A dual timescale model for micro-mixing and its application in LES-TPDF simulations of turbulent nonpremixed f lames

Fang WANG , Rui LIU , Li DOU , Denghuan LIU ,*, Jie JIN

a Aero-Engine Numerical Simulation Research Center, School of Energy and Power Engineering, Beihang University, Beijing 100083, China

b Collaborative Innovation Center for Advanced Aero-Engine, Beijing 100083, China

KEYWORDS Dual time scale model;Large eddy simulation;Sandia methane-air jet f lame;TPDF molecular mixing model;Turbulence combustion model

Abstract The numerical simulation of modern aero-engine combustion chamber needs accurate description of the interaction between turbulence and chemical reaction mechanism. The Large Eddy Simulation(LES)method with the Transported Probability Density Function(TPDF)turbulence combustion model is promising in engineering applications. In f lame region, the impact of chemical reaction should be considered in TPDF molecular mixing model. Based on pioneer research,three new TPDF turbulence-chemistry dual time scale molecular mixing models were proposed tentatively by adding the chemistry time scale in molecular mixing model for nonpremixed f lame. The Aero-Engine Combustor Simulation Code (AECSC) which is based on LES-TPDF method was combined with the three new models. Then the Sandia laboratory’s methane-air jet f lames: Flame D and Flame E were simulated. Transient simulation results show that all the three new models can predict the instantaneous combustion flow pattern of the jet f lames. Furthermore,the average scalar statistical results were compared with the experimental data. The simulation result of the new TPDF arithmetic mean modif ication model is the closest to the experimental data:the average error in Flame D is 7.6%and 6.6%in Flame E.The extinction and re-ignition phenomena of the jet f lames especially Flame E were captured.The turbulence time scale and the chemistry time scale are in different order in the whole flow field. The dual time scale TPDF combustion model has ability to deal with both the turbulence effect and the chemistry reaction effect, as well as their interaction more accurately for nonpremixed f lames.

1. Introduction

Turbulence combustion widely exists in aviation, aerospace,energy, chemical industries and other fields. Turbulence and chemical reaction have strong nonlinear interaction.Turbulence can accelerate the mixing process between fuel and oxygen as well as increasing the f lame surface area,which greatly strengthens the chemical reaction. The chemical reaction meanwhile releases heat, which can reduce the density of mixed gas and make it expanded, lowering the turbulence intensity. On the other hand, chemical reaction heat releasing increases turbulence energy, forms larger temperature gradient in reaction f lame area and affects the pressure fluctuation as well.In turbulence f lame,chemical reaction reacts upon turbulence and vice versa.Hence exact estimation of the interaction between turbulence and chemical reaction is important to the improvement of the simulation accuracy of turbulence combustion.

Direct Numerical Simulation (DNS) can accurately simulate turbulence combustion. However, it can hardly simulate the high Reynolds number flow and complex geometric structure as the industrial applications because of the limitation of computer ability and computational technology at present.The Large Eddy Simulation (LES) can guarantee computational accuracy for average flow parameters, such as temperature,velocity,pressure and so on,by an acceptable spatial and temporal resolution,reappear detailed flow structures,and can be combined with other physical models and execute under regular computing equipment. Hence LES has been the most promising method of applicable numerical solution of turbulence combustion step by step.Meanwhile,Transported Probability Density Function (TPDF) turbulence combustion model can treat the reaction source term accurately without any assumption. Thus, the LES-TPDF method can deal with turbulence fluctuation as well as chemical reaction in reasonable precision. The LES-TPDF method has bright prospect in industrial engineering simulation applications because of its good ability in interpretation on the interaction between turbulence and chemical reaction.

In TPDF equation, the molecular transportation term which is caused by molecular viscosity and molecular diffusion needs to be closed.1In turbulence combustion f lame region,the change of the reaction scalar gradient is very intense due to the very thin thickness of f lame, and the chemical reaction can exert strong influence on the small scale molecular transportation. Hence, in the chemical reaction area, the micomixing in the turbulent combustion is affected by both the small scale turbulence and the local chemical reaction strongly,as well as by the interaction between turbulence and chemical reaction. So the TPDF micro-mixing model should consider the chemical reaction effect.

The first TPDF micro-mixing model which is used in turbulence combustion simulation is the MC (Modif ied Curl’s)model.2Then Dopazo proposed the IEM (Interaction by Exchange with the Mean) model.3,4Many improved mixing models have been developed based on these two models. The model based on IEM model is called the deterministic model and the model based on MC model is called the particle interaction model. The Euclidean Minimum Spanning Tree(EMST) model5,6has been widely employed in TPDF simulations due to the simplicity of their implementation and the guarantee of realizability. Other models such as the binomial Langevin model were also studied.7Li et al.8simulated the methane axisymmetric jet f lame using the scalar dissipation rate and scalar time scale with considering of chemical reaction influence. Pope’s9research showed that the mixing time scale and chemical reaction time scale range overlap. Considering the influence of chemical reaction on mixing, optimizing the time scale that can overall characterize both the chemical reaction and turbulence will improve the mixing model. To accurately simulate the turbulent combustion process, it is necessary to exactly describe the quantitative relationship between turbulence and chemical reaction. Chen et al.10-12studied the Probability Density Function (PDF) mixing models for premixed f lames. Kuron et al.13adopted the weighted average of chemical reaction time scale and turbulence time scale to form a new dual time scale in micro-mixing model of TPDF simulations and evaluated the new model by DNS data of premixed hydrogen-air jet f lame. The new model behaved well in prediction of the f lame structure and f lame propagation.

The exactly expression of TPDF molecular mixing model couldn’t be worked out directly from DNS database or theoretical studies. While from physical concept, the time scale can be a weighted average time scale of turbulence time scale and chemistry time scale. Previous work done by Kuron et al.13demonstrates the potential of using a simple model,which linearly blends the turbulence induced-mixing and chemical reaction induced-mixing, to predict reactive scalar mixing in turbulent premixed f lames better. As for nonpremixed f lame, more dual time scale functions could be tried. In this paper, we made three kinds of modif ication on micro-mixing models for LES-TPDF simulations of nonpremixed turbulent f lame. Based on LES-TPDF method,the numerical simulation is conducted by the Aero-Engine Combustor Simulation Code (AECSC). The chemical reaction time scale is conceptual joint with the turbulence time scale in the TPDF micro-mixing model. Based on three kinds of mathematical relations, the interaction between chemical reaction and turbulence will be considered. The Sandia methane-air jet Flame D and Flame E will be used to test the dual time scale TPDF models.

2. Dual time scale TPDF models

2.1. Scalar TPDF equation

The one-point marginal probability density function of scalar φαcan be defined as:

where

Due to a lot of scalars involved in the turbulent flow, joint probability density function of all scalars can be expressed as:

According to Eq. (4), Favre f iltered density weighted joint sub-grid scale PDF can be obtained:

where GΔ=G（x-x′,Δ（x ）） (G is the f ilter function, Δ is the characteristic f ilter width) and the expression ofuses the sifting property of Dirac-Delta function and describes the probability of scalar within the f ilter volume in interval ψα＜φα＜ψα+dψα. ρ is density and ρ is spatial f iltered density. Ω represents the entire integral flow space.

The f iltered mean scalar and the sub-grid scale variance can be obtained by the following formulas:

where F is the joint probability density function of all scalars.Then the sub-grid scale scalar TPDF equation can be derived from the appropriate conservation equations by standard methods. The result is:

Where ˙ωαis the chemical source term;μ is the dynamic viscosity; σ is the Schmidt number; ujis the velocity component in j direction;xjis the coordinate variable in j direction.The third term on the left side of Eq.(8)represents the chemical reaction term,which is closed and there is no need to model it when the chemical reaction mechanism is specif ied.The first term on the right side of Eq.(8)is the sub-grid scale convection term,which can be approximated by a simple gradient closure directly analogous to the Smagorinsky model in the LES equations:

where μsgsis the turbulence Sub Grid Scale(SGS)viscosity,σsgsis the turbulence SGS Schmidt number. The second term on the right side of Eq. (8) is the molecular diffusion term of PDF, which contains spatial scalar gradient and is discrete.Hence it cannot be expressed byand need to be modeled.The first step is to decompose the term:The first term on the right side of Eq.(10)is closed.The second term can be called micro-mixing term, which contains the f iltered conditional scalar dissipation rate, represents the subgrid scale mixing, and describes the effect of molecular diffusion ofas well. This micro-mixing term, vital in TPDF model, not only serves as a bridge between turbulence and chemical reactions but also ref lects the interaction between them. If using M ψ;x,t（ ） to represent the micro-mixing term,there are following properties:

where

Compared with the MC model, the IEM model is a more ideal choice for high Reynolds number turbulence combustion:

The f iltered sub-grid scale scalar TPDF equation can be finally expressed as following:

where τ′sgsis the SGS mixing time scale.

In Eq. (14), the chemical reaction term is closed without modeling. The sub-grid scale mixing time scale is usually expressed as:

where τtis the turbulence time scale, CDis the micro-mixing constant.

From the dimensional analysis,is equivalent to the square of the grid cell size divided by the dynamic viscosity,which has time dimension. Hence, this time scale can only ref lect the time scale information of turbulent flow and cannot ref lect the characteristics of the chemistry reaction as well as the interaction between chemical reactions and turbulence.Da number (Damkohler number) is usually used to measure the relative size of chemical reaction time scale and turbulence time scale. This number can ref lect whether the chemical reaction belongs to rapid reaction mode or slow mode, which is very important in combustion f lames.According to the expression of Da number,the chemical reaction rate value is adopted as the chemical reaction time scale:

where p and q are the reaction orders of the reactants,CAand CBrepresent the molar concentration of methane and oxygen respectively, k is the chemical reaction rate coefficient. In this paper, τris a representative for chemistry reaction time scale.Eq.(16)can be used for elementary reaction or global reaction mechanism. However, when more complex reaction mechanism was considered, the p and q in the formula will change and are not stoichiometric coefficient any more. As for 15-steps simplif ied mechanism for methane combustion,we choose main reactants concentrations and reaction parameters to calculate the value. According to Arrhenius law, the exact expression in this paper for τris:

Pope’s research9showed that there is a certain degree of overlap between the mixing time scale and the chemical reaction time scale range. Due to the close interaction between chemical reaction and turbulence, it is reasonable and feasible to form a new mixing time scale by combining τtand τrwith certain functional relationship. The new time scale could be a weighted average time scale of turbulence time scale and chemistry time scale.In research of Kuron et al.13the weighted average of chemical reaction time scale and turbulence time scale was adopted to form a new dual time scale TPDF model:

The strong nonlinear interaction between chemical reaction and turbulence makes it difficult to determine in which way they are combined and whether they have the same influence effect on the mixing time scale. Part of the simulation results of nonpremixed f lames in the references indicate that the interaction between turbulence and chemical reaction appears to be: in some places the turbulence is dominant, and in other places the chemical reaction is dominant, thus it is difficult to put forward a universal relational expression at present.In other words, pure conjecture in the form of mathematical relationship is a feasible method to f ind out the relationship between the turbulence and chemistry. So in this paper, hypothetical mathematical relations between the dual time scales were proposed tentatively to form a new time scale which can overall represent chemical reaction and turbulence effect.As for specific averaging method, arithmetic mean, geometric mean and mean square mean of τtand τrare respectively adopted to generate new τsgs:

The new sub-grid scalar TPDF equation can be obtained by replacingin Eq. (14) with τsgs. The new mixing time scale τsgsincludes both turbulence time scale and chemical reaction time scale,which can ref lect the interaction between turbulence and chemical reaction in some degree.

If there are DNS data which can be used for detailed turbulence and chemical reaction time scale dynamic testing, better coupling relationship results can be obtained.Due to the complexity of turbulence combustion phenomenon and the limitation of current research level, quantitative expression which is based on definite physical mechanism and reveals the credible interaction between turbulence and chemical reaction cannot be obtained. Previous work done by Kuron et al.13demonstrated the potential of using a simple model, which linearly blends the turbulence and reaction induced mixing, to better predict reactive scalar mixing in turbulent premixed f lames.So it is assumed that the chemical reaction and turbulence have the same influence level on the sub-grid mixing process,and three kinds of average methods were adopted as an exploration for mixing process.

2.2. Numerical solution of TPDF equation

Monte Carlo method14can be used to solve multivariate problems and then can deal with multivariate joint TPDF equations, in which the relationship between calculation amount and the dimension of TPDF is only linear instead of index.The calculation program structure of Monte Carlo method is relatively clear. And it is easy to get some intermediate results in the process of solving the transport equation. However, the Monte Carlo method also has some disadvantages, such as slow convergence speed and probabilistic property of the error.In order to reduce the error, more particles will be introduced which may result in an increase in computational cost.

Most of the existing studies on TPDF models are converting transport equations into Lagrange equations and solve them by Monte Carlo method. However, in the case of complex chemical reactions,the calculation amount of this method is huge and the compatibility with existing flow calculation software is not good. Pope proved that the statistical error of Monte Carlo method is inversely proportional to the square root of the fluid particle number.15To get accurate results,many fluid particles must be used and will lead to a dramatic increase in computational cost. Therefore, TPDF equation of Euler random method was proposed.16They obtained a scalar random Euler field by deriving stochastic partial differential equations which is equivalent to the joint TPDF equations.This random field is continuous and differentiable in space,while continuous but non-differentiable in time. So, one advantage of the random Euler field method is that there is no sampling error of space when calculating the statistical moment. Another advantage is that this approach is easier to combine with existing turbulence combustion simulation code.As for the forms of the stochastic differential equations, there are mainly two forms. And Ito equation, the computational procedure the paper is based on, is widely investigated at present. Ito used stochastic differential equations with Brownian motion disturbance term to describe Wiener process:

where Amis the drift coefficient, Bmis the diffusion coefficient of the m th stochastic process,Xmis the m th stochastic sample,d Wmis the Wiener process of the m th stochastic process. This generalized Wiener process also can be called Ito process,whose generalization is ref lected in viewing the Brownian movement as a random interference. Sheikhi et al.17gave the final derivation:

where

3. Evaluation of new models using LES

The new dual time scale TPDF models and Euler random field method were tested by experimental data of Sandia methaneair piloted jet f lames: Flame D and Flame E, both with high Reynolds number at the same level of the Reynolds number in the aero-engine combustor. Hence the simulation results may have some practical value for engineering.

3.1. Brief introduction of AECSC code

The original author of this code is William Jones, in Imperial College London.And the code used in the paper is a modified spray f lame simulation parallel version. The code is a FORTRAN program based on LINUX environment with modular programming. The program uses LES method to solve the f iltered Navier-Stokes equations and the relevant scalar transport equations to simulate two-phase turbulence flow and turbulence combustion with a variety of built-in chemical reaction mechanisms.And the combustion model is TPDF model.For the solution algorithm of TPDF model, Euler stochastic field method is adopted.By data structure and transport acceleration, the AECSC gets faster than before.

3.2. Flame simulated

Barlow and Frank18measured the temperature and component concentration of the piloted methane-air jet f lames in 1998.Schneider et al.19measured the gas velocity of the piloted methane-air jet f lames in 2003 with Laser Doppler Velocimetry(LDV). Fig. 1 is a schematic of the piloted jet f lame.

Fig. 1 Schematic of piloted jet f lame.

The jet fluid (the volume ratio of methane to air is 1:3) is sprayed into the combustion chamber through a center nozzle with the diameter of 7.2 mm.Fuel initial velocities of Flame D and Flame E are 49.9 m/s and 74.4 m/s respectively, and stoichiometric ratios are both 0.351. The piloted f lame is a lean(mixing ratio φ=0.77) mixture of C2H2, H2, O2, CO2, and N2with the same nominal enthalpy and equilibrium composition as mixture of methane/air at this equivalence ratio. The temperature of jet and piloted f lame is 290 K and 1880 K respectively. By calculation, the proper mixing fraction is about 0.351. Table 1 shows the inlet velocity and Reynolds number of both Flame D and Flame E.

3.3. Related parameters of numerical simulation

3.3.1. Chemical reaction mechanism

In this paper,the 15-steps simplif ied chemical reaction mechanism (ARM) proposed by Sung et al.20is adopted based on detailed methane reaction mechanism of GRI3.0.

3.3.2. Grid information

The radial dimension of solution domain is 20 times the diameter of the central nozzle, and the axial dimension of solution domain is 50 times the diameter of the central nozzle.The total number of grid is 1.289 million. Free slip boundary condition is adopted in transverse boundary and convective boundary condition is adopted in outlet. The grid features are shown in Table 2.

Jones et al.21have studied the influence of the number of stochastic fields on the simulation results (with the numbers of fields set to 1 and 8) when carrying on spray combustion LES in a gas turbine combustor with the stochastic fieldssolution method.It was found that both the f lame instantaneous structure and the mean or Root Mean Square(RMS)profiles of the temperature and velocities did not change signif icantly when the number of fields increased from one to eight.Theoretically,the calculation accuracy improves with the random field number increasing,which will meanwhile prolong the calculation time.In consideration of the calculation efficiency and accuracy,the random field number is set to 8 in this paper.

The King, when he rose and saw the miracle that had been performed, was beside himself with amazement11, and didn t know what in the world he was to do

Table 1 Inlet velocity and Reynolds number.

Table 2 Grid features.

Previous DNS investigations of non-reacting shear flows22and non-premixed shear driven turbulent f lames23demonstrate that CDis approximately 2.0 for a conserved scalar. And the Refs.17,24,25pointed out that the micro-mixing constant has great influence on the results when using Reynolds Average Navier-Stokes (RANS) method, which is taken usually in the range of 1.0-3.0. From some studies on TPDF simulations of turbulent non-premixed f lames, the reasonable predictions for the combustion characteristics have been reported with a slight variation of CDfrom 1.5 to 3.0.26-28Considering that Flames D and E are non-premixed f lames, we firstly set the CDvalue to 2.0. However, some studies show that, to obtain reasonable predictions for turbulent premixed combustion,CDhas to be varied over a much wider range,from 1.5 to over 20.29-32Then CDfrom 0.2 to 20 were tested during the simulation process of Flames D and E.Finally,2.0 was selected as the CDvalue after comprehensive comparison, which means the prediction results are good enough for TPDF mixing modeling study. The turbulent Schmidt number: σsgs=1.0. The f lame sensitivity and effects of micro-mixing and chemistry on the f lames33,34were carried out before.

The name of each case is shown in Table 3, where Case D represents cases of Flame D,Case E represents cases of Flame E, and mixing time scale is the average value of τtand τr.

3.3.3. Discretization schemes

The calculation program is based on finite volume method,and the implicit low Mach number equation, the pressure smoothing algorithm and Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) are adopted. The discretizationform of scalar transport equation of time and space is shown in Table 4.

Table 3 Information of all cases.

Table 4 Discretization form of scalar transport equation.

3.4. Grid independence verif ication

The total number of grid was selected as 0.922 million, 1.289 million and 1.656 million respectively to calculate Case D1.The instantaneous temperature, velocity, turbulence kinetic energy and CH4mass fraction contour were shown in Fig. 2.Then the calculated average temperature distribution was compared with experimental data in Fig. 3 (r/d represents the radial location, z/D represents the axial location). It can be seen from Fig. 3 that the result of the total grid number of 1.289 million is closer to the experimental data (EXP) compared with the result of the total grid number of 0.922 million.The result of the total grid number of 1.289 million is similar to the calculation result of 1.656 million. Hence, the calculation accuracy can be guaranteed when using the grid with a total grid number of 1.289 million.

4. Results and discussion

4.1. Transient results

Fig. 4 shows the transient temperature field distribution of Flame D and Flame E using arithmetic mean dual time scale TPDF model. These f igures are consistent with the results in references. The simulation results indicate that the extinction possibility increases with the increase of the inlet velocity. As can be seen from Fig. 4, there is scarcely any quenching phenomenon in Flame D. But there are extinction pockets in Flame E.The quenching process happens where the imbalance happens between chemical reaction heat release local and heat transfer/dissipation. When the turbulence time scale is the same as or smaller than the chemical reaction time scale, heat dissipation caused by the turbulent fluctuation is less than heat release of chemical reaction or in equilibrium to it. The local f lame can keep on and exist. When the Reynolds number is high,the turbulence fluctuation is strong,then the heat release of chemical reaction is unable to offset the heat dissipation.With the chemical reaction rate becoming slower and slower,quenching occurs in some areas eventually. When the strain becomes weak in the extinction pockets, the turbulent fluctuation decreases, and the heat release of the chemical reaction and heat dissipation gradually reach a balance again. Reignition will happen if the local mixture fraction and temperature are high enough.

4.2. Scalar statistical results

Fig. 2 Instantaneous contours.

Fig. 3 Average temperature radial distribution of Flame D.

Fig.5 shows the radial distribution of the mean temperature of Flame D and Flame E respectively.As for Flame D,generally the simulation temperature value is close to the experimental data. In z/D=15 and z/D=30 sections, the prediction temperature distribution of Case D2 is the closest to the experimental results. For Flame E, the predicted temperature distribution is mostly close to the experimental results.In general, the calculated temperature distributions of Case D2 and Case E2 are in good agreement with the experimental data.

Fig.4 Transient temperature field distribution of FlamesD and E.

The root mean square errors of the mean temperature radial distribution of Flames D and E are shown in Table 5.The simulation results of the dual time scale TPDF model which considers the chemical reaction time scale (Cases 2, 3,4)are better than that of the TPDF model without considering chemical reaction time scale (Case 1). And among Cases 2, 3,4,the error of calculated results of Case 2 which used the arithmetic average dual time scale TPDF model is minimum.

The root mean square errors of the radial distribution of CH4and CO mass fraction are shown in Tables 6 and 7.The result error of Case 2 is the minimum among four cases.So the arithmetic average dual time scale TPDF model is the most accurate in the simulation of CH4and CO mass fraction.

Fig. 5 Radial distribution of mean temperature of Flames D and E.

Table 5 Root mean square error of mean temperature radial distribution.

Fig. 6 Radial distribution of CH 4 average mass fraction.

Fig. 7 Radial distribution of CO average mass fraction.

As mentioned before, the interaction between turbulence and chemical reactions may appear in many forms,such as linear superposition, geometric non-linear superposition, and other more complex nonlinear relations. At present it is unknown which relationship is more in line with the actual situation and is more universal. From Figs. 5-7 and Tables 5-7,the arithmetic average dual time scale TPDF model behaves the best in simulation of both Flame D and Flame E.In addition, from the research of Kuron et al.13using the method of weighted means to get mixing time scale can get relatively satisfactory simulation results in premixed f lames. These indicate that linear superposition may be a simple but feasible method for dual time scale TPDF model for nonpremixed turbulence f lame.

The dispersion of scatter plots ref lects the diffusion property of the f lame. The higher the dispersion is, the stronger the diffusion is. Fig. 8 shows the scatter plot distributions of temperature-mixture fraction (T-ξ) of the simulation results using arithmetic mean dual time scale TPDF model. Both the temperature-mixing fraction scatter plots of Flame D and Flame E have some degrees of dispersion, indicating that both the two f lames have the characteristics of diffusion f lame.Some of the temperature near the position where the mixture fraction is at stoichiometric ratio is lower than the highest temperature. The position of these points is corresponding with the extinction location. And the number of such points can ref lect the degree of difficulty of extinction. Therefore, from Fig. 8, extinction hardly exists in Flame D but in Flame E at the positions of z/D between 7.5 and 15.

For Flame E,because its jet velocity is high,local f lameout is more and intense. The Flame E simulation results by some turbulence combustion models such as Eddy Break-Up(EBU)model are not satisfactory enough.The transient simulation results above show that dual time scale TPDF models successfully captured the nonpremixed f lame characteristics as well as the local extinguishment and re-ignition phenomena.In the scalars statistical results above, the calculation error of Flame E is smaller than that of Flame D. Besides, it can be seen from Fig. 8’s scalar statistics of Flame E: in z/D∈[7.5,15]there exist distinct extinguishment phenomena.The simulation of local extinguishment and re-ignition phenomena needs good turbulent combustion model which can well ref lect the influence of chemical reactions. It is precisely because dual time scale TPDF models consider chemical reaction factor that the new TPDF model behaves better for Flame E. The new TPDF models are promising in simulation of high Reynolds number turbulent combustion.

The turbulence-chemistry interaction in Flame E is stronger due to higher turbulence intensity. Therefore, the effect of chemical reaction on scalar mixing timescale is of more importance. The improvement of the newly proposed model is accomplished by accounting for both turbulence and chemical reaction effect. This explains why the improvement by applying the newly proposed model is more signif icant in Flame E.

In general, the arithmetic mean dual time scale TPDF model makes the most improvement of simulation accuracy.The reason is that the time scales of chemical reaction and turbulence in the f lame region are in similar magnitude. The new τsgsobtained by means of averaging is relatively close to the original value. To conf irm this view, the size comparison of the two time scales and the distribution of τr/τtare studied.In Fig.9,Part A(left part of the f igure)shows the distribution of τr/τtand Part B (right part of the f igure) shows the distribution of the temperature. Considering that τris the reciprocal of the chemical reaction rate, the chemical reaction time scale will be very large in the region where the chemical reaction is very weak.Outside the f lame,the chemical reaction time scale is much bigger than turbulence time scale due to the slow chemical reaction rate caused by low fuel concentration.Inside the f lame, the chemical reaction rate is also slow due to low oxygen concentration and low temperature. To sum up, the time scales of chemical reaction in these two regions are much larger than turbulence time scale. Hence in the two regions it is the turbulence time scale that plays a decisive role in turbulent combustion.

The area of f lame can be determined by the temperature contour plot (Part B of Fig. 9). By comparison with the τr/τt(the left contour plot), the chemical reaction time scale and the turbulence time scale are similar in magnitude in f lame area. In vicinity of section z/D=30, the chemical reaction time scale is 3-40 times the turbulence time scale, and the difference is relatively signif icant. In this case, the new τsgsobtained by means of average is signif icantly different from the original value,naturally the final simulation results are different. In z/D=15 section, the difference in size between the two time scales is within three times. In this case, the new τsgsobtained by means of average is not signif icantly different from the original value.Therefore,the distributions of the scalar averages in the simulation results of the modified models and original model are slightly different. In z/D=1 section,the chemical reaction rate is 0,and the turbulence time scale is the determinant. And the τsgshere equals to τt. It can be seen from this point that although the tentative proposal of dual time scale TPDF models in this paper has some signif icance,in practical application, the effect of dual time scale modif ication on improvement of simulation accuracy is very limited.So, it is necessary to derive a more rigorous dual time scale TPDF model with definite physical mechanism and meaning from the basic physical model.

Table 6 Root mean square error of radial distribution of CH 4 mass fraction.

Table 7 Root mean square error of radial distribution of CO mass fraction.

Fig. 8 Scatter plot distribution of temperature-mixture fraction of Case 2(left for the calculated values,right for the experimental data).

In our paper, the inter-coupling of the chemical reaction and turbulence is ref lected from the perspective of three mathematical relationship: arithmetic mean, geometric mean, and root mean square respectively. The relationship imaginations of the four kinds of sub grid scale time scale are shown in Fig. 10, which can be a clue for dual time scale relationship understanding. The interaction between turbulence and chemical reactions may exist in many forms. One treatment of turbulence and chemistry time scale is minimum value comparison, which follows the EBU model’s strategy in dealing with the interaction between the turbulence and chemical reaction. Liner superposition (the arithmetic mean method)may be a relatively satisfactory method choice for a dual time scale TPDF model. According to the imaginations, the four kinds of time scales have different characteristics in magnitude and change law, which could be the reason for different time scales with different performance in predictions of f lame characteristics such as temperature profiles.

Fig. 9 Distribution of dual time scale values.

From the inspiration and enlightenment of predecessor,the interaction between the turbulence and chemical reaction should be considered. If the arithmetic mean method performed well, the physical interpretation of the interaction between turbulence and chemistry reaction could be mostly linear for nonpremixed f lame, which is consistent with the Kuron’s premixed f lame simulation.

In the paper, for simple calculation we take the reciprocal of chemical reaction rate as the chemical reaction time scale,that is, using Eq. (16) to calculate the chemical reaction time scale. Because the reaction of methane to oxygen is a complex reaction, p and q in the formula are no longer stoichiometric coefficient, and are derived from the law of each elementary reaction rate in detailed mechanism. Thus the chemical reaction time scale calculated in this way actually involves the information of reaction rate of each elementary reaction in detailed reaction mechanism, which can be regarded as representative of the chemical reaction process to some degree.

There can be different forms of expression of chemical time depending on which respect you care about. In the paper, the purpose to compare the two time scales is mainly analyzing who is dominant in different location of flow field. So in our studies, this form of chemical time is selected as a representative chemical reaction time scale.

Fig. 10 Relationship imaginations of four kinds of sub grid scale time scale.

Pope and Kuron et al.’s research indicates that it is benef icial and necessary to take chemical reaction time scale into account. We proposed new dual time scale TPDF models by using weighted average of chemical reaction time scale and turbulence time scale and validated them by turbulence f lame simulation. By three kinds of test, simulation results in our paper indicates that new dual time scale TPDF model can be used in high Reynolds number turbulence f lame simulation.The simulation results of the arithmetic mean dual time scale TPDF model are the closest to the experimental data. The tentative proposal of turbulence-chemistry dual time scale TPDF models is a meaningful exploration.

5. Summary and conclusions

In this paper, the time scale of molecular mixing model in TPDF combustion model was modified tentatively by superposition of turbulence time scale and chemical reaction time scale in three mathematical relations. And three new kinds of turbulence-chemical dual time scale TPDF models were proposed. Then the methane-air jet piloted Flame D and Flame E were simulated using the Aero-Engine Combustor Simulation Code (AECSC) which is based on LES-TPDF method.At last, the availability and simulation accuracy of the new models were tested by comparing the simulation results with the experimental data. The following conclusions are gotten:

(1) The transient temperature fields of the three kinds of dual time scale TPDF models are similar. They all successfully capture the characteristics of partial premixed f lame and the phenomena of extinction and re-ignition of methane jet piloted f lame especially for Flame E.These indicate that the new dual time scale TPDF models perform better in simulating turbulent combustion especially with high Reynolds number in which the influence of chemical reaction on micro-mixing is signif icant.

(2) Compared with the conventional TPDF model without considering the chemical reaction time scale,all the three new dual time scale TPDF models have certain improvement in the simulation accuracy of the jet f lames. This indicates that the interaction between turbulent flow and chemical reaction has certain impact on turbulent combustion and considering the chemical reaction time scale in micro-mixing model is reasonable and preferable. Besides, in this paper the simulation results of the arithmetic mean dual time scale TPDF model are the closest to the experimental data,which indicates that linear superposition may be a simple and feasible method to get dual time scale TPDF models in nonpremixed turbulent f lames.

(3) In this paper, the chemistry reaction time scale and the interaction between turbulence and chemical reaction is only ref lected from the prospective of hypothetic mathematical relations. To simulate turbulent combustion more accurately, it is necessary to conduct more detailed study on that duel time scale and their interaction, and develop a more reliable dual time scale TPDF model based on definite physical mechanism.

Acknowledgements

This study was co-supported by the National Key R&D Program of China (Nos. 2017YFB0202400 and 2017YFB0202402), the National Natural Science Foundation of China (No. 91741125) and the Project of Newton International Fellowship Alumnus from Royal Society (No.AL120003).