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1 Introduction

The Large eddy simulation (LES) approach is considered to be the most promising approach for the computation of turbulent flows in applications of engineering interest. LES solves large scales of turbulent flows up to grid sizes using spatial filtering and models subgrid scales using Smagorinsky model. CMC is applied for a turbulent combustion modelling using mixture fraction as a conditional variable. A non-conservative LES-CMC has provided predictions of major and minor species for different flames in a last decade. However, inaccurate predictions occur in CMC cells which have large temporal variations of the mixture fraction field. A lack of FDF-weighting (filtered density function) ratios in a convective term of the non-conservative CMC is believed to be the main reason for inaccurate predictions. In contrast to non-conservative LES-CMC, the present conservative formulation is inherently mass conserving. It considers FDF-weighting ratios in convective term so that improved predictions of local conditional scalars can be obtained.

In this work, investigations of turbulent jet flames (Sandia Flame D, E and F) are performed by the conservative LES-CMC approach. Flame D is used as the first test case to validate the numerical results by comparison with well-established experimental data. Subsequently, Flames E and F are investigated for extinction and reignition phenomena. Computational results of the present work from using HLRS resources are given first. Subsequently, the computational resources which have been used to simulate Sandia Flames series will be addressed.

2 Results of the Present Work

In this section, the computational setups are described, followed by the summaries of parametric CMC studies in Sect. 2.2. Section 2.3 presents the simulation results for Sandia Flame D, which is carried out as the first test case in order to validate the LES-CMC simulation models as a reference. Predictions of Flame D are also chosen as representatives in this paper, since the simulation results of Flames E and F follow the same tendency as predictions of in Flame D. This section closes with the discussion and conclusions about the performance of various CMC model parameters in Sect. 2.4.

2.1 Computational Setup

Computational grid was generated with dimensions of 80D in z-direction and 8D in x- and y-directions at the flame base increasing to 60D at the outlet of the domain. Fine grid simulations of Sandia Flame series, having 112 ×112 ×320 cell grids for LES and 8 ×8 ×80 cell grids for CMC (reference case), have been carried out with HLRS resources. The regions above the jet and pilot are captured by 28 and 40 LES cells (for a dimension), respectively. Due to grid independence studies by Navarro et al. [1], these computational cells satisfy a condition that the largest fraction of energy spectrum is resolved after the initial break-up of the jet. The CMC grid has 100 nodes in mixture fraction space which has refinement at η = 0 and 1.

2.2 Parametric Studies

The results of the LES-CMC modelling are presented in three main parts. Parametric studies of flow and mixing field, parametric studies of combustion model and parametric study of CMC grid resolution are investigated in order to study the effects of each parameter on simulation results. Details of these studies for each parameter are given in the following sections.

2.2.1 Parametric Studies of Flow and Mixing Field

The parametric studies of flow and mixing field are the inflow velocity variances in turbulent inflow generator, the Schmidt number, Sc, the turbulent Schmidt number, Sc t , and constant value, C ξ, which is used in modelling of the variance of mixture fraction (\(\widetilde{\xi _{sgs}^{{\prime\prime}2}} = C_{\xi }{\Delta }^{2}\bigg{(} \frac{\partial \tilde{\xi }} {\partial x_{j}} \frac{\partial \tilde{\xi }} {\partial x_{j}}\bigg{)}\)). Besides C ξ, all of them are fluid properties which actually are not allowed to be changed. However, the velocity variance levels are adjusted to yield suitable values for the inflow generator which creates the oscillation of the velocity field in this work. The Schmidt number and the turbulent Schmidt number, which are the ratios of momentum transfer rate to mass transfer rate for resolved and unresolved scales, are varied to test a sensitivity of the flow and mixing field. Sandia Flame D is used as a reference case and thus simulation results from these optimal values are reported in Sect. 2.3.

2.2.2 Parametric Studies of Combustion Model

The evaluation of the LES-CMC combustion model is in the focus of the present research project. The parametric studies concerning the combustion model carried out here comprise the evaluation of the CMC formulation, the approximation of the CMC convective fluxes and the model for the conditionally filtered turbulent diffusivity.

  • CMC Formulations

    The difference of the two CMC formulations (the non-conservative form in Eq. 1 and the conservative form in Eq. 2) is the inclusion of the FDF information into the transport equation, in particular into the convective term (the second term on the LHS) of the conservative CMC formulation.

    $$\frac{\partial Q_{\alpha }} {\partial t} +\widetilde{ u}_{j,\eta }\frac{\partial Q_{\alpha }} {\partial x_{j}} -\frac{1} {\gamma } \frac{\partial } {\partial x_{j}}\bigg{(}\gamma D_{\eta }\frac{\partial Q_{\alpha }} {\partial x_{j}} \bigg{)} =\tilde{ w}_{\alpha ,\eta } +\widetilde{ N}_{\eta }\frac{{\partial }^{2}Q_{\alpha }} {\partial {\eta }^{2}} ,$$
    (1)
    $$\gamma \frac{\partial Q_{\alpha }} {\partial t} + \frac{\partial } {\partial x_{j}}(\gamma \widetilde{u}_{j,\eta }Q_{\alpha } - \gamma D_{\eta }\frac{\partial Q_{\alpha }} {\partial x_{j}} ) = \gamma \tilde{w}_{\alpha ,\eta } + \gamma \widetilde{N}_{\eta }\frac{{\partial }^{2}Q_{\alpha }} {\partial {\eta }^{2}} + Q_{\alpha } \frac{\partial } {\partial x_{j}}(\gamma \widetilde{u}_{j,\eta }),$$
    (2)

    where γ denotes \(\overline{\rho }\widetilde{P}(\eta )\) and \(\widetilde{P}(\eta )\) is the Favre filtered probability density function (FDF). \(Q_{\alpha } =\widetilde{ Y _{\alpha }\mid \eta }\) is the conditionally filtered mass fraction. \(\widetilde{u}_{\eta }\) is the conditionally filtered velocity, \(\tilde{w}_{\eta }\) is the conditionally filtered reaction source term and \(\widetilde{N}_{\eta }\) is the conditionally filtered scalar dissipation rate. Term \(D_{\eta }\frac{\partial Q_{\alpha }} {\partial x_{j}}\) is the modelled -by using of a gradient diffusion approximation- of the subgrid-scale conditional scalar flux. Based on finite volume method, both conditional species transport equations can be applied to each control volume (CV) of the computational domain. It is believed that including the FDF information in convection will make the CMC conservative form more precise than the traditional one [2]. Therefore, the study of two different formulations of the combustion model is performed in this work to reveal the results of the assumption.

  • Flux Approximations

    Since each CMC cell comprises of a number of LES cells, two methods can be applied to approximate the convective flux between CMC cells. The first method calculates convective flux over the CMC cell face from the convective fluxes of the LES cells adjacent to the CMC cell face. The second method computes convective flux from the values at the CMC cell centre, which takes into account the values of all LES cells within the CMC cell. The convective flux from the first method is shown as the summation of the small arrows in Fig. 1, while the convective flux from the second method is shown as the big arrow in the same figure.

  • Conditionally Filtered Turbulent Diffusivity Models

    Fig. 1
    figure 1

    A schematic of the two approximations of the CMC convective flux

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    There are three methods to model the conditionally filtered turbulent diffusivity, D η. However, all of them are based on the Smagorinsky model for the subgrid-scale kinematic viscosity, ν t and the relation of unconditionally filtered turbulent diffusivity, \(D_{t} = \nu _{t}/Sc_{t}\). In the first method, D η is calculated based entirely on CMC cell (named D η, 1). Therefore, ν t in this method is calculated based on CMC grid resolution. In the second method, D η, 2 is calculated based on D t from every LES cell which locate inside that CMC cell. The ensemble averaging over a CMC cell can be computed by weighting with FDF. In the third method D η, 3, the ratio of the size of CMC cell to LES cell is included into the second method in order to adjust the length scale during modelling D η value. This additional value is hoped to predict more accurately since the D η model should be based on the filter width of CMC instead of on the filter width of LES.

2.2.3 Parametric Study of the CMC Grid Resolution

Another main parametric study for all Sandia Flame series is the CMC grid resolution. For a simple and stable flame, this parameter might not reveal any effect. However, it is believed that a high number of CMC cells may capture the extinction and reignition phenomena due to the turbulence-chemistry interactions in Flames E and F. Thus, three CMC grid resolutions are performed in this study topic. These are 4 ×4 ×80, 8 ×8 ×80 and 16 ×16 ×80 CMC cells for the same LES resolution.

To summarize, all parametric studies are shown in Table 1 and they were varied for Flame D. The values resulting in the best agreement between simulation and experiments of parametric studies of flow and mixing field of Flame D were chosen for further simulations of Flames E and F.

Table 1 Summary of parameters studies
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2.3 Results of Sandia Flame D

Results of Sandia Flame D are composed of three principal parts. Firstly, results of the parametric studies of flow and mixing field are reported. Subsequently, results of the parametric studies of combustion model are shown and discussed. Finally, results of the parametric study of CMC grid resolution are given and discussed.

2.3.1 Parametric Studies of the Flow and Mixing Field

Using a reference case of parametric studies in combustion model (CMC-1, flux-1 and D η, 2) and CMC grid resolution of 8 ×8 ×80, the best results of parametric studies in flow and mixing fields are variance-2, Sc 2, Sc t, 1 and C ξ, 1. The meaning of each parameter can be found in Table 1. Overviews of best results in flow and mixing filed which use these optimum values can be observed in Figs. 2 and 3.

Fig. 2
figure 2

Snapshots of the temperature field in total computational domain (left) and in the upstream region (right) for Sandia Flame D. The iso-contour of stoichiometric mixture fraction is presented by the black lines

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Fig. 3
figure 3

Radial profiles of mean and RMS axial velocity and mixture fraction at three downstream locations for Flame D. Symbols denote experimental values [3, 4], while the solid and dashed lines present the mean and RMS values of LES-CMC (reference case of Table 2)

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Table 2 Summary of different parameters in combustion model study. The meaning of each numerical method can be found in Table 1
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Figure 2 shows a snapshot of the instantaneous temperature field along a 2D plane through the burner centerline for the whole computational domain (left) and focused on the upstream region (right). The black lines indentify the isoline of stoichiometric mixture fraction. It can be observed from a temperature profile (Fig. 2 (right)) that there is a high level of turbulence which comes from the digital turbulent inflow generator at the inlet. Moreover, the local extinction, which would be characterized by discontinuous red color of the temperature along the isoline of stoichiometric mixture fraction, hardly occurs in Flame D, in accordance with the experimental findings.

Radial profiles of mean and RMS axial velocity and mixture fraction at three downstream locations are shown in Fig. 3. Both mean and RMS of axial velocity and mixture fraction agree properly with the experiments [3, 4], since the effects of initial inflow from inflow generator are previously checked and adjusted. Moreover, it can be seen form Fig. 3 that the jet spreading is captured well. Small overpredictions of the mean mixture fraction in the range of 1. 2 < r ∕ D < 2 at position \(z/D = 3\) may come from the influences of lateral boundary conditions. Small overpredictions of the mean axial velocity and mixture fraction around 1 < r ∕ D < 2 at position \(z/D = 15\) may require more simulation time for more precision. However, the current predictions provide a good basis for the parametric studies of combustion model.

2.3.2 Parametric Studies of the Combustion Model

As shown in Table 2, five case studies are performed to show the effects of each case in CMC model. All cases are based on the optimal conditions from parametric studies of flow and mixing field (variance-2, Sc 2, Sc t, 1 and C ξ, 1) and use 8 ×8 ×80 for the number of CMC cells. A reference case (case-1) includes the models CMC-1, flux-1 and D η, 2, while other cases have at least one varied parameter compared with the reference case.

2.3.2.1 Preliminary Studies

Preliminary studies of parametric studies of combustion model are required to choose the cases which may predict the simulation results different from the reference case (case-1 from Table 2). These case studies will be further examined for the simulation results in next sections. In the first step, a representative direction has to be defined. Having the highest convective value among three directions, the convective flux in z − direction is chosen as a representative. Note that a consideration of FDF profile is required, since it is applied to transfer the values from a mixture fraction space to a physical space. The low value of FDF means only a small influence of any property can appear in the physical space.

Subsequently, the convective fluxes in z − direction are shown in radial and axial distributions. The instantaneous predictions of CH4 fluxes in Fig. 4 show that the radial distribution exhibits larger differences of the convective fluxes between each case than the axial distribution. Thus, the radial distribution is applied for investigations of convective flux comparisons for the other species. It should be reminded that a consideration of FDF is necessary as previously discussed.

Fig. 4
figure 4

Axial (left) distribution along the centerline and radial (right) distribution at \(z/D = 7.5\) of convective fluxes in z-direction of CH4 for a time step (Sandia Flame D)

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The investigations of fluxes in other species show that fluxes in different species follow the same tendency of CH4 fluxes in Fig. 4. It can be observed that case-2, case-3 and case-4 produce different convective fluxes compared with the reference case (case-1). However, case-3 and case-4 produce similar flux, which means their conditional scalar predictions should not be different. Therefore, case-3 is chosen for the further parametric studies. Fluxes of case-1 and case-5 are similar and also case-3 and case-4 are similar because of low effects of different D η models on the convective term. Since case-1, case-2 and case-3 produce different convective fluxes, the conditional scalar predictions from these cases should differ from each other.

In next step, the statistical results of three cases (case-1, case-2 and case-3) are sampled over 30,000 time steps for the statistics to investigate the different effects of combustion model parameters and to validate with the experiment.

2.3.2.2 Conditionally Filtered Reactive Scalars

As in CMC methodology, species mass fraction are analyzed in mixture fraction space, the efficiency of the combustion model can be decoupled from flowfield predictions. Therefore, the performance of each case of combustion model can be directly considered from conditional profiles. Initial values for the conditional reactive species are obtained from the SLFM solution.

A good agreement of case studies with experiments is shown in Figs. 5 and 6. The conditional mean temperature and the conditional mean mass fraction of CO, CH4 and H2 are given in both figures as a representative of intermediate products, fuel and radicals. Note that the error bars indicate the conditional RMS and they are only plotted to illustrate the turbulent level of each scalar. The reason of different predictions between case-1 and case-2 on the rich side (η > 0. 35) belongs to two different sets of convective fluxes which are calculated from two CMC formulations. Because of the lack of FDF-weighting function in convective term, the low convective fluxes on the rich side are generated in the upstream positions of case-2. These can be observed in Fig. 7. Therefore, case-2 (non-conservative CMC) usually overpredicts on the rich side of mixture fraction in temperature, radical and intermediate product, while the underpredictions occur in the fuel.

Fig. 5
figure 5

Conditional profiles of cross-sectionally averaged temperature and CO at two different downstream positions in mixture fraction space for Flame D. Symbols are experimental data [4], while the solid, dashed and dotted lines present the results of LES-CMC in different cases in combustion model (Table 2)

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Fig. 6
figure 6

Conditional profiles of cross-sectionally averaged CH4 and H2 at two different downstream positions in mixture fraction space for Flame D. Symbols are experimental data [4], while the solid, dashed and dotted lines present the results of LES-CMC in different cases in combustion model (Table 2)

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Fig. 7
figure 7

Radial distribution of mean convective fluxes in z-direction of CH4 at \(z/D = 7.5\) (Sandia Flame D). The solid, dashed and dotted lines present the results of LES-CMC in different cases and FDF (reference case) in combustion model (Table 2)

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It can be observed from the Figs. 5 and 6 that case-1 (conservative CMC) shows more accurate results than case-2. Cross-sectional simulation results from case-3 hardly differ from case-1, even though the convective fluxes from both cases are different (Fig. 7). The reason can be explained by using FDF value for each CMC cell in the same cross section to calculate the cross-sectional averages. If the conditional predictions between two cases of any CMC cell have a difference where the low FDF is calculated in mixture fraction space, the conditional predictions in cross-sectional averages will be similar.

2.3.3 Parametric Study of CMC Grid Resolution

As described in Table 1, three cases of CMC grid resolution are varied, while the same conditions of flow and mixing field (variance-2, Sc 2, Sc t, 1 and C ξ, 1) and CMC combustion model (CMC-1, flux-1 and D η, 2) are set up. The variations of the CMC cells in each x − and y − direction are 4 cells for case-1, 8 cells for case-2 (reference case) and 16 cells for case-3 with the same 80 CMC cells in z − direction, as summarized in Table 3.

Table 3 Summary of different cases in CMC grid resolution study
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Since the CMC resolution varies in the radial distribution for three case studies, the radial distribution of conditional value should show more prominent features. Therefore, the radial distribution of mean scalar are investigated at position z ∕ D = 3, 7.5 and 15. The mean temperature and CO predictions in radial distributions are shown in Fig. 8.

Fig. 8
figure 8

Radial profiles of mean temperature and CO for Flame D. Symbols are experimental data [4], while the solid, dashed and dotted lines present the results of LES-CMC from the different CMC grid resolutions (Table 3)

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It can be seen from position \(z/D = 3\) that res-2 and res-3 perform better than res-1 since there is an underprediction of the temperature for res-1 in this position. Predictions of res-3 can capture the highest value of CO at position \(z/D = 7.5\). Moreover, predictions from res-3 (16 ×16 ×80 for CMC cells) match better with the experiments than the others at position \(z/D = 15\) which show a great advantage of small CMC cells in this resolution. A reason may relate to the size of CMC cell which the big size of CMC cell may predict inaccurately in which a high level of mixture fraction gradient occurs. However, an increasing CMC resolution from res-2 to res-3 requires more computational time than 60 %. Considering the computational time and results from all CMC resolutions, the appropriate resolution is res-2 (8 ×8 ×80 for CMC cells) for Flame D.

2.4 Summary

In this section, parametric studies of LES-CMC are carried out for the Sandia Flame D. The parameter studies comprised investigations of flow and mixing field, variants of the CMC combustion model parameters and CMC grid resolution.

Flame D is used to investigate the influences of various flow and mixing field parameters on the simulation results. These parameters, which are Sc = 0. 7, Sc t  = 0. 4 and C ξ = 0. 2, are optimal values and thus, they are used for further studies of Flame D, as well as for Flames E and F. Values of velocity variance of inflow generator, however, depend on the physical inflow of each flame. It should be noted that an adjustment of the velocity variances is carried out to reduce the high level of turbulence which may come from the implementation of inflow generator. Suitable inflow variances for Flame D are found to be \(\frac{2} {3}\) of the measured variances at \(z/D = 0.14\) of these flames, respectively.

Parameter studies of different variants of the CMC combustion model are carried out to find the most suitable model. Initial studies of the CMC fluxes have shown that the effect of turbulent diffusivity modelling is negligible. However, a comparison of CMC models, which varies the CMC formulation (conservative vs. non-conservative), reveals considerable differences. Moreover, some slight differences between two methods of CMC convective flux approximation (cell face vs. cell centre based) are detected. Therefore, three dominant cases which differ in both numerical aspects are investigated for further studies. Conditional mean scalars, which are averaged in the same cross section, show that the conservative CMC formulation with computing convective fluxes based on LES cells located at the CMC cell faces (case-1) is similar to the one with computing convective fluxes based on CMC cell centers (case-3). This can be explained by the low FDF values where the differences of predictions occur in a CMC cell. Consequently, the conditionally averaged predictions with FDF weighting create the similar results over a cross section. Generally, conditional predictions reveal that case-1 can capture better mean measurements than case-2 (the non-conservative formulation using the same flux approximation). This is because the variation of FDF-weighted convective fluxes in different directions of case-1 allows the predictions to be more accurate.

Three different CMC grid resolutions (4 ×4 ×80, 8 ×8 ×80 and 16 ×16 ×80) are examined in order to find the appropriate number of CMC cells for Sandia Flame D. Basically, the best predictions are found in CMC grid resolution of 16 ×16 ×80. However, the reasonable resolution for Flame D is 8 ×8 ×80 CMC cell due to the computational cost with efficient performance.

3 Future Study Cases

In order to show advantages of conservative CMC, more complicated flames are required for the simulations. Therefore, future test cases will be the Sydney bluff-body flame, HM1, with 218 ×218 ×300 cells for LES for two CMC mesh sensitivity studies. Moreover, four lifted flames investigated at Berkeley [5, 6] and Calgary [7] will be examined (two Berkeley flames and two Calgary flames). The summation of all test cases can be found in Table 4.

Table 4 Test cases for the LES-CMC model
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4 The Usage of Computational Resources

The fine grid simulation results, which are presented here, have been performed on NEC Nehalem Cluster with 80 processors due to scalability tests for LES-CMC. The using of parallel program MPI and vectorization compiler let the code runs faster. An approximate wall time is around 48 h per summited job. The analysis of all parametric studies from Table 1, of Flames D, E and F corresponds to 450,000 CPU hours. The amount of CPU use for Sandia Flame series is appropriate to the requirements of computational resources, 450,000 CPU hours for all test cases in future.