# NOx Formation Mechanism and Emission Prediction in Turbulent Combustion: A Review
## 6 Turbulent Combustion Simulation Approach
### Direct Numerical Simulation Method  
The direct numerical simulation method is a method that is based on the continuity equation, Navier-Stokes equations, energy equation, and transport equations of species, without considering any turbulence model. Compared with the other methods that contain turbulence models, the direct numerical simulation method introduces errors via the truncation error and round-off error. Although the DNS method is straightforward in principle, it is computationally intensive. The number of grid scales depends on the flow pattern, chemical reaction, and thermodynamics. From the view of the flow pattern, the length scale of the mesh would range from the largest length scale to the smallest one, namely from the maximum turbulence integration length scale to the minimum length scale of turbulence, which is the Kolmogorov scale Lk, relating to viscosity. These length scales span a wide number of magnitudes, which is a function of the Reynolds number. Not only the flow effect should be considered, but also the chemical reaction led to refining the mesh, increasing the quantities of mesh. From the view of the laminar flame-let concept, which was widely applied in turbulent combustion, chemical time and length scales are sufficiently smaller than the smallest length and time scales of the turbulence [61]. Hence, to ensure the accuracy of this simulation calculation, not only the length scale of the mesh should consider the Kolmogorov scale, but also the ones corresponding to the chemistry scales. This would enlarge the amount of mesh inevitably. In addition, the interaction between flow and thermodynamics that introduced the Karlovitz number should also be considered. The number of grid-point scales, N, could be evaluated by this formula:  
\[\text{N}=\left(\frac{L}{2l_{T}}\right)^{3}\text{Da}\times\text{Ka}\times\text{ Re}^{7/4}, \tag{21}\]  
where \(L\) represents the one-dimensional turbulence length, \(l_{T}\) represents the integration length scale of the fluctuating field, Da represents the Damkohler number, and Ka represents the Karlovitz number. With the computational capacity of current computers, it is imperative to regulate the mesh size within an acceptable range; the number of Damkohler, Karlovitz, and Reynolds should not be huge [62]. The DNS approach is exclusively applicable under conditions where the Reynolds number and Damkohler number are moderate, while the Karlovitz number is high. For the high Reynolds number condition, the DNS method could not be satisfied with the current computing ability [63]. The DNS method could be directly used to simulate the production of NOx and achieve the detailed structure of flames. [PERSON] et al. [64] applied the DNS method to simulate premixed combustion with methane. The authors attempted to apply three-step reactions to simulate the NO generation to reduce the calculation. When comparing the simulation results with the detailed mechanism, it was found that the abstract mechanism could not replace the detailed ones. [PERSON] et al. [65] used the DNS method to simulate three-dimensional compressible mixing layers with non-premixed hydrogen/air combustion with a detailed chemical reaction mechanism with NOx production. The results show that in the mixing layers, the formation and expansion of the combustion region by the roller vortices, and the baroclinic torque had a significant impact on NO production. The chemical reaction consists of 31 chemical species and 241 elementary reactions. [PERSON] et al. [66] applied a 3D direct numerical simulation of the syngas non-premixed jet flame to discover the NO pollutant formation at a Reynolds number of 6000, and the GRI-Mech 3.0 mechanism was applied. Not only the DNS method could evaluate NOx production, but also its generated data could be applied to build a lookup table for query and proofreading. In the research of [PERSON] et al.