## 2 Materials and Methods  
To describe a continuous reactive fluid first of all the Navier-Stokes equations, containing the mass conservation Equation (1) as well as the momentum conservation Equation (2), are necessary. \[\frac{\partial\rho}{\partial t}+\
abla(\rho u_{i})\ =\ 0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\
abla(\rho u_{i}u_{j})=\rho g+\
abla \tau_{i,j}-\
abla p \tag{2}\]Thereby is \(\rho\) the density, \(u\) the velocity, \(\sigma_{i,j}\) the stress tensor, \(p\) the pressure, and \(g\) the gravity force. For the calculations, a compressible Newtonian fluid is assumed and the pressure is determined by the ideal gas law. During the combustion process, various gas components are consumed and created. To describe this process of conversion mathematically, the species mass conservation Equation (5) is implemented in OpenFOAM. The correction velocity \(V_{i}^{c}\) is introduced to ensure mass conservation since the Hirschfelder-Curtiss approximation is used to determine the diffusion velocity of the specific component in the gas mixture with an effective diffusion coefficient of the species k into the rest of the mixture. In the first step, the component mass conservation equation is modified concerning the Hirschfelder-Curtiss approximation [20]. The Hirschfelder-Curtiss assumption approximates the diffusion mass flux as follows:  
\[j_{k,i}\ =\ -\rho D_{k}\frac{W_{k}}{W}\frac{\partial X_{k}}{\partial x_{i}} \tag{3}\]  
\(W\) here is the molar mass of the total gas mixture, while \(W_{k}\) is the molar mass of the respective component. \(X_{k}\) is the molar fraction of the component \(k\) of the total mixture and is calculated via:  
\[X_{k}\ =\ \frac{W}{W_{k}}Y_{k} \tag{4}\]  
By satisfying the continuity condition, the result for the component mass conservation equation in the modified model is:  
\[\frac{\partial\rho Y_{k}}{\partial t}+\frac{\partial}{\partial x_{i}}\rho(u_{ i}+V_{i}^{c})Y_{k}=\frac{\partial}{\partial x_{i}}(\rho D_{k}\frac{W_{k}}{W} \frac{\partial X_{k}}{\partial x_{i}})+\bar{w}_{k} \tag{5}\]  
Here the correction velocity \(V_{i}^{c}\) is calculated by:  
\[V_{i}^{c}\ =\ \sum_{k=1}^{N}D_{k}\frac{W_{k}}{W}\frac{\partial X_{k}}{\partial x _{i}} \tag{6}\]  
The continuity equation of the components is then solved for \(N-1\) gas molecules. The last component is calculated via the normalization condition for numerical stability reasons. In addition to the stability aspect, the application of the component-specific diffusivity is automatically taken into account in both the physical and chemical domains when calculating the numerical combustion. In the chemical domain, it is taken into account by the calculation of the chemical production term or the reaction rate of the individual elements. In the physical domain, it is included due to the solution of the transport equation or the component mass removal equation according to the respective component [17]. Based on the assumption of a Lewis number unequal to one, the calculation of the energy flow considering the diffusion velocity results in [20]:  
\[q\ =\ -\kappa\frac{\partial T}{\partial x_{i}}+\sum_{k\ =\ 1}^{N}h_{k}\rho Y_{k}V_{k,i} \tag{7}\]  
where \(h_{k}\) the specific enthalpy. The thermal conductivity \(\kappa\) is calculated by the modified Euchen formula and is given by Equation (8), where \(c_{v}\) is the specific heat capacity at constant volume and \(R\)  
\[\kappa\ =\ \mu c_{v}(1.23+1.77(R/c_{v})) \tag{8}\]  
\(V_{k,i}\) is the diffusion velocity of the \(k\)-th component in direction \(i\).