Abstract
Different combustion models for large eddy simulation, including the quasi-laminar (QL), Eddy dissipation concept (EDC), and partially stirred reactor (PaSR) models, are assessed at various filter widths using direct numerical simulation (DNS). The DNS database is lean hydrogen-air turbulent flame across a wide range of Karlovitz numbers (5–239). Overall, the PaSR model performs best, except for small filter width and medium Karlovitz number conditions. The performance of the EDC model is very similar to the QL model at a relatively low turbulent Reynolds number. It is highlighted that both the EDC and PaSR models are suitable for high turbulent Reynolds number and medium Karlovitz number conditions. Theoretical analysis is carried out to explain the current observations and predict the models’ behaviors with the variation of turbulent intensity, combustion intensity, and grid resolution. Implications of the present results for modeling are highlighted.
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Acknowledgements
The numerical computations were performed using \(\pi \)-2.0 at the Center for High Performance Computing, Shanghai Jiao Tong University. The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (No. 91841303 and No. 91941301).
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This work was supported by the National Natural Science Foundation of China (No. 91841303 and No. 91941301).
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Appendix: Validation and verification
Appendix: Validation and verification
Validation and verification of the numerical simulations in the current study are given here. The solver, chemical kinetics, and transport properties for the accurate simulation of combustion are validated first. Then the linear forcing method employed to maintain statistically steady turbulence is validated. Finally, the numerical accuracy is verified.
Our customized OpenFOAM solver was used to simulate a one-dimensional laminar flame at different equivalent ratios and a two-dimensional laminar flame with initial perturbation, which induces the thermo-diffusive and Darrieus-Landau instabilities (see ref. Burali et al. (2016) for detailed descriptions of the two-dimensional case). The results are shown in Fig. 9. It can be seen that the simulated laminar flame speed agrees with the PREMIX solver (Kee et al. 1985) and the experiment (Krejci et al. 2013) results perfectly at various equivalent ratios. For the two-dimensional case, the variation of the flame position agrees well with the simulated results of the NGA solver (Desjardins et al. 2008) with mixture-averaged transport properties.
The linear forcing method is validated by simulating a corresponding non-reactive case of the present study, which is statistically steady homogeneous isotropic turbulence with inflow and outflow. The computational domain, mesh, and forcing parameters are the same as Case 4. The time variation of the ensemble-averaged turbulence properties \(k, l_t\), and the energy-spectrum are demonstrated in Fig. 10. The turbulence properties are averaged over the whole computational domain while the energy spectrum is calculated in different (y, z) planes at the end of the simulation. The turbulent properties reach statistical stationery quickly. The integral length settles to nearly 20% of the computational domain width, corresponding to previous studies in a cubic computational domain (Carroll et al. 2013; Rosales and Meneveau 2005). The scaling of \(l_t\) with respect to the computational domain length (Klein et al. 2017) is not observed in the current study. This may be attributed to the differences in the streamwise boundary conditions. In this study, the Dirichlet velocity inflow and the advective outflow avoid the accumulation of the interaction between different wavelengths. The energy spectrum at different (y, z) planes is almost identical, with an apparent Kolmogorov \(-5/3\) power decay in the inertial range. Figure 10 verifies the capability of the forcing method used in this study to produce statistically stationary one-dimensional homogeneous isotropic turbulence.
In this study, the turbulence is forced across the flame, like the previous simulations (Savard et al. 2015; Aspden et al. 2019). Such setup has raised some concerns regarding the physical fidelity of the turbulent combustion region under forcing. Klein et al. (2017) argued that forcing inside the flame may lead to an unrealistically high level of turbulence on the burned side combined with too small turbulent scales. In this sense, forcing only on the unburned side to make the turbulence freely penetrate the flame is more recommended, allowing the increase of integral length scales across the flame front.
Although forcing on the unburned side would seem more appropriate, the authors have encountered some practical difficulties in the present study. When the forcing ends, the turbulence decay too fast (in the distance \(\sim \)0.001 m) to get desired turbulent properties in the flame region (with the length scale \(\sim \)0.008 m), especially at high turbulence intensity. This disadvantage of the regional forcing was also mentioned by Klein et al. (2017). On the other hand, we estimate the relative effect of forcing in turbulent evolution according to the vorticity transport equation (Bobbitt et al. 2016). The viscous dissipation term \(T_v\) and vortex stretching term \(T_s\) scales as \(\tau _{\eta }^{-3}\) while the forcing term \(T_f\) scales as \(0.5\tau _t^{-1} \tau _{\eta }^{-2}\), where \(\tau _t\) and \(\tau _{\eta }\) are large scale eddy turnover time and Kolmogorov time scale, respectively. This leads to \(T_v / T_f \propto T_s / T_f \propto 3\sqrt{Re_t} \). In our simulations, this ratio is 14 for Case 1 and 51 for Case 4. Thus, the effect of the forcing is relatively negligible in our simulations, given that our purpose is to examine the turbulent combustion models.
The current DNS is carried with OpenFOAM and the overall numerical accuracy is \(2^{\mathrm{nd}}\) order. There are studies that use \(2^\mathrm{nd}\) order code (Aspden et al. 2011; Savard and Blanquart 2015) or OpenFOAM (Zhang et al. 2015; Vo et al. 2016) to perform DNS of turbulent combustion. The reasonable turbulence energy spectrum demonstrated in Fig. 10 also confirms our capability to perform DNS with OpenFOAM.
The grid quality can be considered in two aspects: resolving the flame and the turbulence. There are more than 20 cells in the thermal flame thickness for Cases 1-3 and more than 40 for Case 4, sufficient for resolving the flame according to previous DNS studies (Vo et al. 2016; Aspden et al. 2011; Savard and Blanquart 2015). As for the resolving of turbulence, it is suggested that the grid of DNS is good enough at \(\Delta x / \eta < 2.1\) (Pope 2001). In our simulations, this criterion is satisfied in all the cases.
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Liu, H., Yin, Z., Xie, W. et al. Numerical and Analytical Assessment of Finite Rate Chemistry Models for LES of Turbulent Premixed Flames. Flow Turbulence Combust 109, 435–458 (2022). https://doi.org/10.1007/s10494-022-00329-7
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DOI: https://doi.org/10.1007/s10494-022-00329-7