## 3 Numerical Analysis
### Governing Equations  
The spark is often under-resolved on the CFD mesh because the initial spark size is typically modest in comparison to the cell size. Modelling the spark by burning a few cells close to the spark position reveals considerable grid and time-step sensitivity, and the diffusion flame speed may be inaccurate owing to the poor space and time resolution. Additionally, ignition happens too soon when the initial spark is lower than the size of the cell. The ANSYS IC engine k-\(\epsilon\) fluent calculates a sub-grid equation for the spark evolution to lessen this sensitivity. It is believed that the flame front of the spark is infinitely thin and completely spherical. The spark radius, \(r\), increases with time, \(t\)[18]. Continuity equation:  
\[\
abla\cdot\overset{\rightarrow}{V}=0 \tag{1}\]  
Momentum equation:  
\[\frac{\partial\overset{\rightarrow}{V}}{\partial t}+\overset{\rightarrow}{V} \cdot\
abla\overset{\rightarrow}{V}=\frac{1}{\rho}\left(-\
abla P+\mu\
abla^{ 2}\overset{\rightarrow}{V}+\rho\overset{\rightarrow}{g}\beta(\mathrm{T}- \mathrm{T}_{\mathrm{ref}})\right)+S_{m} \tag{2}\]  
Energy equation:  
\[\frac{\partial h_{\mathrm{sens}}}{\partial t}+\frac{\partial h_{\mathrm{lat}}} {\partial t}+\
abla\cdot\left(\overset{\rightarrow}{V}h_{\mathrm{sens}} \right)=\
abla\cdot\left(\frac{k}{\rho c_{p}}\
abla h_{\mathrm{sens}}\right) \tag{3}\]  
Figure 4: CI engine domain and mesh generated. \[\frac{dr}{dt}=\frac{\rho_{u}}{\rho_{b}}S_{t} \tag{4}\]  
where \(S_{t}\) is the turbulent flame speed, \(\rho u\) is the density of the burned fluid behind the flame, and \(\rho u\) is the density of the unburned fluid in front of the flame front. Through a representative volume of CFD cells, the sub-grid spark model is translated to the CFD grid. The local turbulent length scale at the spark site is estimated as the fixed diameter of this spherical container. However, the radius of the representative sphere, \(r_{t}\), is determined as follows to make sure that it is neither too big (in relation to the size of the combustor) nor too tiny (in relation to the size of the cell),  
\[r_{t}=\max\left(r_{0}+34.3\ r_{0},\min\left(\frac{1}{2}l_{t},r_{0}+10\Delta \right)\right) \tag{5}\]  
where \(l_{t}\) is the turbulent length scale and \(r_{0}\) is the initial spark radius that the user specifies [14]. An alternative is to use the spark-model text interface to provide \(rt\) as a fixed value. Since the flame speed is influenced by all of the turbulent scales present after the spark diameter reaches this size, \(l_{t}\) is used to define the representative volume. Even if the simulation period is longer than the user-specified time limit, the spark radius will rise until the length scale is achieved. It should be noted that the period supplied is solely utilized to determine the rate of the spark energy input and the moment at which this input stops. Ansys Fluent turns off the spark flame speed model at this point, modelling the flame speed across the domain using the flame speed model you chose in the species dialog box. The typical spherical volume's temperature and species composition are determined as follows [18]:  
\[\varphi=c\varphi^{b}+(1-c)\varphi^{u} \tag{6}\]  
where \(\varphi^{u}\) is the unburned composition and the equilibrium burnt composition is represented by \(\varphi^{b}\). These musical arrangements are constant in time and consistent in space. Spark energy is not needed to ignite the mixture because the thermo-chemical condition behind the spark flame front is instantly equilibrated as the spark propagates. All combustion models start off with the spark energy set to zero and the burned temperature as the equilibrium temperature. The temperature behind the spark, however, may be changed via the user interface to a positive number, in which case the equilibrium temperature will be greater [18; 19].