## 2 Materials and Methods
For \(V_{k,i}\) holds :  
\[V_{k,i}\ =\ -D_{k}\frac{\
abla X_{k}}{X_{k}}+V_{i}^{c} \tag{9}\]This results from Equation (7) leads to the corrected energy conservation equation, where \(h_{s}\) is the sensible enthalpy:  
\[\frac{\partial\rho h_{s}}{\partial t} +\frac{\partial\rho K}{\partial t}-\frac{\partial p}{\partial t}+ \
abla(\rho uh_{s})+\
abla(\rho uK) \tag{10}\] \[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]0  
The thermal diffusion is calculated by the ratio of the thermal conductivity \(\kappa\) and the specific heat capacity \(c_{p}\)[16]. \[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]1  
For a more accurate calculation, the specific heat capacity \(c_{p}\) is assumed not to be constant in this work, but is calculated as a temperature-dependent polynomial according to Equation (12). \[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]2  
The coefficients \(a_{0}\),...,\(a_{4}\) are based on the Burcat database [21]. For improved prediction of the heat capacity, the coefficients are divided into a low and high-temperature range. While for the calculation without considering component-specific diffusion, the diffusion coefficient \(D_{k}\) is assumed to be equal to the thermal diffusion. In this model, \(D_{k}\) is based on the Schmidt number. It is not calculated as a binary diffusion but it approximates the mixing process of the specific species into the rest of the gas mixture and is given by:  
\[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]3  
where \(\mu\) is the dynamic viscosity and \(S_{c}\) the specie specific Schmidt number. \(Sc_{k}\) is the specific Schmidt number. For this, according to [PERSON] et al. [22] and [PERSON] et al. [16] the following values N\({}_{2}\) = 0.81, H\({}_{2}\) = 0.21, O\({}_{2}\) = 0.76 and H\({}_{2}\)O = 0.63 as been selected. The dynamic viscosity depends on the temperature and is calculated by the Sutherland's law (14), where \(A_{s}\) and \(T_{s}\) are the Sutherland coefficients  
\[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]4  
Only the H\({}_{2}\)O molecules could theoretically increase hydrogen flame radiation. However, this contribution was found to be very limited [16; 18], and is accordingly neglected in this work. Since the combustion of hydrogen is involved, a 1-step mechanism is assumed for the reaction kinetics. In addition, the reaction is considered irreversible. One reaction model in OpenFOAM that uses this method to calculate the reaction rate is the Eddy Dissipation Diffusion model. In this modeling, the magnitude of the reaction rate is based on the time scaling of the diffusion. Here, the reaction rate is given by Equation (15). \[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]5  
\(r_{s}\) is the stoichiometric oxygen-fuel mass ratio and the diffuse mixing time scalar \(\tau_{diff}\) is based on the equation:  
\[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]6  
The model constant \(C_{d}\) is based on experimental results [23]. For the investigation purposes, a 3-dimensional cylindrical numerical domain (Figure 1) was defined. The micro-nozzle with parallel jet flows of hydrogen and air was placed at the bottom of the domain. The edges of the combustion chamber, except for the base side, are defined as open. The entire computational domain, as shown in Figure 1, is meshed with 41,000 nodes. Conducted mesh studies indicate that a further increase in the mesh resolution does not affect the results. For more efficient calculation, the area with the high required accuracy, such as that of the reaction zone, is more finely meshed than the area before the outlet.