New Dynamic Scale Similarity Based FiniteRate Combustion Models for LES and a priori DNS Assessment in Nonpremixed Jet Flames with High Level of Local Extinction
Abstract
In this work, the performances of two recently developed finiterate dynamic scale similarity (SS) subgrid scale (SGS) combustion models (named DB and DC) for nonpremixed turbulent combustion are a priori assessed based on three Direct Numerical Simulation (DNS) databases. These numerical experiments feature temporally evolving syngas jet flames with different Reynolds (Re) numbers (2510, 4487 and 9079), experiencing a high level of local extinction. For comparison purposes, the predicting capability of these models is compared with three classical nondynamic SS models, namely the scale similarity resolved reaction rate model (SSRRRM or A), the scale similarity filtered reaction rate model (SSFRRM or B), and a SS model derived by the “test filtering” approach (C), as well as an existing dynamic version of SSRRRM (DA). Improvements in the prediction of heat release rates using a new dynamic model DC are observed in high Re flame case. By decreasing Re, dynamic procedures produce results roughly similar to their nondynamic counterparts. In the lowest Re, the dynamic methods lead to higher errors.
Keywords
Dynamic scale similarity combustion model Finiterate SGS combustion model A priori DNS analysis LES Nonpremixed jet flame Extinction reignition1 Introduction
In the governing equations of Large Eddy Simulation (LES) of reactive flows with detailed kinetics, transport equations of the filtered species mass fractions are solved. The equation for k th species, has a filtered production/consumption rate, i.e. \(\overline {\dot {\omega } }_{k}\left (\boldsymbol {\varphi } \right ) \), which needs to be modeled. In \(\overline {\dot {\omega } }_{k}\left (\boldsymbol {\varphi } \right ) \), φ is the composition vector together with the temperature (T) and pressure (p), and \(\overline {(.)} \) a filtering operator in LES. Since the net formation rates of species are nonlinear functions of φ,the equality \(\overline {\dot {\omega }}_{k}\left (\boldsymbol {\varphi } \right )=\dot {\omega }(\overline {\boldsymbol {\varphi }}^{f} ) \), which is called the “no model” or “quasi laminar” approach, in many conditions does not hold. In the previous statement, \(\overline {(.)}^{f} \) is a Favre filtering operator defined as \(\overline {\rho (.)}/\overline {\rho }\) with ρ the density. Subgrid scale (SGS) combustion models try to include the effects of turbulence on chemistry in such a way that \(\overline {\dot {\omega }}_{k}\left (\boldsymbol {\varphi } \right )\) can be computed by using \(\overline {\boldsymbol {\varphi }}^{f}\).
Subgrid scale (SGS) combustion models may be divided into two main classes [1, 2], namely the flameletbased and finiterate combustion models. The assumption of mixed is burnt (in nonpremixed flames) leads to the widely used flameletbased combustion models, treating a turbulent flame as an ensemble of thin laminar flames called flamelets. The consideration of turbulent conditions is achieved by using a probability density function (PDF) of parametrized progress variables, e.g., a conserved scalar like mixture fraction where a presumed form is usually applied. A transport equation for the PDF can also be solved but remains computationally expensive. The main challenges in the flameletbased models are in the two steps of building the table and accounting for the effects of turbulencechemistry interactions (TCI) for which a priori knowledge of the flow and the flame behavior is needed. Furthermore, for concurrent nonpremixed and premixed combustion regimes, the flamelet approach requires suitable but complex modifications, as reported in [3, 4, 5].
In contrast to the flameletbased models, the finiterate combustion models in LES try to evaluate \(\overline {\dot {\omega } }_{k}\left (\boldsymbol {\varphi } \right )\) as a function of filtered quantities which are available after LES solution. In finiterate combustion models, which are the focus of the present study, there is no assumption about the flow or flame, but instead, they attempt to model the lowpass filtered net formation rates [1]. Models in this class are the Thickened Flame Model (TFM) [6, 7], the Transported PDF (TPDF) models [8, 9], the Eddy Dissipation Concept (EDC) model [10], the Partially Stirred Reactor (PaSR) model [11, 12], and the Scale Similarity (SS) models [13, 14]. The TFM was developed primarily for turbulent premixed flames. The TPDF models can deal with premixed, nonpremixed and multiregime combustion, but they are reported to be computationally very expensive. The EDC and PaSR, which have been developed for Reynolds Average Naiver Stokes (RANS), are now being extended to LES.
In their general form (i.e., Bardina’s approach [15] for the closure of the SGS stress field), Scale Similarity models are soft deconvolution methods [16, 17], which use low order approximations to reconstruct the exact field based on filtered fields. In Bardina et al. [15] approach, the exact velocity field in the SGS tensor is replaced by its “grid filtered” counterpart. In other words, the SGS stress tensor is replaced by the modified Leonard stress in Germano’s decomposition [18]. Liu et al. [19], based on the decomposition of the velocity field in logarithmic bands, used test filters (filters with larger withds than the grid) instead, in the definition of the modeled SGS stress.^{1} SS models of these types were found to capture well the locations where the contribution of the SGS stress field is high, using a priori assessments (see, e.g. [15]), as well as experimental data analyses (see, e.g. [19]). However, the drawback of the original models was their slight dissipative characters. It is argued that the SS models are actually low order soft deconvolution models and the information which are lost by the inherent grid filter is unrecoverable in LES [17]. So each soft deconvolution model needs a complementary model to handle the lost data. This is the idea behind “mixed models” which take the advantages of both the SS models and the eddy viscosity type models.
In SGS combustion modeling, first, DesJardin and Frankel [13] used the SS idea to model \(\overline {\dot {\omega }}_{k}\left (\boldsymbol {\varphi } \right )\) in LES. They proposed two SS combustion models, namely the scale similarity resolved reaction rate model (SSRRRM), hereafter model A, and the scale similarity filtered reaction rate model (SSFRRM), hereafter model B, which will be presented in detail in Section 3. In their formulations, they were inspired by the original approach proposed by Bardina et al. [15] however, the models also contain complementary parts to account for the lost data. Germano et al. [20] proposed the third model, model C, which will be better explained in Section 3. The three mentioned SS models hereafter will be called nondynamic SS models. In [21], a comprehensive a priori study of the three mentioned nondynamic SS models was performed using a DNS database of a temporally evolving jet flame experiencing a high level of local extinction. The analysis was carried out using different filter widths in two instants of extinction and reignition. It was observed that the nondynamic SS models can capture correctly the locations where the SGS combustion effects prevail. In particular, the SS models following Bardina’s “grid filtering” approach (models A and B) were found to have lower errors than the SS model derived according to Germano’s “test filtering” approach (model C).
In the nondynamic scale similarity formulations, the model coefficient is by default set to one (see e.g., [13, 21, 22]). Jaberi and James [14] extended the nondynamic model A by proposing a dynamic version. In the dynamic model, hereafter called DA2, the similarity coefficient is evaluated based on Germano’s identity. The model was tested merely using a DNS database of homogeneous isotropic compressible reacting turbulent flow with one step Arrhenius reaction. The results were reported to be in a good agreement with those obtained by the DNS.

The nondynamic finiterate SS SGS combustion models were found to yield good predictions for the direct closure of the filtered species production/consumption and heat release rates [13, 21]. One way to check if the models can be improved is through the dynamic evaluation of the SS coefficients. The first version of Jaberi and James [14] (the dynamic SSRRRM or DA2) was tested in an ideal test case of homogeneous isotropic turbulence with one step reaction. There has been no comprehensive study of the performance of the model in practical combustion regimes like jets with multispecies/multireactions.

Of particular interest is the development of dynamic version of model B since the nondynamic B was found to predict well minor species [21].

Since the dynamic SS models include explicit filtering (sometimes up to 4 stages), which is computationally not easy to implement and also timeconsuming in LES, it is important to understand whether the dynamic procedures are effective or not.

developing dynamic versions of finiterate SS SGS models B and C,

assessing their prediction capability by using 3 DNS databases of complex 3D temporal nonpremixed jets in which the flames experience a high level of local extinction. A skeletal mechanism (similar to the reference DNS) with 11 species and 21 reactions [23] is used in the current study, which made possible the direct assessment of the models in prediction of the filtered source terms of major species as well as radicals.
The paper is organized as follows. The DNS databases are introduced in the next section. The nondynamic SS SGS finiterate combustion models are presented in Section 3, followed by a detailed derivation of new dynamic models. The comparison metrics used in the current study are introduced in Section 4 and an a priori DNS assessment is carried out in Section 5. Finally, conclusions will be drawn in the last section.
2 A priori Analysis and the DNS Databases
DNS databases of reactive flows with relatively detailed chemistry, which are now available thanks to massively large parallel computational resources, can be utilized to assess combustion models for LES of reactive flows. The two main ways of DNS data utilization are through a priori and a posteriori tests [24]. In this work, an a priori analysis is adopted by comparing “modeled” targets (i.e., filtered combustion and heat release rates) with the “exact” filtered ones from DNS databases. Modeled targets make use of directly filtered quantities from DNS databases. The main drawback in a priori analyses is their inability to predict the time properties of subgrid closures [25]. It is not guaranteed that if a model performs well in a priori analyses it will also perform well in a real LES. The opposite is also possible, this means that some models may fail in a priori DNS analyses while in LES giving acceptable results (see, e.g. [26]). On the other hand, the advantage of a priori analyses is that target models can be assessed in an isolated system, to a good extent, free from errors or uncertainties caused by other models. In a special case of SGS combustion modeling, uncertainties regarding applied turbulence models are skipped and one can focus directly on the performance of a combustion model itself. If a model is considered as a system with inputs and outputs, it is of interest to study the outputs while the inputs are free from errors. Considering a relatively small computational time required for the a priori analysis, it is a very good choice for comparing the models’ performance. Many previous studies have been performed a priori analyses to study the performance of SGS combustion models [27, 28, 29, 30, 31, 32]. Scale Similarity combustion models have been a priori tested in a 2D DNS/LES of a spatial jet [13] and a 3D DNS/LES of isotropic decaying reactive flow [14], both using small LES grid filters \({\overline {\Delta }}\) (\({\overline {\Delta }}=3{\Delta }_{\text {DNS}}\) in [13] and \({\overline {\Delta }}=4{\Delta }_{\text {DNS}}\) in [14]) and also a single step chemistry. In the present study, SS models (will be introduced in the next Section) are tested using a larger filter width of \({\overline {\Delta }}=12{\Delta }_{\text {DNS}}\) and also in challenging test flames experiencing a high level of local extinction.
The specification of the DNS databases [23] used in the current study
Case L  Case M  Case H  

Initial jet width (H) [mm]  0.72  0.96  1.37 
Initial fuel and oxidizer temperature [K]  500  500  500 
Mesh (n_{x} × n_{y} × n_{z})  576× 672 ×384  768× 896 ×512  864× 1008 ×576 
DNS mesh size (Δ_{DNS} × 10^{− 6}[m])  15  15  19 
Filter size (\(\overline {{\Delta }}\times 10^{6} [m]\))  180  180  228 
Initial streamwise velocity difference (U[ms^{− 1}])  145  194  276 
Reynolds number (Re)  2510  4478  9079 
Maximum turbulent Reynolds numebr (Re_{t})  92  172  318 
Comparing the scales in Fig. 1b, c and d and the filter size from Table 1, one can conclude that the filter size is about the Taylor length scale. Based on Fig. 1, the \({\Delta }/\underline {\lambda }_{f}\) increases from the low Re case (L) to the high Re case (H). In [33], an a posteriori analysis carried out using the DNS databases L and H. They chose \({\overline {\Delta }=8{\Delta }_{DNS}}\). In [34] they did LES of the H case and used two filter sizes of \({\overline {\Delta }=8{\Delta }_{\text {DNS}}}\) and \({\overline {\Delta }=16{\Delta }_{\text {DNS}}}\). In [35], they studied LESLEM of the M case DNS, using nonuniform grids with the minimum resolutions of \(5\underline {\eta }_{f}\) and \(2.5\underline {\eta }_{f}\), translating these to our notations gives Δ ≈ 5Δ_{DNS} and Δ ≈ 2.5Δ_{DNS}, respectively. In [21], the turbulent kinetic energy (TKE) spectrum of the H case DNS was constructed at the mid plane of the jet at the same time instant used in this study. The locations of the spectral cutoff filters with the same filter width as tophat kernels are shown on a loglog diagram of the compensated energy spectrum for \({\overline {\Delta }=8,12,18{\Delta }_{\text {DNS}}}\). The location of the cut off filters found to lie in the inertial range, although the range is very narrow because the Re of the DNS cases is not high. Also, the fraction of the resolved Favre mean turbulent kinetic energy was depicted using the mentioned filter widths. It was observed that using \({\overline {\Delta }/{\Delta }_{\text {DNS}}=8}\), more than 80% of the TKE is resolved. This fraction is reduced by increasing the filter width to 70% for Δ/Δ_{DNS} = 12 and 60% for \({\overline {\Delta }/{\Delta }_{\text {DNS}}=18}\). With this brief review, \({\overline {\Delta }/{\Delta }_{DNS}=12}\) seems to be a proper choice for the current study. The filter width is also suitable for the future a posteriori analyses. The chemical kinetic mechanism is the same as the one used in the DNS and has 11 species and 21 elementary reactions [23]. The computational domain of the DNS is a box with lengths 12H × 14H × 8H in Ox (streamwise), Oy (transverse), and Oz (spanwise) directions, respectively. The DNS mesh is a uniform grid with the size of Δ_{DNS} which is mentioned in Table 1. Periodic boundary conditions are used in Ox and Oz directions so that the flame is statistically 1D and Oxz planes at each Oy location can be considered as the statistically homogeneous planes to extract the statistical first and second moments.
3 Scale Similarity Closures for Reactive Flows
3.1 Nondynamic finiterate scale similarity SGS combustion models
3.2 Dynamic finiterate scale similarity SGS combustion models
To derive the dynamic versions of the previous models, the generalized Germano identity [37] can be used. A nonlinear operator \(\mathcal {N}(\boldsymbol {\varphi })\) is defined, where in general \(\mathcal {N}(\boldsymbol {\varphi }) \neq \mathcal {N}(\overline {\boldsymbol {\varphi }})\). The difference between these two is the contribution of the subgrid scales [38]. One may try to write the difference in additive form i.e., \(\overline {\mathcal {N}}(\boldsymbol {\varphi }) ={\mathcal {N}}(\overline {\boldsymbol {\varphi }})+{}_{F}\), where the “model” \({}_{F}\) can be of any type and the subscript F is used to remark that the model is defined at the F filter level, \(\overline {(.)}\), with cutoff \(\overline {{\Delta }}\). A second filter called “test filter”, at G level, with cutoff larger than F level, i.e. \(\widehat {{\Delta }}>\overline {{\Delta }}\), is introduced, which is denoted by \(\widehat {(.)}\). If one filters the nonlinear operator using the test filter, the operator at the FG level is derived, i.e., \(\widehat {\overline {\mathcal {N}}}(\boldsymbol {\varphi }) =\widehat {\mathcal {N}}(\overline {\boldsymbol {\varphi }})+\widehat {{}_{F}}\). On the other hand, the nonlinear operator can be directly defined at the FG level, i.e., \(\widehat {\overline {\mathcal {N}}}(\boldsymbol {\varphi }) ={\mathcal {N}}(\widehat {\overline {\boldsymbol {\varphi }}})+{}_{FG}\). The generalized Germano identity is defined by equating the two expressions of \(\mathcal {N}(\boldsymbol {\varphi }) \) at the FG level, i.e., \(\widehat {\mathcal {N}}(\overline {\boldsymbol {\varphi }})+\widehat {{}_{F}}={\mathcal {N}}(\widehat {\overline {\boldsymbol {\varphi }}})+{}_{FG} \) [38]. The identity can be used to evaluate the coefficients in the “model” part of the filtered operator. Here it should be noted that \({}_{FG}\) is by definition the scale similarity model defined at the FG level [39]. Some authors proposed to use the information at the F level instead of FG level to reduce model complexity [14, 40]. Below, this will be further investigated.
In the context of scale similarity models in reactive flows, \( \mathcal {N}(\boldsymbol {\varphi } )=\dot {\omega }(\boldsymbol {\varphi }) \) and \({}_{F}=C{}_{\dot {\omega }} \) defined above. Replacing the definitions in the generalized Germano identity, dynamic models will be derived.
3.2.1 DA: Dynamic formulation of SSRRRM (model A)
It should be mentioned that in Eq. 16, Λ_{A1} is a similarity closure for FG level based on the fields defined at FG level. This is the mathematically consistent formulation suggested by Vreman [39] in computations of SGS stress fields using SS models. In this paper, it is applied to nonlinear chemical formation rates to see its effects.
3.2.2 DB: Dynamic formulation of SSFRRM (model B)
3.2.3 DC: Dynamic formulation of model C
It should be emphasized that Υ_{C} = Υ_{A} for two models A and C. This is because both models decompose the nonlinear term \(\overline {\mathcal {N}}\left (\boldsymbol {\varphi }\right )\) in a similar way i.e. \(\overline {\mathcal {N}}\left (\boldsymbol {\varphi }\right )=\mathcal {N}\left (\overline {\boldsymbol {\varphi }}\right )+{}_{F}\). The difference between the models comes from the way they treat the residual field i.e. \({}_{F}\).
Summary of the SGS combustion models used in the current study
model  Definition  Residual field (\({{}}_{\dot {\boldsymbol {\omega }}}\))  Similarity constant  Υ  \(\boldsymbol {\mathcal {X}}\) 

Nondynamic A  
(DesJardin and Frankel [13])  \({\overline {\dot {\omega }}}^{A}(\boldsymbol {\varphi })=\dot {\omega }\left ({\overline {\boldsymbol {\varphi }}}^{f}\right )+C^{\overline {{\Delta } }}_{A}\ {{}}_{{\dot {\omega }}^{A}}\)  Eq. 7  \(C^{\overline {{\Delta } }}_{A}=1\)  –  – 
Dynamic A1  
(Present work)  \({\overline {\dot {\omega }}}^{DA1}(\boldsymbol {\varphi })=\dot {\omega }\left ({\overline {\boldsymbol {\varphi }}}^{f}\right )+C^{\overline {{\Delta } }}_{DA1}\ {{}}_{{\dot {\omega }}^{A}}\)  Eq. 7  \(C^{\overline {{\Delta }}}_{DA1}={\underline {{{\Upsilon }}_{\mathrm {A}}{\mathcal {X}}_{A1}}}/{\underline {{\mathcal {X}}_{A1}{\mathcal {X}}_{A1}}}\)  Eq. 15  Eq. 22 
Dynamic A2  
(Jaberi and James [14])  \({\overline {\dot {\omega }}}^{DA2}(\boldsymbol {\varphi })=\dot {\omega }\left ({\overline {\boldsymbol {\varphi }}}^{f}\right )+C^{\overline {{\Delta }}}_{DA2}\ {{}}_{{\dot {\omega }}^{A}}\)  Eq. 7  \(C^{\overline {{\Delta } }}_{DA2}={\underline {{{\Upsilon }}_{\mathrm {A}}{\mathcal {X}}_{A2}}}/{\underline { {\mathcal {X}}_{A2}{\mathcal {X}}_{A2}}}\)  Eq. 15  Eq. 23 
Nondynamic B  
(DesJardin and Frankel [13])  Eq. 9  \(C^{\overline {{\Delta } }}_{B}=1\)  –  –  
Dynamic B1  
(Present work)  Eq. 9  \(C^{\overline {{\Delta } }}_{DB1}={\underline {{{\Upsilon }}_{\mathrm {B}}{\mathcal {X}}_{B1}}}/{\underline { {\mathcal {X}}_{B1}{\mathcal {X}}_{B1}}}\)  Eq. 26  Eq. 33  
Dynamic B2  
(Present work)  Eq. 9  \(C^{\overline {{\Delta } }}_{DB2}={\underline {{{\Upsilon }}_{\mathrm {B}}{\mathcal {X}}_{B2}}}/{\underline { {\mathcal {X}}_{B2}{\mathcal {X}}_{B2}}}\)  Eq. 26  Eq. 34  
Nondynamic C  
(Inspired from [20])  \({\overline {\dot {\omega }}}^{C}(\boldsymbol {\varphi })=\dot {\omega }\left ({\overline {\boldsymbol {\varphi }}}^{f}\right )+C^{\overline {{\Delta } }}_{C}\ {{}}_{{\dot {\omega }}^{C}}\)  Eq. 7  \(C^{\overline {{\Delta } }}_{C}=1\)  –  – 
Dynamic C1  
(Present work)  \({\overline {\dot {\omega }}}^{DC1}(\boldsymbol {\varphi })=\dot {\omega }\left ({\overline {\boldsymbol {\varphi }}}^{f}\right )+C^{\overline {{\Delta } }}_{DC1}\ {{}}_{{\dot {\omega }}^{C}}\)  Eq. 7  \(C^{\overline {{\Delta } }}_{DC1}={\underline {{{\Upsilon }}_{\mathrm {C}}{\mathcal {X}}_{C1}}}/{\underline { {\mathcal {X}}_{C1}{\mathcal {X}}_{C1}}}\)  Eq. 37  Eq. 39 
Dynamic C2  
(Present work)  \({\overline {\dot {\omega }}}^{DC2}(\boldsymbol {\varphi })=\dot {\omega }\left ({\overline {\boldsymbol {\varphi }}}^{f}\right )+C^{\overline {{\Delta } }}_{DC2}\ {{}}_{{\dot {\omega }}^{C}}\)  Eq. 7  \(C^{\overline {{\Delta } }}_{DC2}={\underline {{{\Upsilon }}_{\mathrm {C}}{\mathcal {X}}_{C2}}}/{\underline { {\mathcal {X}}_{C2}{\mathcal {X}}_{C2}}}\)  Eq. 37  Eq. 40 
4 Metrics for Statistical Analysis
5 Results and Discussions
5.1 Performance of different variants of dynamic models
5.2 Comparison of different SS SGS combustion models for flows with different Re
In this section, the data of three flames at t = 20t_{j} with different Reynolds numbers are used to assess the performance of dynamic and nondynamic scale similarity models introduced in Table 2 to model filtered production/consumption rate of species. As mentioned before, increasing Reynolds number from case L to H is equivalent to the increase of \({\overline {\Delta }}/\underline {\lambda }_{f}\). The analysis is a priori testing of models using filter width \({\overline {\Delta }}/{\Delta }_{\text {DNS}}\)= 12. The time instant for case H and M is when the flame experiences maximum local extinction and for case L is close to it (as also shown in Fig. 1a). The flames at this time instant are in the fully turbulent, selfsimilar regime [43]. H_{2} (fuel) is selected as major specie, whereas O and OH are selected to assess the performance of models in prediction of radicals net production rates. The results for other species are presented in Online Resource 1. Different metrics have been used to compare the performances of different SS models. These include the pointwise metrics together with conditional means in composition space. The data were confined to a region where \({\underline {Z}}_{f}\ge 0.02\), with \({\underline {Z}}_{f}\) being the Favre mean mixture fraction. The “quasi laminar” or “no model” approach is also used as the base model. In the “no model” approach, SGS effects are neglected and \({\dot {\omega }}^{noModel}\left (\boldsymbol {\varphi }\right )=\dot {\omega }({\overline {\boldsymbol {\varphi }}}^{f})\).
Regarding model A, the error of its dynamic version becomes comparable to its nondynamic version by increasing Reynolds number (increasing \({\overline {\Delta }}/\underline {\lambda }_{f}\)), especially in the core of the jet. It is seen that in some locations DA2 produces larger mean local errors. Moreover, the error of DA2 is comparable to the “no model” approach in low and medium Reynolds numbers. The results show that the dynamic version of model A has no advantage over its nondynamic version for these test cases.
Regarding model B, the performance of DB2 is worse than nondynamic model B, especially in low and medium Reynolds number. The error of model DB2 is even higher than the “no model” approach, while nondynamic model B has lower error than the “no model” approach in the shear layers. In the core of the jet, both models DB2 and B show large errors, higher than the “no model” approach. Again, the results show that the dynamic version of model B has no advantage over its nondynamic version.
Regarding model C, in the low and medium Reynolds numbers, the SS models C and DC1, produce large local errors. However, in the high Reynolds number case (Fig. 4c) the error of model DC1 becomes comparatively low. This shows that using Germano’s test filtering approach in derivation of SS model C is more suitable for high Reynolds numbers.
Finally, as it can be seen for the fuel (H_{2}) net production rate, regardless of the Reynolds number, the nondynamic model A has the lowest locally incurred error among all tested models. All dynamic models (except model DC1 in high Reynolds case H) locally produce larger errors than their nondynamic counterparts. The same behavior is observed for H_{2}O (product). The analysis is also carried out for CO_{2}. The comparisons are extensively reported in Online Resource 1. In summary, in agreement with what observed for H_{2} and H_{2}O, there is no major improvement observed by applying dynamic versions of the three similarity models studied in this paper. Furthermore, among all the models, the nondynamic model A has the lowest mean of locally incurred errors among all the models studied. The same behavior is observed for CO (fuel) and O_{2} (oxidizer).
To sum up, the error of DC1 decreases by increasing the Re. In case H (the highest Re in the three cases), for some major species, the error becomes lower than nondynamic model C. It is interesting to see that in the flame regions (the regions between vertical blue and red lines in the plots) DC1 produces less local errors for medium and high Re cases almost for all species (see Figs. 4 and 5). DB2 shows local errors higher than (for radicals) or at most equal (for major species) to model B. This can be due to four filtering levels in the mathematical formulation of DB2; little information is remained to be used for the prediction of the true coefficient (see Eq. 33). Similar to DC1, the mean local error of DA2 decreases by increasing the Re, however, unlike DC1, here the local errors approaches to those of model A.
The scatter plots (see Online Resource 1), comparing the modeled source terms for different species plotted versus the exact filtered source terms from the three DNS databases, also show the same results. By increasing the Re, improvements can be observed. The improvement is more pronounced in major species than radicals (see Online Resource 1.). In most species, between a nondynamic model and its nondynamic counterpart, except for model C, no visible improvement was observed. For major species the correlation coefficients are already high. However, for radicals it seems that non of the models perform well when looking at the scatter plots.
6 Summary and Conclusions
In this paper the finiterate dynamic Scale Similarity (SS) SGS combustion models for LES were developed and a priori tested using the DNS databases of nonpremixed turbulent syngas jet flames with varying degrees of extinction and different Reynolds numbers. An explicit tophat filter with relatively large filter width (\({\overline {\Delta }}\)/Δ_{DNS} = 12), compared to the previous studies on SS models, was applied to compute the exact LESlike filtered quantities from the DNS databases. Two variants of dynamic versions for each existing nondynamic model were derived. In particular, three classical nondynamic models were: (i) the SS models derived following Bardina’s grid filtering approach [15], namely the scale similarity resolved reaction rate model (model A), (ii) the scale similarity filtered reaction rate model (model B) proposed in [13], and (iii) a SS model derived using the test filtering approach [20, 21] (model C). Two versions of dynamic models (DA1, DA2, DB1, DB2 and DC1, DC2) resulted from using different filtering levels in the mathematical formulations. Comparing the two dynamic versions of each SS model, it was found that DA2, DB2 and DC1 produce better results. This demonstrates that the mathematically consistent formulation of the SS models for combustion does not always improve the results like what was seen before for SGS stress field [39]. It should be mentioned that among the three dynamic models, only DC1 was derived using the mathematically consistent formulation (see Section 3.2.3).
Comparisons were made with the nondynamic models as well as the existing dynamic one (model DA2) [14]. The focus was on the assessment of the ability of different SS models in the prediction of filtered net formation rate of major species, radicals and also the filtered heat release rates in flames with extinction.
Considering the conditional mean heat release rates (see Fig. 8), the SS models could predict correctly (in mean) the filtered heat release rates, while in all three flames, the “no model” approach predicted higher heat release rates. It was observed that by increasing the Re (increasing \({\overline {\Delta }}/\underline {\lambda }_{f}\)), the error of both dynamic and nondynamic models decreased. DC1 produced better results than the nondynamic C for case H (see e.g. Fig. 8b and c). However, for the other two models, in the best case the dynamic procedure produced results similar to their nondynamic counterparts. It should be mentioned that in these DNS cases, there is more extinction in the higher Re conditions than the lower Re one, showing progressively more flameletlike behavior. So it is possibly the case that it is the flameletlike behavior that is more challenging rather than lower Re. This needs to be further studied in the future. It seems that the specific test cases considered here are not suitable to reveal the true potentials of the new dynamic procedures. One may conclude that the dynamic models can at least converge to the best predictions which here resulted from the default similarity coefficient of 1. The optimal estimators concept [44] can be exploited to find the minimum achievable error by the SS models for these specific databases.
 (i)
The error of DC1 decreased by increasing the Re and interestingly, in the flame regions, produced less mean local errors for medium and high Re cases. This is consistent with observations regarding the prediction of filtered heat release rate.
 (ii)
The mean local error of DA2 decreased by increasing the Re, however, unlike DC1, the local errors became similar to the nondynamic A model. It seems that for the specific test cases considered in this study, using the default value of 1 for the similarity coefficient of model A produces the best results and the dynamic evaluation of the coefficient leads to the same mean local error.
 (iii)
DB2 showed local errors higher than or at most equal to B. No improvement was observed by increasing the Re. This can be due to four filtering levels in the mathematical formulation of DB2.
 (i)
It is expected that the dynamic procedures presented in the current study, produce acceptable results in higher Reynolds than the ones considered here. The highest Re in this study is around 9000. One may argue that the encouraging results of the SS SGS combustion models observed in the current study may be due to the specific DNS test cases in which the scales are overlapped and there is possibly not enough scale separation. It is true that these DNS databases lack a distinct scale separation, however, one should also take this into account that in this study it was observed that the performance of the models are better in higher Reynolds cases M and H. Furthermore, by increasing the Re, the \({\overline {\Delta }}/\underline {\lambda }_{f}\) is also increasing. So it is reasonable to expect better performances in higher Reynolds numbers. It is so suggested to do this analysis to draw a solid conclusion on the effect of the Re on the performance of the new models.
 (ii)
It is of interest to study directly the effect of the width of the filter on the performance of the new dynamic models. Furthermore, the analyses should be expanded into the other regimes of nonpremixed flames like reignition phases, or flames without any extinction.
 (iii)
The application of Germano’s identity for pure SS models seems to be unsuccessful in improving the performance in low Re flames. It will be interesting to use a combination of mixed models and Germano’s identity to compute the similarity coefficient dynamically like what is done for flame surface density closure in [45].
 (iv)
To complete the assessment, it is also suggested doing a posteriori DNS analyses of the new dynamic models. Although the models (especially DA2 and DC1) produced acceptable results in the a priori analysis, as mentioned earlier, it is not guaranteed that they show the same performance in a real LES.
Footnotes
 1.
The terms, “grid” and “test” filters will be defined more precisely later.
 2.
Strictly speaking, the name “double Favre filtered” may not be suitable in mathematical point of view since the second operator (thick bar) uses the filtered density in the formulations (see also Table 4). However, the operation is similar to what is done in LES codes where only filtered density is available. This is the reason the term “double Favre filtered” has been used.
Notes
Acknowledgements
The authors would like to thank Professor Evatt Hawkes from The University of New South Wales for providing us the DNS validation databases.
Funding
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No 643134.
Compliance with Ethical Standards
Conflict of interests
The authors declare that they have no conflict of interest.
Supplementary material
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