Yifan DUAN, Zhixun XIA, Likun MA, Zhenbing LUO, Xu HUANG,Xiong DENG

College of Aerospace Science and Technology, National University of Defense Technology, Changsha 410073, China

KEYWORDS

Abstract In this paper,the Eulerian Stochastic Field (ESF)model in the Transported Probability Density Function(TPDF)class model is combined with the Flamelet Generated Manifolds(FGM)model.This method solves the joint probability density function transport equation by ESF method that considers the interaction mechanism between flame and turbulence with high precision.At the same time,by making use of the advantage of the FGM model,this model is able to incorporate the detailed chemical reaction mechanism (GRI 3.0) with acceptable computational cost. The new model has been implemented in the open source CFD suite-OpenFOAM. Validation of the model has been carried out by simulating the Sandia flame series (three turbulent piloted methane jet flames) issued by the National Laboratory of the United States. The accuracy and advancement of the ESF/FGM turbulent combustion model are verified by comparing the LES results of the new model with the rich experimental data as well as the RANS results. The results demonstrate that the model has a strong ability in capturing combustion phenomena such as extinction and re-ignition in turbulent flame, which is essential in the accurate prediction of the combustion process in real combustion devices, for example, aircraft engines.

1. Introduction

As the most common and main combustion mode closely related to human production and life, turbulent combustion has always been a hot topic and important issue for researchers in many countries. With the rapid advancement in the computer technology, numerical simulation technology is becoming an increasingly important research tool for studying turbulent combustion problems. The turbulent combustion process is an extremely complex physical and chemical process involving turbulent flow, heat transfer, mass transfer, and chemical reactions. The fundamental mechanism of turbulent flow has not been fully understood yet, and it is one of the remaining unsolved issues in classical physics. The turbulent flow has a strong nonlinear coupling relationship with the simultaneous chemical reaction,1–4which makes the numerical simulation of turbulent combustion very difficult.

At present, the Large Eddy Simulation (LES) method has become a commonly used numerical simulation method for studying complex turbulent combustion problems due to the continuous increase of computer resources. In this method,the turbulent flow associated with large-scale eddy is directly numerically simulated. By establishing a corresponding model to simulate the influence of small-scale vortices on large-scale vortex motion, the method achieves the best compromise between computational accuracy and computational complexity compared to Direct Numerical Simulation (DNS) and Reynolds-Averaged Navier-Stokes (RANS) simulation.5The LES method is capable of capturing unsteady processes associated with turbulent combustion, but this method does not fully solve all turbulence and scalar structures.For the simulation of the turbulent combustion process, the LES method needs to establish a corresponding model to describe the turbulent-flame interaction at the sub-grid scale.For turbulent combustion processes involving complex chemical reactions,the LES method requires appropriate models and a large amount of computational resources for accurate numerical simulation. At present, for LES, the mainstream turbulent combustion model can be divided into Transported Probability Density Function (TPDF) method, flamelet-based model,moment closure model and other combined models.6The TPDF model group consists of Eulerian Stochastic Field(ESF)model and Monte Carlo TPDF model.The most prominent advantage of this model is that the chemical reaction source term can be directly closed by the joint Probability Density Function (PDF) without the need to introduce a closure model, which makes the calculation accuracy very high and has the advantage of being unrestricted by the combustion mode.However,for the single-point-single-time PDF,it is difficult to account for the molecular diffusion process, and the computational cost of the simulation is enormous. The flamelet model considers that, under certain conditions, the turbulent combustion field consists of a series of flamelets and a non-reactive turbulent flow field surrounding these flamelets,which realizes the decoupling calculation of turbulent flow and chemical reaction process, has high computational efficiency and can take into account the advantages of detailed chemical reaction mechanism.

In recent years, the Flamelet Generated Manifolds (FGM)proposed by van Oijen and de Goey7has been widely used in numerical simulation of various turbulent flames, and the calculation results that agree well with the experimental data have been obtained.8–10The FGM model establishes a multidimensional flamelet look-up table by calculating a series of one-dimensional laminar flames that consider the detailed chemical reaction mechanism. The traditional FGM model expands the laminar flamelet look-up table into a turbulent flamelet look-up table by considering the interaction between turbulence and flame by a Presumed-Probability Density Function (P-PDF) method. However, P-PDF itself has many limitations. Firstly, the model assumes that the control variables are independent of each other, but this is not the case under many circumstances. Chen et al.11found in their study of partial premixed flames that the Beta PDF (a widely used P-PDF method) distribution of the independence variables is inconsistent with the DNS data, proving that the statistically independence assumption of control variables is questionable.Secondly,with the increase of the number of control variables used in the FGM model or the increase of the reaction mechanism,the size of the look-up table increases exponentially and the memory requirements are huge. In addition, the assumed shape coefficients in the P-PDF method are obtained from empirical data,and the PDF distribution required for different types of flames cannot be universally described by these coefficients with good accuracy. Bray et al.12found in the study of turbulent diffusion flames, that the assumed shape coefficients of the three most widely used models in the P-PDF method have a great influence on the prediction of chemical reaction rate. Finally, the P-PDF method is derived from the empirical data of the gas phase,which limits the further application of the FGM model in multiphase combustion,e.g.spray combustion,where the controlling variables are no longer conserved variables.Ge and Gutheil13found in the study of a turbulent spray with TPDF method that the actual probability density of the mixture fraction, the gas temperature and the enthalpy has a large difference from the distribution obtained by the Beta model.

The coupled ESF/FGM model for turbulent combustion developed in the current study still decouples the chemical reactions and turbulent flow by means of tabulated chemistry method, i.e. FGM model, but directly solves the PDF of the control variables by the ESF model. It maximizes the advantages of the two types of models,accomplishing a class of turbulent combustion model that can achieve high precision at low computational cost, and is able to consider the detailed chemical reaction mechanism, facilitating the accurate simulation of, for example, pollutant emission as well as ignition/extinction processes. The model treats the turbulent flow field in a random way. Through multiple real-time solutions to the PDF transport equation of the control variables under local flow field conditions, the statistical average gets the local control variables under the influence of turbulent fluctuation, and then the other scalar fields of the simulated flame are obtained by retrieving the FGM look-up table. Similar application has been applied in Ref.14.In this paper,we report the implementation of the new model in the open source CFD suite-OpenFOAM and its validation. We select the Sandia flame D-F15as target cases to validate the accuracy and advancement of the new model.

2. Theoretical calculation model

2.1. FGM model

In the FGM model, there is no need to solve the transport equations for all components and energy, and the chemical reactions in turbulent combustion are thought to occur in low-dimensional manifolds,which means that only a few independent variables are required in the entire component space to characterize chemical reaction in turbulent combustion. In the model, the ‘‘mixture fraction”,Z, that characterizes the mixing state of fuel and oxidizer, and the ‘‘progress variable”,C, that characterizes the progress of chemical reaction, are usually selected as independent variables. Of course, depending on the physical model being simulated, variables such as pressure and enthalpy loss can be added as supplementary independent variables.8,10,16,17The FGM turbulent combustion model under the LES context can be expressed as18

whereWis molar mass andYis mass fraction.The normalized progress variableCis defined as

where superscripts ‘‘b” and ‘‘u” represent the burned and unburned states respectively. The variablesZandCare the independent variables of the look-up table. The equations of the unnormalized progress variableYCare directly solved in the simulation because there are additional source terms that need to be modeled in the transport equation of the normalized progress variableC, which increases the complexity of the model.

The influence of turbulence on the local flame structure is considered by the joint PDF of the control variables (ZandC) in the FGM model. The Favre average of the scalar φ in the turbulent flow field(e.g.temperature,mass fraction of each components) can be calculated as

2.2. ESF model

where ζ（0，1 ） represents a Gaussian random variable with a mean of 0 and a variance of 1, and Δtis the time step size.Eq.(10)is a general formula for the joint PDF transport equations of arbitrary control variables, which provides a good basis for the new model to add more control variables.

3. Validation cases and numerical setup

This paper uses a series Sandia flame D-F released by the Sandia National Laboratory, which is rich in measurement data,as validation cases.The fuel of this series of flames is a mixture of methane and air. The fuel is 25% methane diluted in 75%air by volume, and the diameter of the jet inlet is 7.2 mm.The average velocity of the fuel jets of the Sandia D, E and F flames are 49.9, 74.4 and 99.2 m/s respectively. The pilot flame of this series of flames is high-temperature combustion product. The equivalent ratio is set to 0.77 and the diameter is 18.9 mm. The average speed of the pilot flames of Sandia D, E and F are 11.4, 17.1 and 22.8 m/s respectively. The temperature of fuel jet and pilot flame are 290 K and 1880 K respectively. Other flow field parameters can be found in Ref. 15.

Fig.1 Transient and time-averaged simulation for Sandia flame D, E and F.

Fig.2 Instantaneous scatter plots of temperature vs mixture fraction for Sandia flame D, E and F (x/d=7.5).

Fig.3 Central profiles of mean and RMS temperature and main components mass fraction for Sandia flame D.

Fig.4 Central profiles of mean and RMS temperature and main components mass fraction for Sandia flame E.

This series of flames, especially the Sandia flame D, has been widely used as validation cases for many new models.24–28Among them, Mustata et al.21applied the ESF method to simulation of the Sandia flame D, Renzo et al.29applied the Flamelet/Progress Variable (FPV) model to the Sandia flame D, and Jones and Prasad30applied a systematic study of the Sandia flames series D-F using the ESF method combined with the simplified mechanism of the Augmented Reduced Mechanism (ARM). The authors31used the ESF/FGM model studied in this paper to study the Sandia flame D under the RANS context, and achieved good results.

Fig.5 Central profiles of mean and RMS temperature and main components mass fraction for Sandia flame F.

Fig.6 Radial profiles of mean and RMS temperature at six axial locations for Sandia flame D.

Fig.7 Radial profiles of mean and RMS temperature at six axial locations for Sandia flame E.

Fig.8 Radial profiles of mean and RMS temperature at six axial locations for Sandia flame F.

The open source computational fluid dynamics software(OpenFOAM) is used to complete the program development.The velocity inlet boundary condition of the fuel and pilot flames of Sandia flames D-F adopt the measure profiles in the current study.The solution region extends 17 jet diameters(d)in the radial direction and 60 jet diameters in the axial direction.The grid used is refined in the direction of the fuel jet,with a total mesh size of about 3.5 million, and all lateral boundaries are free-slip. This paper uses eight stochastic fields to describe the effect of turbulent flow on the flamelet.The simulation was performed on the Tianhe-1 supercomputer.One simulation of the case requires 132 CPUs to run for about 4 days.

Fig.9 Radial profiles of mean and RMS CH4 mass fraction at six axial locations for Sandia flame D.

Fig.10 Radial profiles of mean and RMS CH4 mass fraction at six axial locations for Sandia flame E.

The flamelet databases (FGM table) of flame were generated by solving the non-premixed counterflow flame with Chem1D software. The boundary conditions for generating the FGM table were given according to the boundary conditions of the Sandia flame series.

4. Simulation results and analysis

Fig.11 Radial profiles of mean and RMS CH4 mass fraction at six axial locations for Sandia flame F.

Fig.12 Radial profiles of mean and RMS O2 mass fraction at six axial locations for Sandia flame D.

In this section,by analyzing the scatter plot of the temperature,the distribution of species mass fraction along the central axis,and the radial profiles LES results for each typical location,and compared with the experimental results32and RANS given in Ref. 31, the characteristics of the LES model of ESF/FGM are validated and analyzed.Since the results of the ESF/FGM model and the original FGM model have been analyzed in the literature,it will not be described in detail in this paper.Based on the average velocity of the jet flow, the results given in this section are extracted from the average of the 10–15th flowthrough cycles after the steady flow of the jet is achieved.Six typical axial positions are selected to analyze the radial profile results, which are the ignition positions near the jet inlet:x/d=1,x/d=2,x/d=3,the position where partial extinguishing occurred is:x/d=7.5,and there is a position of re-ignition combustion phenomenon:x/d=15,in accordance with the typical characteristics of the diffusion flame:x/d=30.

Fig.13 Radial profiles of mean and RMS O2 mass fraction at six axial locations for Sandia flame E.

Fig.14 Radial profiles of mean and RMS O2 mass fraction at six axial locations for Sandia flame F.

Fig.15 Radial profiles of mean and RMS CO2 mass fraction at six axial locations for Sandia flame D.

Fig.16 Radial profiles of mean and RMS CO2 mass fraction at six axial locations for Sandia flame E.

Fig.17 Radial profiles of mean and RMS CO2 mass fraction at six axial locations for Sandia flame F.

Fig.18 Radial profiles of mean and RMS CO mass fraction at six axial locations for Sandia flame D.

As shown in Fig.1,it is an instantaneous result of the temperature (T)field and the average result of 10–15 time periods after the full development of the Sandia series of flame flow fields. It can be clearly seen that the greater the velocity in the three flames is, the thicker the thickness of the simulated flame is and the higher the height is, which is in line with theoretical analysis.As shown in Fig.2,it is the position of the partial flameout phenomenon of the Sandia series flame, and the scatter plot of the experimental value and the simulated value of the temperature and the mixing fraction atx/d=7.5. We can clearly see that from the Sandia flame D to the Sandia flame F,the phenomenon of partial flame flameout continues to increase, the Sandia flame E and Sandia flame F are more obvious,while the Sandia flame D has almost no partial flameout,which is a typical diffused flame characteristic.In general,the simulation results are basically consistent with the experimental results.

Fig.19 Radial profiles of mean and RMS CO mass fraction at six axial locations for Sandia flame E.

Fig.20 Radial profiles of mean and RMS CO mass fraction at six axial locations for Sandia flame F.

Fig.21 Radial profiles of mean and RMS H2O mass fraction at six axial locations for Sandia flame D.

Fig.22 Radial profiles of mean and RMS H2O mass fraction at six axial locations for Sandia flame E.

Fig.23 Radial profiles of mean and RMS H2O mass fraction at six axial locations for Sandia flame F.

Fig.24 Radial profiles of mean and RMS OH mass fraction at six axial locations for Sandia flame D.

As shown in Figs.3–5,the results of the LES simulation of the temperature, CH4, CO, CO2, H2O, and O2mass fractions of the Sandia series flame at the central axis positions agree well with the experimental values (Exp) and are better than that the best 48 stochastic field results obtained by the authors in the previous stage.31It can be seen that, in general, the simulation average results and Root Mean Square(RMS)values of the LES of the Sandia series flames are in good agreement with the experimental results, especially the LES simulation results of the Sandia flame D are in good agreement with the experimental results. Compared with the simulated average results of LES and RANS of Sandia flame D, LES results effectively improved the problems of overestimation of chemical reaction rate, high peak value and ignition delay,especially the prediction of CO concentration. However, it has also been found that the accuracy of the simulation decreases with the increase of the velocity of the flame jet,such as the mass fraction of CH4(fuel) and O2(antioxidant) consumption position and the mass fraction of CO2and CO(combustion products) production position are relatively greater than the experimental results, because the simulated ignition process is slightly delayed compared with the actual one.

Fig.25 Radial profiles of mean and RMS OH mass fraction at six axial locations for Sandia flame E.

As shown in Figs. 6–8, the LES simulation results of the temperature of Sandia series flame at six axial positions were compared with the experimental values and the best 48 stochastic field results of the RANS obtained by the author in the previous stage. We can see that on the whole the simulated average and RMS values of the LES of the Sandia series flames are generally consistent with the experimental results.As the distance from the inlet position increases, the simulation accuracy of RANS becomes lower and lower.LES,by virtue of the advantages of the model,can effectively improve the prediction of the impact of turbulent flow on chemical reaction, making the simulation results of LES higher. Compared with Sandia flame D and E,the simulation accuracy of Sandia flame F is relatively low, especially for the prediction of temperature peak and flame thickness in the downstream of flame.This may be because the jet velocity of Sandia flame F is high,and the peak speed will reach 120 m/s, which makes the influences of the small scales on the flame more significant and the standard Smagorinsky method used in the current study may not be sufficient for the accurate description of this process.It is suggested that more advanced SGS models should be used in future studies.

Figs. 9–26 show that the LES simulation results for the mass fraction of each component for the Sandia series of flames at six axial positions, including CH4, O2, CO2, CO,H2O and OH. It can be seen that, in general, the simulation average results and RMS values of LES of Sandia series flames are in good agreement with the experimental results.In particular, the prediction results of minor species such as OH are surprisingly good, and the results are better than those given in Refs. 28,30,33,34. This is supposed that the detailed chemical reaction mechanism and the direct solution of the joint PDF of the control variables adopted in the current study can achieve high-precision simulation of turbulence/flame interaction. For the prediction of CO2mass fraction, the LES model results effectively improved the peak prediction and solved the problems caused by the use of simplified chemical reaction mechanism in Ref. 21. However, in the comparison of the simulation results of the three flames, the model also has the common problem with Refs.28,30,33,34, namely,the result of Sandia flame F is not accurate enough.The reason for this is uncertain though it has to be acknowledged that Sandia flame F is very close to blow-off and it represents a severe test of all aspects of the simulation.

Compared with Sandia flame E and F, the result of the RANS model is unstable. With the increase of fuel jet speed,affected by the influence of turbulence intensity,the simulation result of RANS model is larger and larger than the actual value. LES model simulation result is stable, and the model can more accurately simulate the Sandia flame E and Sandia flame F ignition location. From the mass fraction evolution rates of fuel CH4, oxidant O2, combustion products H2O,CO2and CO, it can be found that LES model effectively improves the overestimation of chemical reaction rate by RANS model, because LES model effectively improves the mixing effect of fuel and air.

Fig.26 Radial profiles of mean and RMS OH mass fraction at six axial locations for Sandia flame F.

As mentioned in Ref.35 for studying of the Sandia flame D by the Euler random field method alone,the results of 16 and 8 random fields are similar. Therefore, we can think that the model adopts more random number and more accurate solution of PDF,and cannot increase the simulation accuracy of Sandia flame D. However, Ref. 35 does not use the same method to study the Sandia flame F.Due to the complex turbulence characteristics of the Sandia flame F,the results of this flame simulation which uses more random fields may be more accurate.It is also possible to influence the simulation results with the selection of micro-mixing constants.Ref.36 shows that when solving the joint probability-density function transport equation, the selection of this constant is important for the micro-mixing term.In the next work,a similar approach will be used to study the role of micromixing constants in the LES method. Above are the possible effects of selecting the simulation model.Similarly, the error of the measured data released by Sandia National Laboratory can also lead to the deviation of the simulation results.As shown in Ref.14,the temperature measurement error for the pilot flame is plus or minus 50 K, and the temperature used in this paper is the standard value in the reference,1880 K.The selection of the inlet boundary temperature has a direct impact on the process of building the flame surface model and the sensitivity of the chemical reaction mechanism.These parameters will be checked in further study.

5. Conclusions

In this paper,the Eulerian stochastic field method is combined with the flamelet generated manifolds model for large eddy simulation modeling of turbulent combustion. The combined model enables the consideration of detailed chemical reaction mechanisms and high-precision simulation of the interaction between flame and turbulence under low computational cost,and the program implementation of the combined turbulent combustion model was based on OpenFOAM software. In the numerical simulation of the Sandia series of turbulent jet flames using this model, we found that the LES model can accurately predict the shape of the flame and capture the phenomenon of extinction and re-ignition under different Reynolds numbers. At present, this combined model can accurately simulate the evolution of temperature, the mass fractions of methane, carbon dioxide and hydroxyl groups of Sandia flame D and E.However,for the Sandia flame F which contains severe extinction and blow-off state, the accuracy of the model simulation is not satisfactory without changing the parameter settings of the model and example. In the future study, we will investigate the effects of the number of random fields, micro-mixing constants, and inlet temperatures on the simulation results of Sandia flame F.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 51706241).