Application of LES/PDF and RANS/PDF approaches for simulation of spray combustion

This paper is addressing of a coupling Large-eddy simulation (LES) and RANS turbulence models with mixture fraction/probability density function as a combustion model. The two models have been implemented to simulate ethanol-air spray combustion. The gas phase is described with the Eulerian approach while the liquid phase is designed using a Lagrangian framework. The LES/PDF approach is obtained statistically. The sub-grid scale energy equation is used with the LES/PDF approach. The numerical results are validated with experimental data. Both LES/PDF and RANS/PDF approaches are compared with the experimental data. The LES/PDF approach shows a good agreement in predicting the average gas temperature compared with RANS/PDF approach. The LES/PDF shows a better prediction of both turbulence intensity profiles and the vortices which are generated in the turbulent flow in comparison with the RANS/PDF approach.


Introduction
Liquid fuels are widely used in industrial combustion systems like the internal combustion engine, liquid-fueled rockets, and gas turbine. It plays a significant role in our energy supply. Typically, liquid fuels are provided as a turbulent spray into the combustion chamber. The efficiency of combustion, stability, and pollutant formation depend heavily on the turbulent spray characteristics [1][2][3][4][5][6]. Therefore, attention must be paid to understand the mechanism of turbulent spray combustion and developing methods that would give a clearer picture of the behavior of such phenomena. Modelling and the numerical simulation are considered a challenging because it involves complex processing such as turbulence, phase change, heat transfer, and chemical reactions. In recent years, largeeddy simulation of liquid spray combustion has received a lot of attention. LES can provide more accurate statistical results and detailed dynamic flow and flame structures compared to Reynolds-averaged modelling (RANS). In the future, it is predicted that the LES turbulence model will become a widely used computational fluid dynamics method that will replace RANS modelling [7,8]. There are many pieces of research in LES of the turbulent two-phases flow and the liquid spray combustion were reviewed by the current authors. Jones et al. [9] CFD simulation of swirl stabilized flames fueled by liquid kerosene were provided by large-eddy simulation (LES). There were two flames studied for which experimental data was available. Flame (A) was a stable flame, while flame (B) was unstable due to its lower air fuels ratio. The results of a combined LES/ PDF technique were applied to high swirl reacting spray flows and compared to empirical data. The formulation took into account sub-grid turbulence-spray-chemistry interactions, as well as the coupling between the gas and liquid phase. Dodoulas et al. [10] Three pre-mixed piloted turbulence CH4/air flames at different Reynolds numbers were subjected to Large-eddy simulation/Filtered Density Function model. In contrast to experimental data using the same closures used in non-premixed combustion with no changes to the combustion regime, the results showed strong agreement. The effect of heat release on the flow domain was investigated and correctly captured. The results showed that pre-mixed combustion could be modeled using LES/PDF methods, at least under work conditions, and that approach could reliably capture spray conditions where combustion was not significant and large pockets of extinction emerged. Heye et al. [11] The experimental data of a pilot-established ethanol spray flame was analysed using the LES/ODF method. A Lagrangian Monti-Carlo model was used to solve the PDF equation. The droplet evaporation occurred way from the flame front, separation two processes. Owing to the high flow rate of the droplet-laden air, the front flame was unable to propagate. As compared to the others, there were significant differences in the conditions of the droplet inflow. Ukai et al. [12] A combined Large-Eddy Simulation (LES)/Conditional Moment Closure approach applied to model spray combustion. Additional source terms in the transport equation of CMC originating from fuel evaporation were modelled, and their effect on flame structure and global quantities including tentatively averaged temperature profile had been investigated. The laboratory spray jet flame prediction was generally good. The evaporation source term caused a flux in mixture fraction space, which shifted peak temperatures to higher mixture fraction regions. Jones et al. [13]In the GENERIC combustor, LES was used to simulate kerosene/air. The liquid phase was defined using the Eulerian method, which was fully coupled with the Lagrangian method. Reacting and non-reacting environments were simulated in two separate test cases. In terms of mean statistics and flame structure estimation, the flow field simulations were satisfactory. Ukai et al. [14] To model spray combustion, a hybrid Large-Eddy Simulation (LES)/Conditional Moment Closure method was used. The CMC method was coupled with a chemistry table that allowed for spatial and temporal variations of the tabulated chemical composition and provided a fair reliable estimation of the chemical source term of the LES/filtered reaction development variable. The numerical results were validated with experimental of two dilute acetone spray with preevaporation. An axial spray velocities and temperature predictions were noticeably enhanced, while RMS velocity showed reasonable agreement. Dodoulas et al. [15] For simulating a non-premixed flame with high extinction, a Large Eddy Simulation/PDF method was used. This method is effective in predicting the local flame extinction at different flame locations, and the major species predictions were found to be in good agreement with the practical data. The flame structure was analyzed by using chemical explosive modes. Due to the turbulence combustion model nature, the statistical data on the sub-grid flame structure can be obtainable. Sub-grid explosive models, for example. According to the results, sub-grid structures were only essential near the inlet nozzle in the current flame, and downstream extinction was controlled by Large-Scale interaction. Wang et al. [16] On the acetone flame structure, three combustion models (finite-rate chemistry model, flame-prolongation of intrinsic low dimensional manifold model, and flamelet-progress variable model) were applied to achieve large-eddy simulation (LES) of a turbulence spray structure. In comparison to experimental results, the use of a finite-rate chemistry model combined with a reduced chemical mechanism results in improved prediction of spray flame structure and heat release among the three models. Sacomano Filho et al. [17] In the form of a C2H5OH/air turbulence flame, a turbulence flame interaction and evaporative cooling effect were investigated in LES turbulence. The artificially thickened flame method (ATF) was combined with the mixture adaptive thickening extension technique to account for enthalpy variations. The Eulerian/Lagrangian system was used to track the droplets. The flamelet-generated manifold method has been tabulated by chemistry (FMG). Variations in enthalpy were integrated into the outcomes FGM database in a universal manner that was not limited to heat losses due to evaporative cooling effects. Aside from being novel, the proposed modelling strategy's comprehensiveness allowed for a significant contribution to the understanding of turbulent spray combustion modeling's most relevant phenomena. [17] As a result of the introduction of renewable aviation fuels, alternative methods and models are needed that can predict the combustor performance based on the fuel composition. In the sense of Eulerian/Lagrangian LES, a multi-component vaporization model was coupled with a direct. The burner which used in this work exhibited some of the existing features of aero-engine combustors like the reaction zone position and measured spray combustion were fine reproduced by LES turbulence model. The computational outcomes showed that the evaporation and mixing were the rate-controlling stages in the flame zone. Chemistry can be presumed to be significantly fast in this zone. However, other regions existed where the finite rate chemistry influence available. The approach of the finite rate chemistry demonstrated great potential in terms of pollution formation. Furthermore, the formation of benzene from one specific chemical form in the fuel suggested that to accurately predicted soot emissions, multi-component explanation of the liquid phase and the evaporation process was needed. Paulhiac et al. [18] A spray-swirled burner was subjected to LES/Discrete particle simulation. There are two types of validation: reacting and non-reacting. The velocity field of the two-phases finding was consistent in both cases as compared to experimental data. The LES grid fails to describe the non-premixed flame set near the droplets due to direct interaction between the spray and flame. Instead, the current model referred to pre-mixed combustion, which allowed for a percentage error on both the burning rate and combustor efficiency. It was necessary for using a single droplet combustion model which is appropriated to the LES/DPS framework. [18] Spray reactive flow is used in a variety of technological instruments. A sub-grid scale combustion model is studied. Two different flamelet methods were implemented which were consisting of spray flamelet and multi-regime gas flamelet models. The numerical evaluation was implemented in largeeddy simulations (LES) of a benchmark C2H5OH/air spray flame a partial pre-vaporization. An Eulerian/Lagrangian numerical framework was adopted. Outcomes showed that the spray flamelets developed from counterflow partially-premixed spray flames showed a better agreement with the experimental data. In the present work, Ansys Fluent is applied for simulating two-phase flow in 2D, free shear stress with a certain domain 360mm×460 Ge et al. [19]. Large Eddy Simulation (LES) and Reynolds averaged Navier-Stokes equations (RANS) turbulence model are amplemented to simulate the turbulent spray of C2H5OH/air spray. Both turbulence models are combined with a Probability Density Function (PDF) as a combustion model. Then used for investigating using various parameters in an attempt to find out which of approach is the best in giving results closer to the practical results and closer visualization of the practical reality.

Governing equations
The main physical processes of down-draft evaporative cooling consist of turbulent flow, species mixing, mass, and heat transfer. Differential equations for each species, mass conservation, momentum conservation, and energy conservation must be solved. Also, the transport equations for the turbulence model should be solved. These equations are employed into three sets, continuous phase (Euler phase) equations used for solving the fluid flow and species transport, dispersed phase (liquid phase ) calculate the trajectory of the fuel droplets, and due to strong cooperation between two-phases, the coupled effect must be solved for the complex environment. All equations and constants used based on ANSYS Fluent theory Fluent et al. [20]. The general equations that are using for the solution identified as following:

Euler phase model
FLUENT ANSYS solves gas-phase flow as well as analysis the differential equations of mass conservation, momentum conservation, energy conservation, and species. In the initial stages of the process, the model solves the gas-phase without including the interaction between two phases, when the liquid phase (disperse particle) trajectories are solved, and the source terms are added [20]. FLUENT solves gas-phase with consideration of the source terms in the numerical calculations. The conservation mass equation is where ( ) is the air velocity component in Euler-phase (m/s), defined as an additional mass to gas-phase from liquid-phase (discrete phase) ( / 3 ). The transformation momentum due to the exchange can be expressed: When the species diffusion and the source due to the energy interchange between phases are included, the energy equation is identified in the following part (),where is the volumetric heat source ( / 3 ), the heat transfer is calculating by using: where is the molecular viscosity ( / 2 2 ), is the momentum source ( / 2 2 ), is a gravity ( / 2 ), P is the static pressure (pa), is the internal energy ( / ), is air temperature in ( ) , ℎ is the species enthalpy, Φ is the Rayleigh dissipation function ( / 3 ) , is the thermal conductivity ( / ), is the species (i) diffusion flux( / 2 ). The species conservation equations For predicting the local mass fraction of each phase, i.e., gas-phase of fuel and air, are: where is the local mass fraction of specific phase (i) and , is the diffusion coefficient for a specific phase (i) in the mixture.

Liquid phase model
The discrete droplets in the gas-phase are suggested to be spherical. At the starting of the solutions, the Lagrangian approach is used to calculate the trajectory of the particles (discrete phase) by integrating the force balance, while the gas-phase is recalculated and adjusted, it solves the liquid-phase in the following equations: where is the droplets velocity (m/s), ( − ) is the drag force per unit droplet mass ( / 2 ), ( ( − ) ⁄ ) is the gravitational force on the droplet, the last term on the right hand of the equation represents the additional force due to the pressure gradient in the fuel, ⁄ is the evaporation rate of the droplet, is the gravity ( / 2 ), is the drag coefficient, is the diameter of the droplet (m), and 1 , 2 , and 3 are constants obtain by Morsi and Alexander [21]. The heat balance utilizes for computing the heat transfer between the gas-phase and liquid-phase. Radiant heat transfer is ignoring, is identified as: where is the droplet surface area ( 2 ), is the droplet heat capacity ( / ), , ∞ are the temperature of the droplet and gas-phase respectively in (k), ℎ is the convective heat transfer coefficient ( / 2 ), ℎ is the latent heat of evaporating of fuel ( / ).

Coupling phases
An iterative method is implementing to solve two coupled phases(gas phase and discrete phase). When the computation of the particle stream is done, including the calculation of the gain or loss of mass, momentum, and heat by the particle trajectory, then these quantities are added into the gas-phase (continuous) calculation. The mass exchange between two phases is evaluated as the mass differences between all C.V using.
where is the mass exchange, ∆ is the change of the particle mass in each C.V, is the particles mass flow rate, , is the mass flow rate of the particle at the initial state. Then included in the mass conservation equation. In addition, the gas-phase is added as a source of mass in the species equation. The momentum transfer between two phases can be calculated as: The vaporization rate of the droplets is controlled by the concentration of the vapour between the surface of the droplet and gas-phase. , where is the vapour molar flux ( / 2 ), is the coefficient of the mass transfer ( / ), , is the vapour concentration at the surface of the droplet ( / 3 ), ,∞ is the vapour concentration in gas-phase ( / 3 ), is the vapor pressure at saturated state (pa), is the universal gas constant ( ⁄ ), ℎ is the Sherwood number, is Schmidt number, is the local bulk mole fraction of specific phase . Droplet evaporation is submited to law 2 in FLUENT, and the change of the droplet mass can be found: where , is the molecular weight of specific phase ( ), ℎ is the volumetric source term, can be expressed as where ̃ is the droplet average mass in a control volume ( )

Turbulence model
The turbulence model is a mathematical model used to describe the turbulent flow which is commonly used in a realistic and engineering application .Turbulence models are used to represent turbulence by using simple equations . In this work standard k-and large-eddy simulation are used representing the random and irregular flow.

Standard k-model
In the Standard model derivation ,flow assumes fully turbulent and ignores influence molecular viscosity .It is appropriate just for fully turbulent. Where is the Turbulent viscosity modeling as : is the effect of buoyancy = Where : The Dissemination rate is , the production of turbulent kinetic energy is , the Turbulent velocity is ́, the thermal expansion coefficient is , the turbulent Prandtl number is is ,at default = 0.85, the source term defined by used are , ,and the reverse active prandtle are , for , respectively. is acquired by solving the transport equation Where the filter size is ∆ , the sub-grid scale eddy viscosity is .
The mechanism of the reaction is implemented in one step, the ethanoloxygen reaction is described chemically as For the radiative heat transfer, the P1 model is used where the Stefan-Boltzmann constant denotes by = 5.67 * 10 −8 . −1 , the absorption coefficient is = 0.1 −1 and the incident flux is which is processed from the equation of the radative transfer. when → 0 ,̅̅̅ → 0.

Combustion model
In this research, the probability density function (pdf) was employed to evaluae some combustion properties. Obtained from the mixture fraction to provide the effect of the turbulence fluctuation on the quantities of the conserved scalar, two types of probability density function : a clippsed Gaussian and beta function. In the present research, a beta function commonly is used for simplicity and lower cost .
Beta function mathematically identified : explicit functions of (̃, ) mathematically described : Species concentration of each product temperature and enthalpy were obtained by weighting the amount of these quantities with depending (pdf ) on mixture fraction ,̃( ) . ̃, g is calculated using finite difference method for each grid , may be expressed ,where ̌ is the Favreaveraging quantity where a mixture fraction is , Mean mixture fraction of mass is ̃c oefficients of beta pdf are , , the mass fraction for species element I is , the mass fraction for the fuel stream inlet is , , and the mass fraction for oxidizer stream inlet is , .

Numerical method and condition
The computational domain in the present work was a combustor 360× 460 mm shown in Fig. 1 in which a C2H5OH/air coaxial diffusion flame can be generated at a pressure of 2 MPa atm [19]. Combustor geometry is shown in Fig. 2. The optimal grid size of the cells is 83200×167304× 84105 and a quadrilateral refinement shape is implemented. The numerical conditions that were used are summarized in Table 1. As mentioned in the introductory part, the probability density function (PDF) is coupled with the Large Eddy Simulation (LES) and RANS turbulence model which is representing standard k-turbulence model. For temporal term, the SIMPLE algorithm as a numerical procedure with a secondorder implicit scheme is adopted. For momentum, the second-order upwind is used. For boundary conditions, the inlet velocity of the droplet and the injection angle are implemented according to the experiment [22]. Fig. 3. Shows the comparison between numerical simulation is using FLUENT ANSYS and the measurement.

Results
The injection conditions of C2H5OH/air spray combustion are used for validation related to the measurement data taken [19]. The numerical simulation for the spray combustion using RANS and LES turbulence models. The spray flame wings are predicted by the LES/PDF approach especially at 20 mm and 25 mm better than RANS/PDF approach. According to the outcomes of the experiments, the gradients of the temperature at the flame edge are very high. This phenomenon at section 6 mm is closely predicted by the LES/PDF approach, whereas shows a smoother accurately in various point of the experimental data although it predicts the gas-phase temperature at the tip edge of the nozzle is broader than the measurement data. It is important to improve the sub-grid scale combustion model for large-eddy simulation of two phases [16] while RANS/PDF in comparison with LES/PDF is succeeded in predicting the regions at the tip edge of the nozzle and then the accuracy of the numerical results decreases. In section 10 mm, the numerical results predicted by LES/PDF fail in comparison with RANS/PDF approach.    4 demonstrates the gas phase contours of the numerical simulations that were performed by other authors in comparison with the numerical results of the present work. Difference in flame capturing is attributed to many causes such as mesh type, mesh size, numerical method procedure, approaches performance, and anticipating the missing conditions. Fig. 5 shows the static temperature profiles of the present work and other authors with the experimental data at x=6 mm above the nozzle. Fig. 6 shows a comparison of the reactions zone calculated by the experiment [14] as shown in (a), and the numerical simulation achieved using FLUENT ANSYS 19.2 in (b,c).  Fig. 7 shows the contours of gas-phase temperature profiles. The coherent structure appears on the right side with LES/PDF in which shows the eddy size and energy variation through the domain that can be explained to subgrid scale energy equation model the ability in reinforces the prediction of the small eddies in addition to large eddies which predict by the type of mesh utilizing in comparison to RANS/PDF approach.  Fig. 8 shows a rising in a mean mixture fraction that can be expressed due to the evaporation rate. Mixing is relatively higher and the amount of the injected fuel at the nozzle region is higher too in comparison to the oxidizer in this domain. LES/PDF approach shows an additional consumption in the fuel in comparison with RANS/PDF. Fig. 9 shows a gas temperature phase against the mean mixture fraction. There is a low distribution of mean mixture fraction (pdf) confront with the low temperature at the nozzle region and then demonstrate both static temperature and mean mixture fraction rise along with radial distance, especially in the reaction zones when the mixing is done between the ethanol stream and the oxidizer in which the burnt gas created in the upstream and blended with un burnt gas that is resulting in the generation of the high-temperature zone. In comparison, the gas temperature against pdf that is determined by using the LES/PDF approach is slightly higher than predicted by RANS/PDF approach. Figure 9. Comparison of predicted gas-phase temperature against mean mixture profiles along with the radial distance. Fig. 10 shows the turbulence intensity of gas-phase profiles. At the nozzle region, the turbulence intensity records a high percentage which may be expressed to the turbulence resulting from the hydrodynamic effect, the chemical interaction between two phases, liquid velocity, and the rate of evaporation, then it is eliminated with progressing away from the nozzle region. LES/PDF approach with sub-grid scale energy equation shows an additional fluctuation in the turbulence intensity profiles along with the radial distance of the domain which demonstrates the realistic prediction in comparison with the RANS/PDF approach.  Fig. 11 shows the vortices profiles along with the radial distance. In the near and inlet shear stress of the fuel jet flow, the coherent structures are expected to demonstrate and then are concentrated to be large vortices and eventually are weakly in the downstream area that can be predicted by using the LES/PDF approach while RANS/PDF approach fails in demonstration these phenomena. Figure 11. Comparison of predicted vortices profiles along with the radial distance.

Conclusions
The turbulent spray flame of ethanol/air which is studied in this work using two approaches: Large-eddy simulation coupled with a probability density function (LES/PDF) approach and standard k-turbulence model combined with a probability density function as a combustion model (RANS/PDF) approach. Generally, the statistical results presented that are validated using the LES/PDF approach, they show good agreement when compared to the experimental data are better than those acquired by RANS/PDF approach, especially in estimating the turbulence intensity profile and vortices.