1 Introduction

One of the main challenges for the numerical simulation of combustion is the high computational cost caused by the numerous transport equations and the stiffness due to a wide range of time scales. A mainstream manner to solve the above problem is the dimension reduction technique, e.g., the flamelet generated manifolds (FGM) method ^[1] and the flame prolongation of intrinsic low-dimensional manifold (FPI) method^[2-3]. The basic idea of the FPI or FGM method is to parameterize the whole thermal-chemistry state, which is obtained by solving the one-dimensional (1D) laminar premixed flame equations, by a few reduction parameters, e.g., tabulated scalars to build the look-up table. The progress variables, which are usual parameters representing the reaction evolution, are defined subjectively as a linear combination of the major species and used as the reduction parameters to build the thermo-chemistry table. The FGM method can also be combined with other common turbulent combustion models such as the flame surface density (FSD) model^[4-5] and the artificial thicken flame (ATF) model^[6-7], so as to achieve a reasonable predictive accuracy for the turbulent premixed combustion simulation. For the stratified flame, the effect of the gradient of the mixture fraction is usually ignored, and the flame is usually regarded as a set of unstrained premixed flames^[8]. Generally, two control parameters are included for the chemistry representation in the stratified flame, i.e., the reaction progress variable describing the reaction evolution and the mixture fraction expressing the stratification effect^[8].

However, the strain caused by the velocity gradient or the vortex in turbulent flow field strongly affects the local flame front structure. Tay-Wo-Chong et al.^[9] indicated that the rate of fuel consumption decreased exponentially when the strain rate increased. When the strain rate was small, its influence on the flame structure could be neglected. However, it should be taken into account under a high strain rate. Thus, it was desirable to include other scalars as a progress variable to represent the strain effect on the flame structure in the thermo-chemistry tabulation approach for premixed or stratified flame simulation. However, the transport equation for the strain rate might be very complicated in flame simulation, and thus it is not appropriate to directly select the strain rate as a tabulated scalar. Kolla and Swaminathan^[10-11] selected a conditional reaction progress variable and its dissipation rate as the two tabulated scalars to construct the thermo-chemistry table in the premixed flame modeling strategy. However, the dissipation rate of the progress variable was not monotonic with the strain rate, leading to an inherent problem for this modeling strategy. Moreover, it was also very difficult to model the scalar dissipation rate in the combustion simulation. Edward et al.^[12] selected a commonly used reaction progress variable and mass fraction of species H as the two tabulated scalars for the CH₄/air premixed flame look-up chemistry table. However, it was necessary to confirm whether the mass fraction of the species H or other species was suitable for the second tabulated scalar under different types of fuel and different equivalence ratios. Moreover, the somewhat arbitrarily selected progress variables may not necessarily be linearly independent to form an orthogonal base in the composition space, leading to the loss of accuracy.

The principal component analysis (PCA)^[13-15] is a systematic method based on the eigenvalue analysis^{[8, 16-25]}. It was developed to define the linearly independent progress variables for thermo-chemistry tabulation. With the PCA, Chen et al.^[25] reconstructed the CH₄/N₂/air jet lifted flame structures based on simulation data. It was found that the mass fraction of the major species and the temperature were well reconstructed, while the reconstruction for the minor species such as OH had some deviations. Biglari and Sutherland^[23] used the PCA to deal with the extinction and reignition of the non-premixed CO/H₂/air jet flame, and achieved good results. Najafi-Yazdi et al.^[8] proposed an IL-FGM approach to construct the flamelet-based chemistry table by using the PCA to define the progress variables as the tabulated scalars in unstrained premixed and stratified flame, and showed that PCA could also be extended to strained premixed/stratified flame chemistry tabulation.

The motivation of the current work has three folds. Firstly, 1D laminar premixed flames under several different strain rates are used to generate the thermal-chemistry database for the PCA to define the appropriate two linearly independent progress variables as the tabulated scalars for the chemistry tabulation, and prior tests are carried out to evaluate the developed tabulation approach. Secondly, this approach is extended to strained stratified flames to define three linearly independent progress variables to construct the chemistry table. Thirdly, the Schmidt orthogonalization method is used to construct the new linearly three independent tabulated scalars to keep the mixture fraction as a tabulated scalar in the stratified flame tabulation. The outline of this paper is as follows: the PCA method and its application in the chemistry tabulation are introduced in Section 2. The PCA method for the premixed strained flamelet tabulation method is presented in Section 3. The PCA method and the modified PCA method for the stratified strained flamelet tabulation method are presented in Section 4. Conclusions are drawn in Section 5.

2 PCA method and its applications in combustion

The chemistry database can be analyzed based on orthogonal transformation with the PCA method^[8], through which the possibly correlated variables in the composition space can be converted into a set of values of linearly independent variables, i.e., the principal components (PCs). The greater the variance of the scalar is, e.g., the species mass fraction in the combustion chemistry database, the more information it contains and the greater the weight of the PCs is. The first PC has the largest possible variance accounting for the variability in the database, and each succeeding PC, which is orthogonal to the preceding PCs, has the lower possible variance. The PCs are then used to optimize the progress variables as the tabulated scalars.

In the present research, the OPPDIF program^[26] is used to solve the 1D laminar counter-flow premixed flame in the physical space by changing the strain rate through the jet velocity. At a given equivalent ratio φ and a given strain rate a, there are n thermo-chemical state points in the physical space. The kth species mass fraction in all the thermo-chemical state points can be expressed in a column vector as follows:

(1)

where m is the number of species in the flame. Then, all species mass fractions constitute a species mass fraction matrix Y_{(φ, a)} as follows:

(2)

Under a fixed equivalent ratio φ, a series of species matrices Y_{(φ, a1)}, Y_{(φ, a2)}, ..., Y_{(φ, aq)} of the premixed flamelets by increasing the strain rate gradually from the initial small strain a₁ to the flame quench strain a_q can be obtained. These matrices can be combined to obtain the species mass fraction matrix Y_(φ) as follows:

(3)

For the strained stratified flame, the equivalent ratio is inhomogeneous. Ignoring the effects of the gradient of the mixture fraction^[8], we can establish the chemistry database by increasing the equivalent ratio φ from lean to rich in the flammable region and repeating the above steps. The mass fraction matrix Y_sum containing all Y_(φ) with equivalent ratios in the flammable range can be obtained as follows:

(4)

According to the definition of the PCA, the PC matrix F can be calculated by the product of the species mass fraction matrix Y, such as Y_(φ) in Eq. (3) or Y_sum in Eq. (4), and the weight coefficient matrix U of the PCs as follows:

(5)

where U is an m × m matrix expressed by

(6)

The PC weight coefficient matrix U can be obtained by decomposing the eigenvalues of the covariance matrix S of Y. The covariance matrix S is defined by

(7)

(8)

Take the eigenvalue decomposition of S as follows:

(9)

(10)

where the diagonal values λ₁, λ₂, ..., λ_m of the diagonal matrix Λ are the characteristic roots of S. We rearrange the characteristic roots in the descending order λ₁ ≥ λ₂ ≥... ≥ λ_m, where the corresponding eigenvector of λ_i is the weight coefficient vector of the ith PC. When the PC weight coefficient matrix U is obtained, the PC matrix F can be calculated by Eq. (5).

The PCs are in fact the weighted linear combinations of the species mass fractions, and can be used as the linearly independent progress variables for the chemistry tabulation^[8]. Based on the work of Najafi-Yazdi et al.^[8], the first few PCs obtained from the PCA approach are used to define the progress variables to build the chemistry table for strained premixed or stratified flames to avoid prohibitive memory and computational cost.

3 PCA method for the chemistry tabulation of strained CH₄/air premixed laminar flame

The GRI3.0 mechanism for the CH₄/air combustion, which consists of 53 species and 325 reactions, is used for solving the strained 1D CH₄/air premixed flame. Under the atmosphere pressure and temperature, i.e.,

considering the equivalent ratio φ = 1.0, the species mass fraction matrix Y_(φ=1.0) in Eq. (3) can be obtained under 10 different strain rates by increasing the strain rate a from 3s^-1 to 1 200 s^-1, covering the range between very small and near the flame quench strain rate. Then, Y_(φ=1.0) is taken for the PCA to obtain the PC weight matrix U and PCs by Eq. (5).

Generally, it is desirable to minimize the number of the progress variables to avoid the prohibitive memory and computational requirements. Chen et al.^[25] discussed the reason why only two PCs were used to optimize the progress variables. In practice, the number of progress variables to build the thermo-chemistry table are less than three, and thus the first two independent PC scores F₁^p and F₂^p are used as the tabulated scalars^{[8, 25]}, where the superscript p represents the premixed flame. Table 1 shows the weight coefficients of the major species for the first two PCs. The strained premixed look-up chemistry table can be established as follows:

(11)

In this study, the number of the tabulation nodes in the two directions F₁^p and F₂^p are 200 and 30, respectively, in order to reduce the deviations caused by linear interpolation. Table 2 shows the operating conditions of three laminar flames, i.e., Flame A, Flame B, and Flame C, with the strain rates 2 s^-1, 600 s^-1, 1 200 s^-1, respectively, at the equivalent ratio 1.0. Figure 1 shows the comparison of the results given by the numerical solution in physical space obtained from OPPDIF code and that retrieved by the linear interpolation from the chemistry table given by Eq. (11). The abscissa of the scatter points is the temperature T or the mass fractions of CH₄, CO₂, H₂O, CO, and OH obtained by the numerical solution in physical space, and the ordinate is the corresponding value in the chemistry table of Eq. (11), which is obtained by using the tabulated scalars F₁^p and F₂^p given by the PCA approach. It indicates that the value obtained by looking up the chemistry table is precise. The closer the distance of the scatter point to the red line is, the smaller deviation of the look-up chemistry table is. From Fig. 1, we can see that, the temperature and major species retrieved from the chemistry table are in excellent agreement with those given by the direct solution in the physical space with complex chemistry in all the three flames. By contrast, the minor species OH mass fraction retrieved from the chemistry table with some deviations, which is consistent with the results obtained by Parente et al.^[17], indicating that more PCs are needed to give more accurate results for Y_OH. Chen et al.^[25] also pointed out that the accuracy of the major species is higher than the minor species by generating the chemistry table with only the first two PCs. In the PCA, the values and variances of the major specie mass fraction are higher than those of the minor species, which can lead to more precise results retrieved from the table.

4 PCA method for the chemistry tabulation of strained CH₄/air stratified laminar flame 4.1 PCA method for tabulated strained stratified laminar flamelets

In the present research, the strained CH₄/air stratified flame is selected as a target flame and regarded as a set of strained premixed flames by ignoring the effects of the gradient of the mixture fraction. Under the atmosphere pressure and indoor temperature, i.e.,

the species mass fraction matrix Y_sum in Eq. (4) including the effect of the strain rate could be obtained by solving a series of the strained premixed flames in physical space under twelve equivalent ratios covering the flammable limit, φ ∈ [0.5, 1.6] with the equidistant increment. Then, Y_sum is taken for the PCA to obtain the PC weight matrix U and PCs. The weight coefficients of the major species of the first three PCs, i.e., F₁^s, F₂^s, and F₃^s, are listed in Table 3.

As discussed in the work of Chen et al.^[25], the first three PCs F₁^s, F₂^s, and F₃^s are used as the tabulated scalars to construct the chemistry table, i.e.,

(12)

To reduce the error caused by the interpolation of looking up the table at an acceptable memory, the number of nodes in the three directions of the thermo-chemistry table are 200, 30, 20, respectively. Prior tests could be done by comparing the results of the major species and temperature from the physical solutions and that retrieved from the chemistry table given by Eq. (13). Nine different 1D laminar flames, i.e., Flame A, Flame B, Flame C, Flame D, Flame E, Flame F, Flame G, and Flame I (see Table 4), corresponding to the three equivalence ratios 0.55, 1.05, and 1.55, and three strain rates are calculated by using the OPPDIF code. Figure 2 shows the comparison of the results of temperature and the mass fraction of the species CH₄, CO₂, H₂O, CO, and OH. Similar to the results of the strained premixed flames in Section 3, the temperature, the major species, i.e., CH₄, CO₂, H₂O, and CO, and mass fractions retrieved from the flamelet chemistry table are in good agreement with the OPPDIF solution. The minor species mass fraction Y_OH retrieved from the chemistry table of Eq. (13) has some deviation, which is consistent with the results of the strained premixed flame discussed in Section 3. The above prior tests show that the PCA method could be used to optimize the progress variables for the chemistry tabulation of the strained stratified flames.

4.2 Modified PCA method for tabulated strained stratified laminar flamelets

In the general combustion modeling strategy, a passive scalar such as the mixture fraction is prior to be selected as the tabulated scalars to represent the mixing process in stratified or partially premixed flames. In the above PCA, none of these progress variables, a linear combination of the species mass fractions, is a passive scalar. In this paper, a new approach based on the PCA is proposed to choose new tabulated scalars by transforming one of the PCs, which has the highest correlation coefficient with the mixture fraction, to a passive scalar.

Assume that the combustion contains L elements and m species. The mixture fraction Z_p based on the mass conservation of the element p could be constructed as follows^[27]:

(13)

where a_kp, W_k, and Y_k correspond to the number of the p element in the kth species, the molar mass of the kth species, and the mass fraction of the kth species, respectively. W_p is the molar mass of the p element.

The weight coefficient vector U_Zp for the mixture fraction Z_p is defined by

(14)

The mixture fraction Z_p can be calculated by the inner product of the vector U_Zp and the vector of the mass fraction of m species at each thermo-chemical state point in the physical space. In the present research, element H is selected to define the mixture fraction, i.e., Z_H.

To identify the relationship among F₁^s, F₂^s, F₃^s, and Z_H, the correlation analysis is performed. The results show that the correlation coefficient between the second PC F₂^s and the mixture fraction Z_H is 0.92, and the correlation coefficients between F₁^s and Z_H and between F₃^s and Z_H are 0.06 and -0.17, respectively. Due to the higher correlation between F₂^s and Z_H, F₂^s could be replaced by the mixture fraction and the new orthogonal progress variables for the chemistry tabulation could be obtained by the following Schmidt orthogonalization.

The weight coefficient vector U_ZH of the mixture fraction Z_H is used to define the coefficient vector U_F2^s* of the second tabulated scalar F₂^s* as follows:

(15)

F₂^s* and the mixture fraction Z_H have the following relationship:

(16)

Following the Schmidt orthogonalization process, the new orthogonal vectors U'_F1^s and U'_F3^s can be obtained as follows:

(17)

(18)

By unitizing the vectors U'_F1^s and U'_F3^s given in Eqs. (17) and (18), one can get the normalized orthogonal basis vectors U_F1^s*, U_F2^s*, and U_F3^s* as the new orthogonal variables for the chemistry tabulation as follows:

(19)

(20)

Using U_F1^s*, U_F2^s*, U_F3^s*, and Y_sum in Table 5, we can calculate F₁^s*, F₂^s*, and F₃^s* for all state points and a new chemistry table ϕ^s* = ϕ^s*(F₁^s*, F₂^s*, F₃^s*). The new three tabulated scalars inherit the advantages of the PCA method, and also contain the passive scalar F₂^s*. Considering the same flames as Section 3, Fig. 3 shows the comparisons of the results given by the numerical solution in physical space and that retrieved by the new chemistry table. In comparison with the PCA method and the modified PCA method in the prior result of the strained stratified combustion, it can be found that the accuracy of the two methods is quite equivalent. The modified PCA method preserves a passive scalar that characterizes the mixing without affecting the computation accuracy, and therefore has more advantages in the computation simplification.

5 Conclusions

The PCA is used to analyze the high dimensional chemistry data of the laminar premixed/stratified flames under the effects of strain rates, and the first few PCs, which have larger contribution ratios, are chosen as the tabulated scalars to build the look-up chemistry table. The main conclusions can be drawn as follows:

(ⅰ) PCs can be used to define appropriate linearly independent progress variables as the tabulated scalars for the chemistry tabulation of strained premixed flames. Prior tests show that the temperature and major species given by the direct solution in physical space with complex chemistry are in excellent agreement with those retrieved from the chemistry table.

(ⅱ) This approach can be extended to strained stratified flames by defining three linearly independent progress variables to construct the chemistry table.

(ⅲ) The method for selecting the tabulated scalars is based on the rigorous mathematical analysis, i.e., the Schmidt orthogonalization is proposed to construct the new linearly orthogonal tabulated scalars to keep the mixture fraction as a tabulated scalar in the stratified flame tabulation. The prior test results show that the method can reconstruct the strained flame well, and the selected scalars have clear physical meaning and can be solved readily in the computational fluid dynamics simulation. The method avoids choosing the tabulated scalars subjectively, and is quite promising for premixed and stratified combustion simulations.

The proposed new approach can improve the simulation results of more complex configurations with more complicated strain rate fields, and is applicable for the simulation of real turbulent premixed and stratified flames.