## 4 Numerical Modeling
### Grid Independence Study  
One of the most important steps in numerical simulation is the achievement of an independent mesh. The great challenge is to find a mesh sufficiently refined so that the results generated are in good agreement with experimental or other numerical results without a high computational cost. In general, there is a moment where the numerical results are practically unchanged when compared to a previous mesh (that was coarser). Thus, the coarser mesh solves the problem with a relatively low error and with much less computational effort [32]. It is worth mentioning that, for both models, the near wall region is modeled considering a scalable wall function and the maximum wall plus is y\({}^{+}\leq 30.0\)  
Figure 6: Mesh for the sliding mesh model. (**a**) Top view of the tank. (**b**) Side view of the tank. (**c**) Top view of the impeller. (**d**) Side view of the impeller. Recently, a statistical method was defined for quantifying the influence of grid discretization on numerical problems that has been considered highly effective for quantifying how close any solution for the investigated mesh is to the independent region [32, 33]. This technique is named the grid convergence index (GCI) and it has been successfully used in recent studies to predict suitable meshes for numerical studies [34]. A total of five steps are required to apply this method, and all calculation criteria are defined by the norm ASME V&V [32]:  
Figure 7: Computational domain scheme for the momentum source model. (**a**) View of the two cut interfaces 90\({}^{\circ}\), (**b**) side interface cut 90\({}^{\circ}\) and (**c**) top view. Step 1: Define a representative cell, mesh or grid size, \(h\). Step 2: Select three significantly different sets of grid resolutions and run simulations to determine the values of key variables important (\(\varphi\)) to the objective of the simulation study. It is desirable that the grid refinement factor (\(r=h_{coarse}/h_{fine}\)) should be greater than 1.3. Step 3: Let \(h_{1}<h_{2}<h_{3}\) and \(r_{21}=h_{2}/h_{1}\), \(r_{32}=h_{3}/h_{2}\) and calculate the apparent order, \(p\), of the method by:  
\[p=\left[\frac{1}{\ln(r_{21})}\right]\left[\ln|r_{32}-\gamma_{21}|+q(p)\right] \tag{19}\]  
\[q(p)=\ln\left(\frac{r_{21}^{p}-S}{r_{32}^{p}-S}\right) \tag{20}\]  
\[S=1\cdot\text{sign}(\gamma_{32}/\gamma_{21}) \tag{21}\]  
where sign is a function that returns 1 if the value is positive, \(-1\) if the value is negative or 0 if a value is 0, \(\gamma_{32}=\varphi_{3}-\varphi_{2}\), \(\gamma_{21}=\varphi_{2}-\varphi_{1}\) and \(\varphi_{k}\) denotes the simulation value of the variable on the \(k\)th grid. Note that \(q(p)=0\) for \(r=constant\). This set of three equations can be solved using fixed point iteration with the initial guess equal to the first term (i.e., \(q=0\)). A minimum of four grids are required to demonstrate that the observed order \(p\) is constant for a simulation series. Step 4: Calculate the extrapolated values from the equation:  
\[\varphi_{ext}^{21}=\left(r_{21}^{p}\varphi_{1}-\varphi_{2}\right)/\left(r_{21 }^{p}-1\right) \tag{22}\]  
Step 5: Calculate and report the following error estimates along with the observed order of the method \(p\). Approximate relative error may be cast as a dimensionless form or in dimensioned form respectively given by:  
\[e_{a}^{21}=\left|\frac{\varphi_{1}-\varphi_{2}}{\varphi_{1}}\right| \tag{23}\]  
\[e_{a}^{21}=\left|\varphi_{1}-\varphi_{2}\right| \tag{24}\]  
The fine grid convergence index is calculated as follows:  
\[GCI_{fine}^{21}=\frac{F_{S}\cdot e_{a}^{21}}{r_{21}^{p}-1} \tag{25}\]  
where \(F_{S}=1.25\) is the factor of safety. For the present study, the application of the GCI method is presented in Table 3. The magnitude of GCI between meshes 1 and 2 was below 1.0% and the asymptotic range of convergence was around 1.0, which, according to literature [32; 33], indicates the mesh is in the independent region.