respectively. These dimensions proved to be large enough to avoid edge effects on the numerical results. The base grid size was set to 30𝜇𝑚. Besides, fixed embedding has been used to refine the grid at specific locations especially near the nozzle exit, where the grid size was specified to be 7.5𝜇𝑚. This minimum value is the same refined value used in [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]. At the jet inlet, uniform velocity and temperature profiles for 𝑢 𝐶 6 𝐻 14 and 𝑇 𝐶 6 𝐻 14 are used, as defined in Table 2. In addition, no-turbulent fluctuations are specified at the inlet

RFM model results

Figure 7(a,b), respectively, shows the numerical results of the RFM model, obtained when the LES-Sigma turbulence model [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF] is employed, in terms of mixing temperatures versus AMT for the three different test cases (T480, T560, and T600) defined in Table 2. Specifically, Figure 7(a) demonstrates a good agreement between the predicted phase boundary calculated in this study compared with [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]. The low scattering of the LES mixing temperatures in Figure 7(a)

indicates almost adiabatic mixing injection (between n-hexane and 𝑁 2 ). The graphical insert shows that the LES mixing temperature scatter plots for the T560 case do intersect the dew line, enter the two-phase region and may induce condensation. Figure 7(b) includes the theoretical AMT (defined in Section 3.5) for the three cases and show the same behavior as the RFM model for case T560.

In Figure 8(a-c), the experimental single-shot images of [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF] (left frame 1) are qualitatively compared with instantaneous numerical snapshots (right frame 2) for the three cases (T480, T560, and T600) obtained using the RFM model, when the LES-Sigma turbulence model [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF] is employed. For the experiments, the elastic light scattering (ELS) images superimposed onto shadowgraph images, to visualize both the flow structure and the thermodynamic state. Similarly for the numerical side, the gas volume fraction 𝛼 𝐺𝑎𝑠 distribution is superimposed onto the temperature contours in a plane passing through the axis of the nozzle to indicate regions of two-phase flow and especially the lowest values of 𝛼 𝐺𝑎𝑠 near the periphery of the jet. These numerical results were taken at a point in time so that the jet could be considered as quasi-steady.

In doing so, a direct comparison of the two-phase droplet regions in the experiment and LES is rendered in Figure 8(a-b).

More precisely, in the subcritical Case T480, n-hexane is injected in a liquid state as depicted in blue color in Figure 7(b) indicating tiny 𝛼 𝐺𝑎𝑠 in the liquid core. In this case, a classical atomizing jet is shown experimentally in Figure 8(a). A spray can be seen clearly in this (shadowgraph + ELS) experimental image with distinct droplets at the periphery of the jet (especially for 𝑋/𝐷 > 50). Also, the ELS experiments have revealed a strong phase separation within the jet by showing high scattering intensity, (see I the palettes shown in Figure 8). Turbulent diffusion of 𝑁 2 into the injected n-hexane liquid core causes this mixture to enter the two-phase dome at a mole fraction of 𝑁 2 of about 0.1, as shown in Figure 7. This strong phase change can be seen in the numerical result shown in Figure 8(a) where the liquid core is represented (in blue)

as the low gas fraction region. The rest of jet is in two-phase state representing the droplets spray and highlighting the flow structure at its periphery. This flow structures can be also seen in Figure 9 which demonstrates the temporal evolution of superposed temperature and gas volume fraction (𝛼 𝐺𝑎𝑠 ) contours. It is worth noting on Figure 9 that the two-phase jet can be considered as quasi-steady for t>4 ms. Also, the authors would like to remind here that surface tension is not taken into account in the proposed RFM model and that future work should verify its effect on the numerical results of the subcritical T480 case.

For the second case T560, n-hexane is injected at a supercritical liquid-like state into 𝑁 2 which is ). The ELS image superimposed onto the corresponding shadowgraph are used to visualize both the flow structure as well as the thermodynamic state (adapted from [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]) in quasi-steady state condition. 𝛼 𝐺𝑎𝑠 is the gas volume fraction It is worth noting that the comparison between the simulation and the experiment has shown some differences regarding the jet penetration length and dispersion. This can be attributed to the experimental uncertainty of the velocity and unkown turbulence at the nozzle exit. In addition, there is some probability of uncertainty in the numerical simulation as well. However, Fig. 10. Temporal evolution of (temperature +gas volume fraction (𝛼 𝐺𝑎𝑠 )) contours using Sigma turbulence model for case 560 (see table 2).

the current validation has demonstrated that the RFM model can efficiently capture the main physics in various conditions.

Analysis of droplet formation process for binary systems at supercritical regime

With the phase equilibrium model employed in this study, a detailed analysis of the droplet formation process becomes possible in the condition the metastability relaxation time is smaller than the CFD time step. Obviously, VLE assumption is not valid in case the metastability duration is much longer than the CFD time step. In this case, detailed models for nucleation and droplet growth under non-equilibrium condition are required [START_REF] Luo | On phase transition in compressible flows: modelling and validation[END_REF].

Here, the case T560 is discussed based on Figure 11(a-d) in order to further illustrate the phase transition and droplet formation near to the jet border of an initially supercritical n-hexane fluid (𝑇 𝑖𝑛 𝑗 > 𝑇 𝑐 ) due to the interaction with its surrounding supercritical nitrogen. More specifically,

Turbulence modelling impacts on two phase jets

Turbulence is known to significantly enhance the mixing rate of species, momentum and energy.

This convective process occurs at different length scales as it results from the occurrence of turbulent eddies with different characteristic sizes, in high Reynolds number flows. Currently, only the direct numerical simulation (DNS) method can handle turbulent flows, but only for single-phase gas flows. In general, if a simulation does not use a sufficiently refined grid (DNS), it cannot properly account for turbulent mixing effects. This is still not practical in most of industrial test cases. Thus, turbulence models are often employed to estimate the additional mixing due to turbulence. As Richard Feynman (the American theoretical physicist, Nobel Prize in Physics in 1965) mentioned:" Turbulence is the most important unsolved problem of classical physics", hence developing a general turbulence model is still complicated due to highly unsteady and intricate interactions between various physical phenomena (turbulence, mixing, multi-phase flows, etc.) along with complex geometries and diverse boundary conditions. Traditionally, RANS (Reynolds-averaged Navier-Stokes equations) models [START_REF] Han | Turbulence Modeling of Internal Combustion Engines Using RNG k-e Models[END_REF] employ an effective turbulent viscosity to model the Reynolds stress term. Besides, in the past few years, Large-Eddy Simulation (LES) [START_REF] Nicoud | Using singular values to build a subgrid-scale model for large eddy simulations[END_REF][START_REF] Sagaut | Large-Eddy Simulation for Incompressible Flows, 3rd edn. Scientific Computation[END_REF] as a promising approach for turbulence modeling has been extensively used due to its unique potential to reproduce these unsteady and irregular phenomena. The main difference between LES and RANS models is how the fields are decomposed for modeling. For RANS approaches, the field is decomposed into an ensemble mean and a fluctuating component.

However, LES resolves more of the turbulent spectrum by using a more refined grid in such a way that only the isotropic smallest eddies remain for an easier modelling of sub-grid scale.

Hence, for LES approach, the field is decomposed into a resolved field and a sub-grid field, and numerous LES models have been developed to relate the turbulent stresses to a differential operator acting on the resolved velocity field, as Equations (10 -12). However, recent studies still demonstrate that each approach has some cons and pros, and choosing the best global LES approach for turbulence modeling, especially for internal combustion engines, is a crucial issue due to appearance of various complex phenomena [START_REF] Misdariis | Numerical methods and turbulence modeling for LES of piston engines: impact on flow motion and combustion[END_REF][START_REF] Rieth | Comparison of the sigma and Smagorinsky les models for grid generated turbulence and a channel flow[END_REF]. The oldest and most commonly used approach is the one proposed by Smagorinsky [START_REF] Sagaut | Large-Eddy Simulation for Incompressible Flows, 3rd edn. Scientific Computation[END_REF]. This model, known as a zero-equation LES model, does not solve any additional transport equations for sub-grid scale. It relates the turbulent viscosity to the magnitude of the strain rate tensor and the filter (usually taken as the cell size). Besides, an other promising zero-equation sub-grid model, called the Sigma model, has been developed by Baya-Toda et al. [START_REF] Sagaut | Large-Eddy Simulation for Incompressible Flows, 3rd edn. Scientific Computation[END_REF]. This more recent model shows that it can be a better alternative to the Smagorinsky model. It constructs the turbulent viscosity from the singular values of the resolved velocity gradient tensor. Most of the performance study of Sigma model have been performed for combustion [START_REF] Misdariis | Numerical methods and turbulence modeling for LES of piston engines: impact on flow motion and combustion[END_REF][START_REF] Rieth | Comparison of the sigma and Smagorinsky les models for grid generated turbulence and a channel flow[END_REF], and not for two-phase flows. In addition, two-phase turbulence effects have largely been neglected in the current literature. Indeed, most of the current state of the art two-phase flow simulations still consider one-fluid turbulence modelling approach, i.e. assuming that the turbulent structures behave similarly to single-phase turbulence.

In this study, the objective is to give some insight to two-phase turbulence modelling by analyzing the interaction between the turbulence and two-phase flows, especially when phase transition (evaporation, condensation) occurs. Generally, the turbulent viscosity in LES zero-equation models are defined as the following Equation 10.

𝜈 𝑆𝐺𝑆 = (𝐶 𝑚 Δ) 2 D𝑚 (10)
where 𝐶 𝑚 and 𝐷 𝑚 are the model (𝑚) specific constant, and differential operator of the model respectively. Δ is the filter width (i.e. the cell size in the context of LES with implicit filtering).

𝐷 𝑚 as the differential operator of the Smagorinsky and Sigma models are computed as the following Equations [START_REF] Gorelli | Liquid like behavior of supercritical fluids[END_REF][START_REF] Gorelli | Dynamics and Thermodynamics beyond the critical point[END_REF].

D𝑆𝑚𝑎𝑔 = S = √ 2 S 𝑖 𝑗 S 𝑖 𝑗 , S 𝑖 𝑗 = 0.5( 𝜕 ū𝑖 𝜕𝑥 𝑗 + 𝜕 ū 𝑗 𝜕𝑥 𝑖 ) (11)
D𝑆𝑖𝑔𝑚𝑎 = 𝜎 3 (𝜎 1 -𝜎 2 ) (𝜎 2 -𝜎 3 ) 𝜎 2 1 (12)
The differential operator of the Smagorinsky model is based on the resolved characteristic rate of strain ( S 𝑖 𝑗 ). However, the Sigma model employs the singular values of the resolved velocity gradient tensor so that the singular values 𝜎 𝑖 (𝜎 1 ≥ 𝜎 2 ≥ 𝜎 2 ) are the square roots of the eigenvalues of the matrix S 𝑖 𝑗 * S 𝑗𝑖 [START_REF] Sagaut | Large-Eddy Simulation for Incompressible Flows, 3rd edn. Scientific Computation[END_REF].

As a final comment on the turbulence modelling, it is worth to note that turbulent viscosity (𝜇 𝑆𝐺𝑆 ) is known to be much greater than laminar one (𝜇) in single-phase turbulent flows. However, this is not always the case in turbulent two-phase flows, especially at the liquid-gas interface where the laminar viscosity experiences a high gradient. Another important point is that the rate of strain S 𝑖 𝑗 in D𝑚 equations may vary significantly in the interface region due to a large phase transition. This is also may be the case due to non-linear property (𝐶 𝑝, 𝐷 𝑇 , ...) behaviors at supercritical state in the Widom line region. The latter two effects are somewhat included in the proposed RFM model through the tabulation of the VLE properties and the impulse effect of the phase change on the momentum. However, a more thorough fundamental study of the interaction between turbulence and EoS should be conducted in future work [START_REF] Poormahmood | A methodology for modeling the interaction between turbulence and non-linearity of the equation of state[END_REF].

Figure 12 shows instantaneous contour plot of temperature, and density for case T480 at t=2.5 ms using various turbulent models denoted as RANS, LES-Smagorinsky, and LES-Sigma models, respectively. Noteworthy, it does not make sens to compare ensemble averaged RANS results and relatively to the LES results. This is due to the higher turbulent viscosity and kinetic energy (TKE) generated in the RANS simulation (see also Figure 13(a,c)).

The Sigma-LES model gives an initial spray cone angle that better matches the experimental image shown in Figure 8(a) than the LES-Smagorinsky model. Also, the liquid core obtained with Sigma-LES model is shorter than the one obtained by LES-Smagorinsky model. This demonstrates a higher evaporation rate obtained using the LES-Sigma model, and thus the significant effect of turbulence modeling on such two-phase numerical results. However, more simulations should be performed in future work to assert the superiority of LES-Sigma in two-phase flows. To have a better insight, Figure 13(a-c as also reported by [START_REF] Misdariis | Numerical methods and turbulence modeling for LES of piston engines: impact on flow motion and combustion[END_REF]. Hence, this lower turbulent viscosity level has led to less dissipation of the turbulent energy, TKE, as shown in Figure 13(c). Besides, some interesting points can be observed in Figure 14 in which is plotted dimensionless radial profiles of the density, density gradient, gas volume fraction and the turbulent conductivity, at 𝑋/𝐷 = 10.

First, it is important to note in this figure that the extrema of the turbulent conductivity and density gradient almost coincide with the locus of the liquid-gas interface at 𝛼 𝐺𝑎𝑠 = 0.5. This demonstrates the high coupling of turbulence with flow characteristics such as density gradient.

Indeed, it is well known that the density gradient in a two-phase flow plays a key role in turbulent two-phase flows and particularly on liquid atomization.

Generally, vorticity analysis can be used as an appropriate tool for the flow mixing analysis [START_REF] Chassaing | Variable Density Fluid Turbulence[END_REF][START_REF] Rogers | The three-dimensional evolution of a plane mixing layer: The Kelvin-Helmholtz rollup[END_REF].

the following Equation 13gives the total derivative of vorticity 𝜔 for compressible flows [START_REF] Chassaing | Variable Density Fluid Turbulence[END_REF].

𝐷𝜔 𝐷𝑡 = -𝜔(∇.𝑢) + (𝜔.∇)𝑢 + ∇𝜌 * ∇𝑃 𝜌 2 -𝜈∇ 2 𝜔 (13)

Quantitative validation

To quantitatively validate the tabulated RFM-LES results, the numerical results will be compared in this section with the recent experimental data from Baab et al. [START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF]. They performed quantitative sound speed measurements in jet mixing zones for different configurations with various mixtures at different conditions. The n-hexane case, which corresponds to "Case 1" in Baab et al. [START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF] is chosen for the comparison to our numerical results. The applied numerical setup is identical to the one used in the previous section. The experimental conditions, listed in Table 3, are very similar to the T600 case in Section 4, as shown in Table 2. One point is worth mentioning is that the nozzle exit temperature 𝑇 𝑒 = 627𝐾 and velocity 𝑢 𝑒 = 90𝑚/𝑠 were estimated in that experimental study based on the assumption of an isentropic flow inside the injector [START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF].

In this study, another strategy has been adopted, where inflow turbulence fluctuations have been generated and superimposed on the inflow velocity profile using Fourier method, based on the proposals by [START_REF] Davidson | Hybrid LES-RANS using synthesized turbulent fluctuations for forcing in the interface region[END_REF]. This method has been used in order to generate correlated turbulent fluctuations over the mean inlet profile at the nozzle exit. Using synthetic turbulence at the computational domain inlet has shown significant effect on the spray development as demonstrated in [START_REF] Desantes | LES Eulerian diffuse-interface modeling of fuel dense sprays near-and far-field[END_REF][START_REF] Desantes | Modelling and valida-tion of near-field Diesel spray CFD simulations based on the -𝑌 model[END_REF],

when the in nozzle flow is not included in the simulation. As pointed out in [START_REF] Desantes | LES Eulerian diffuse-interface modeling of fuel dense sprays near-and far-field[END_REF][START_REF] Desantes | Modelling and valida-tion of near-field Diesel spray CFD simulations based on the -𝑌 model[END_REF], this turbulent intensity ranges between 3% and 5% depending on the turbulence model. Hence, the Besides, the LES-Sigma model has been used for the turbulence modeling of this test case due to its performance, as shown in the previous section.

Conclusion

In this paper, a fully compressible tabulated multicomponent real-fluid (RFM) model has been developed in order to explore the interaction between phase separation and turbulent fluid dynamics in multi-species supercritical flow jets. More specifically, the converge CFD solver has been coupled to the IFPEN-Carnot thermodynamic library using a generalized tabulation method to gain computational efficiency and robustness. This modeling approach has been implemented in the CONVERGE solver for the simulation of the turbulent mixing of n-hexane injected in a chamber filled with nitrogen, considering phase separation processes (condensation) at high pressures conditions with the fluids initially at supercritical state.

First, n-hexane has been injected at three inflow temperatures in the same conditions as the experimental and numerical study of Traxinger et al. [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]. Besides, a more quantitative study has been performed using the experimental study of Baab et al. [START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF], which has the same setup configuration as [START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF]. The obtained numerical results have been shown and proved to be in good agreement with the experiments [START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF][START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF]. In addition, the RFM model was shown to be capable of simulating subcritical, transcritical and supercritical jets exhibiting phase separation and condensation processes. As a consequence, if the jet is at a subcritical state, with regard to the mixture critical point, it experiences the classical droplets formation process governed by the liquid atomization. In contrast, if n-hexane is injected at supercritical state in a colder nitrogen have demonstrated the importance of a proper turbulence modeling, and the suitability of the LES-Sigma model, in addition to accurate thermodynamic and transport properties, for these highly complex jet flows. Finally, compared to our previous studies [START_REF] Yi | A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change[END_REF][START_REF] Yang | Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach[END_REF], the computational efficiency, accuracy, and robustness of this proposed tabulated RFM model as a remedy to the direct evaluation of costly phase equilibrium solver have been confirmed.

Fig. 1 .

 1 Fig. 1. Illustration of the phase transition in subcritical states (i.e., evaporation and condensation processes) when the coexistence line is crossed, and gradual phase change processes in the transcritical regime (i.e., pseudo-boiling and pseudo-condensation) when crossing the Widom line. Pressure-Temperature diagram: (a) Density contour mapped by coexistence and Widom lines, and (b) Heat capacity contour for 𝐶 6 𝐻 14 . Nonlinear behavior is present along the Widom line but also along the liquid-vapor coexistence line.

Fig. 2 .

 2 Fig. 2. Comparison of PR-EoS and ideal gas EoS with reference data obtained from NIST [51] for n-hexane at 𝑃 = 5𝑀 𝑃𝑎 showing a) Density, and b) Isobaric heat capacity profiles. The results, using PR-EoS, Peng Robinson equation of state, are in agood agreement with NIST data[START_REF] Linstrom | The NIST Chemistry WebBook: A Chemical Data Resource on the Internet[END_REF], however the fluid properties specially for isobaric heat capacity (outside the range) significantly differ from an ideal gas at this high pressure 𝑃 = 5𝑀 𝑃𝑎.

1 𝑑

 1 (𝑋,𝑋 𝑖 ) , 𝑋 denotes an interpolated (arbitrary) point, 𝑋 𝑖 is an interpolating (known) point, 𝑑 is the given distance from the known point 𝑋 𝑖 to the unknown point 𝑋, 𝑁 is the total number of known points used in each local interpolation. 𝑁 = 8 for uniform 3D table.

Fig. 3 .Fig. 4 .

 34 Fig.3. Vapor-Liquid equilibrium of 𝐶 6 𝐻 14 -𝑁 2 calculated in this study using PR-EoS compared to the experimental data[START_REF] Eliosa-Jimenez | High-pressure vapor-liquid equilibria in the nitrogen + n-hexane system[END_REF]

  Figure5(a,b) displays the AMT temperatures (dashed lines) for two different chamber pressures at 𝑃 = 1𝑀 𝑃𝑎 and 𝑃 = 5𝑀 𝑃𝑎 in a T-𝑥 𝑁 2 phase diagram (triangle) obtained using PR-EoS. As shown in Figure5(a), the AMT, dash lines, are plotted with the same inflow temperature, 𝑇 𝑖𝑛 𝑗 = 554.8𝐾 and chamber temperature, 𝑇 𝑐ℎ = 293𝐾. For the low pressure case, 𝑃 = 1𝑀 𝑃𝑎, the AMT does not cross the corresponding two-phase boundary, and hence the jet is supposed to stay in the single-phase mixing regime. However, for the higher pressure case, 𝑃 = 5𝑀 𝑃𝑎, the AMT crosses the corresponding two-phase boundary very close to the 𝑁 2 mole fraction of one and it is therefore expected that the jet enters the two-phase region near the periphery of the jet as shown experimentally in Traxinger et al.[START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]. In addition, it can be seen in Figure5(a) that the slope (dT/dx) of the phase diagram (near 𝑥 𝑁 2 = 1) can control the phase separation process (condensation) and the formation process of spray droplets for certain temperature and pressure ranges of the supercritical injection regime. Besides, Figure5(b) shows and confirms the same behavior for a constant pressure, 𝑃 = 5𝑀 𝑃𝑎 but at two different n-hexane inflow temperatures, 𝑇 𝑖𝑛 𝑗 = 554.8𝐾, and 𝑇 𝑖𝑛 𝑗 = 595𝐾. Indeed, this figure

Fig. 5 .

 5 Fig. 5. Condensing and non-condensing supercritical regime (a) at different pressures, (b) at different temperatures. AMT denotes adiabatic mixing temperature, plotted by dash lines, and by crossing the phase boundary(dew points line), the droplet formation process, and the condensation begins and develops at the periphery of the jet. If the AMT doesn't cross the two-phase dome, it will remain at supercritical regime without any droplet formation process, non-condensing supercritical regime (single phase mixing regime).

Fig. 6 .

 6 Fig. 6. Schematic of the injector setup as (a) Geometry setup, and (b) Computational domain grid discretization. Fixed embedding has been used to refine the grid at specific locations in the domain. (Local grid size = (Base grid size) / 2 𝑒𝑚𝑏𝑒𝑑 𝑠𝑐𝑎𝑙𝑒 ).The base grid size was set to 30𝜇𝑚 so that the minimum grid size near the nozzle exit has been specified to be 7.5𝜇𝑚 using an 𝑒𝑚𝑏𝑒𝑑 𝑠𝑐𝑎𝑙𝑒 = 2.

Fig. 7 .

 7 Fig. 7. Temperature 𝑁 2 -composition phase diagram for binary mixture of 𝐶 6 𝐻 14 -𝑁 2 at P= 5 MPa, calculated by PR-EoS along with the adiabatic mixing temperature (AMT). (a) includes the comparison with [21] along with the three scatter plots of the numerical LES mixing temperature results. The low scattering of the LES mixing temperatures indicate almost adiabatic injection flows. The graphical insert shows that the LES mixing temperature scatterplots for the T560 case do intersect the dew line and enter the two-phase region. (b) includes the theoretical AMT for the three cases and show the same behavior as the RFM model for case T560. The two-phase region background is colorized by the gas volume fraction values.

Fig. 8 .

 8 Fig. 8. Comparison of experimental (left(1)) and numerical snapshots (right(2)) for the three test cases (T480, T560, and T600). The ELS image superimposed onto the corresponding shadowgraph are used to visualize both the flow structure as well as the thermodynamic state (adapted from[START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]) in quasi-steady state condition. 𝛼 𝐺𝑎𝑠 is the gas volume fraction

Fig. 9 .

 9 Fig. 9. Temporal evolution of (temperature +gas volume fraction(𝛼 𝐺𝑎𝑠 )) contours using Sigma turbulence model for case T=480.

Figure 11 shows

 11 Figure 11 shows the contour plots of a) the temperature, b) the 𝐶 6 𝐻 14 mass fraction in the mixture, c) the phase indicator PHI= 1, 2 which designates the gas state and the two-phase state, respectively, and finally d) the 𝐶 6 𝐻 14 mass fraction in the liquid phase defined as follows: The mass fraction of each component (for a system with (𝑘) components), in any phase ( 𝑝), (𝑌 𝑘, 𝑝 ) is defined as the ratio of component (𝑘) density in the phase ( 𝑝) considered with respect to the phase density (𝜌 𝑝 ), as 𝑌 𝑘, 𝑝 = 𝛼 𝑝 𝜌 𝑘, 𝑝 𝑘 𝛼 𝑝 𝜌 𝑘, 𝑝 .As the phase indicator shows in Figure11(c), the periphery of the jet goes to two phase regions. This can also be seen in the mass fraction contour of 𝐶 6 𝐻 14 , in Figure11(b). In this figure, a small mass fraction of 𝐶 6 𝐻 14 (black color) is present near the boundary of the jet at near room temperature (light gray color in the temperature contour in Figure11(a)). More importantly,

Figure 11 (

 11 Figure 11(d) reveals the mass fraction of 𝐶 6 𝐻 14 in the liquid phase formed by condensation, mostly near the jet periphery. It worth to mention that as 𝑘 𝑌 𝑘,𝑙 = 1, the mass fraction of 𝑁 2 in the liquid phase formed by condensation is revealed to be very small in Figure 11(d).

Fig. 12 .

 12 Fig. 12. Impacts of turbulence modellings of the jet structure for various properties for two phase flows using (𝑘 -𝜖) RANS, LES-Smagorinsky, and LES-Sigma models, respectively, from top to down, at t=2.5 ms.

  ) display a qualitative comparison of some turbulent properties profiles at 𝑋/𝐷 = 10 (see the arrow in Figure 12(b)), as a) Turbulent viscosity (𝜇 𝑇 ), b) Turbulent conductivity (𝜆 𝑇 ),and c) Turbulent kinetic energy (TKE) using the different turbulence models, respectively at t=2.5 ms. Figure13(a,b) demonstrates the huge values obtained with the RANS model relatively to the LES models. In addition, lower turbulent viscosity and conductivity levels are obtained using the Sigma model compared to the Smagorinsky model in the liquid-gas interface region. This lower turbulent viscosity is expected since Sigma model has been developed to avoid overestimating the SGS turbulence in shear layers,

Fig. 13 .

 13 Fig. 13. Impacts of turbulence modellings on different turbulent properties using different (𝑘 -𝜖) RANS, LES-Smagorinsky, and LES-Sigma models, respectively, from top to down, at t=2.5 ms at 𝑋/𝐷 = 10. Due to the much higher values from the (𝑘 -𝜖) model for the different turbulent properties, they have been divided by 100, 100, and 20 for turbulent viscosity, TKE, and turbulent conductivity, respectively, to be in the same range as LES data.

Fig. 14 .

 14 Fig. 14. Impacts of density gradient on the turbulent properties, at t=2.5 ms at 𝑋/𝐷 = 10, using LES-Sigma model.

Fig. 15 .

 15 Fig. 15. (a) Density gradient, (b) Pressure gradient and (c) Vorticity contours at the interface at t=2.5 ms. The misalignment of spatial pressure and density gradients is at the origin of the baroclinic torque source term.

Fig. 16 .

 16 Fig. 16. (a) Mean velocity field ( Ū), (b) The resolved velocity (𝑈) at 3 ms, and (c)The velocity profile plotted along the jet axis where the resolved velocity is at 3 ms and the mean velocity is computed using the LES results in the time interval between 2 and 3 ms after the start of injection. For this case, a fluctuating boundary condition has been superimposed on the inflow velocity profile using Fourier method with 5% intensity.

Figure 16

 16 demonstrates (a) The mean velocity contours computed using the LES results in the time interval between 2 and 3 ms after the start of injection, (b) The resolved velocity, in addition to (c) The velocity profile along the jet axis obtained with an inlet fluctuating boundary condition with 5% intensity. Unfortunately, no experimental velocity and turbulence data are available in this case. Therefore, these numerical results can be used as a reference for future work.

Fig. 17 .

 17 Fig.17. Temperature-𝑁 2 mole fraction diagram for binary mixture of 𝑁 2 -𝐶 6 𝐻 14 , colorized with the mixture sound speed, using Wood formula for two phase region[START_REF] Wood | A Textbook of Sound. Third Edition[END_REF] at P= 5 MPa, using PR-EoS, along with the phase boundary (in black). The scatter plots (white points) show the locus of the mixing temperature obtained by the RFM-LES model for the case with 𝑇 𝑒 = 627𝐾 simulation.

Figure 17

 17 Figure17shows the Temperature-𝑁 2 mole fraction diagram for the binary mixture of 𝑁 2 -𝐶 6 𝐻 14 , colorized and mapped with the mixture sound speed, computed using the Wood formula[START_REF] Wood | A Textbook of Sound. Third Edition[END_REF] for the two-phase region at 𝑃 = 5𝑀 𝑃𝑎, calculated by PR-EoS, along with the scatterplots (white points) locus of the mixing temperature obtained by the RFM-LES model. In this figure, we can observe the sound speed variations along the mixing temperature (white dots). This demonstrates that the flow in the current supercritical condition experiences relatively small sound speed variations. This could be seen more precisely in Figure18(c) which depicts the instantaneous fully developed 𝑁 2 -𝐶 6 𝐻 14 LES results in terms of sound speed contour after t=3 ms. Indeed, Figure18presents the instantaneous mixture sound speed at axial center-line and at radial direction at 𝑥/𝐷 𝑖 = 35, extracted from the RFM-LES results. An intensely turbulent mixing process of n-hexane with the supercritical nitrogen ambient can be observed in Figure18(a,b) in terms of the mixture temperature and the hexane mass fraction. The supercritical fuel jet with higher density (𝜌 ≈ 115𝑘𝑔/𝑚 3 ) and temperature (𝑇 = 627𝐾) enters to the chamber filled with 𝑁 2 with lower density (𝜌 ≈ 58𝑘𝑔/𝑚 3 ) and temperature (𝑇 = 296𝐾), and then n-hexane jet gradually mixes with 𝑁 2 inside the chamber. Moreover, as Figure18(c) shows, the sound speed varies from 𝐶 𝑠,𝐶6𝐻14 (𝑃 𝑐ℎ , 𝑇 𝑒 ) = 197𝑚/𝑠 at the exit of nozzle and 𝐶 𝑠, 𝑁 2 (𝑃 𝑐ℎ , 𝑇 𝑐ℎ ) = 360𝑚/𝑠. Finally, quantitative comparisons of the mixture sound speed with the experimental data from Baab et al.[START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF] are shown in Figure19along the axial and radial axis shown in 18(c). Ten numerical profiles shown as thin gray lines denotes the RFM-LES results in the time range of t=2-3 ms. Furthermore, Figure20(a) shows the instantaneous contours of the n-hexane mole fraction at Z=0 cut section at 𝑡 = 3𝑚𝑠. Besides, Figure20demonstrates the instantaneous contours of the n-hexane mole fraction and the validations of LES n-hexane mole fraction with the analytical self-similar solutions calculated by Gerber et al.[START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF] using REFPROP. Specifically, some instantaneous LES results on the jet axis, as shown in Figure20(a), are shown (colorized with gray thin lines) in the range of t=2-3 ms. Also, LES simulation results at t=3 ms are highlighted. Overall, a good agreement are found in the sound speed as well as hexane mole fraction predicted results, compared to the reference data, which demonstrate that the RFM-LES solver is able to predict the mixing process of 𝐶 6 𝐻 14 and 𝑁 2 accurately.

Fig. 18 .

 18 Fig. 18. The top LES results at 𝑡 = 3𝑚𝑠 are: (a) Instantaneous contours of the mixture temperature, (b) n-hexane mass fraction at Z=0 cut section. The bottom LES results (c) at 𝑡 = 3𝑚𝑠 are the mixture sound speed at 𝑧 = 0 (left), and at 𝑥/𝐷 𝑖 = 35 cut sections (right). The sound speed in the chamber, filled initially with the 𝑁 2 at 𝑇 𝑐ℎ = 296𝐾, is 𝐶 𝑠,𝑐ℎ ≈ 360𝑚/𝑠 (orange color). At the nozzle exit where 𝑇 𝑒 = 627𝐾, the sound speed is 𝐶 𝑠,𝑒𝑥𝑖𝑡 ≈ 197𝑚/𝑠 (blue color).

Fig. 19 .

 19 Fig. 19. Validations of mixture sound speed with experimental data of Baab et al. [62], (a) Along the jet axis, and (b) At radial direction at 𝑥/𝐷 𝑖 = 35 determined and plotted at Figure 18. LES results colorized with gray thin line denotes the LES-sigma resultsfor some instantaneous times in the range of t=2-3 ms. LES simulation @ t=3 ms shows the variations of the mixture sound speed at t=3 ms, as shown in Figure18.

Fig. 20 .

 20 Fig.[START_REF] Roy | Disintegrating supercritical jets in a subcritical environment[END_REF]. Instantaneous contours of (a) n-hexane mole fraction, at Z=0 cut-section at 𝑡 = 3𝑚𝑠, and (b) Validations of LES n-hexane mole fraction, along the jet axis, with the analytical self-similar solutions calculated by Gerber et al.[START_REF] Gerber | Fluid injection with supercritical reservoir conditions: overview on morphology and mixing[END_REF]. They hired PR EoS, and REFPROP data to analytically calculate fuel mole fraction at different distances from the nozzle exit. LES results colorized with gray thin line denotes the LES-sigma results for some instantaneous times in the range of t=2-3 ms. LES simulation @ t=3 ms, with blue color shows the variations of the n-hexane mole fraction, shown in contour plot in (a).

Table 1 .

 1 Parameters for PR equation of state (index c stands for critical value)

	𝛿 1	𝛿 2	(𝑎 𝑐 ) 𝑖	𝑚 𝑖	𝑏 𝑖
	1 +	√	2 1 -	√	2 0.45727	𝑅 2 𝑇 2 𝑐 𝑃 𝑐	0.37464 + 1.54226𝜔 -0.26992𝜔 2 0.0778 𝑅𝑇 𝑐 𝑃

𝑐

critical point value for a pure component when it appears as in 𝑇 𝑐 or 𝑃 𝑐 . Also, it is worth noting that local mixture critical point values are denoted by a (𝑐, 𝑚𝑖𝑥) index.

Table 2 ,

 2 where 𝑇 𝑡 ,𝐶 6 𝐻 14 is the temperature in the injector reservoir. The temperature 𝑇 𝐶 6 𝐻 14 as well as the mean velocity at the nozzle exit 𝑢 𝐶 6 𝐻 14 were calculated based on the assumption of an isentropic nozzle flow as explained by Traxinger et al.[START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF].

𝐶 6 𝐻 14 ) injected into supercritical nitrogen (𝑁 2 ). More precisely, following the investigations of Traxinger et al.

[START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF]

, n-hexane is injected at three temperatures (480𝐾, 560𝐾, 600𝐾) into a chamber initially filled with 𝐺 𝑁 2 at an ambient pressure of 𝑃 𝑐ℎ = 5𝑀 𝑃𝑎. The critical pressure and temperature of pure 𝐶 6 𝐻 14 and 𝑁 2 are (𝑃 𝑐 ≈ 3𝑀 𝑃𝑎, 𝑇 𝑐 ≈ 507𝐾) and (𝑃 𝑐 ≈ 3.4𝑀 𝑃𝑎, 𝑇 𝑐 ≈ 126.2𝐾), respectively. Hence, the ambient pressure, 5𝑀 𝑃𝑎 correspond to a reduced pressure 𝑃 𝑟 ,𝐶 6 𝐻 14 ≈ 1.67 for pure n-hexane, and 𝑃 𝑟 , 𝑁 2 ≈ 1.47 for pure 𝑁 2 . However, due to the mixing of two components, the mixture critical point will vary locally as function of the mixture composition. Therefore, with respect to the mixture critical temperature of 𝐶 6 𝐻 14 -𝑁 2 at 𝑃 𝑐ℎ = 5𝑀 𝑃𝑎, n-hexane is injected in subcritical state when it is at 480𝐾 and it is injected in a supercritical state when it is heated to 560𝐾 or 600𝐾. The three simulation cases are summarized in

Table 2 .

 2 Operating conditions used for the current numerical study similar to the experimental study of[START_REF] Traxinger | Experimental and numerical investigation of phase separation due to multicomponent mixing at high-pressure conditions[END_REF].𝐶𝑎𝑠𝑒 𝑃 𝑐ℎ [𝑀 𝑃𝑎] 𝑇 𝑐ℎ [𝐾] 𝑇 𝑡 ,𝐶 6 𝐻 14 [𝐾] 𝑇 𝐶 6 𝐻 14 [𝐾] 𝑢 𝐶 6 𝐻 14 [𝑚/𝑠] Injection state

	𝑇480	5	293	480	479.3	51	subcritical
	𝑇560	5	293	560	554.8	72.1	supercritical
							(condensing)
	𝑇600	5	293	600	595	90.3	supercritical

Table 3 .

 3 Operating conditions used for the current numerical study taken from Baab et al[START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF] experimental study. The nozzle exit temperature 𝑇 𝑒 was estimated based on the assumption of an isentropic flow inside the injector calculated by[START_REF] Baab | A quantitative speed of sound database for multicomponent jet mixing at high pressure[END_REF].Injection fuel Chamber gas 𝑃 𝑐ℎ [𝑀 𝑃𝑎] 𝑇 𝑐ℎ [𝐾] 𝑇 𝑡 ,𝐶 6 𝐻 14 [𝐾] 𝑇 𝑒,𝐶 6 𝐻 14[𝐾]

	𝐶 6 𝐻 14	𝑁 2	5	296	630	627

Acknowledgements

This project has received funding from the European Union Horizon 2020 Research and Innovation programme. Grant Agreement no. 861002 for the EDEM project. The authors would like to acknowledge Dr. Angela Di Lella for her supports, and for providing the thermodynamic library, IFPEN-CARNOT. The authors also would like to thanks Dr. Christoph Traxinger from the Institute for Thermodynamics, Bundeswehr University Munich, Germany, for the explanations about the sound speed measurements.