# Thrust Augmentation of Micro-Resistojets by Steady Micro-Jet Blowing into Planar Micro-Nozzle
## 3 Numerical Setup and Methodology
### Grid Sensitivity Study  
In the present work, the definition of the proper refinement level resulted from a grid independence study (GIS) based on the analysis of water vapor flow at \(\dot{m}\) = 5 mg/s, \(T_{inlet}\) = 505.58 K and \(p_{out}\) = 500 Pa. It was split into two steps:  
* a 2D GIS in the planar mid-section of the micro-nozzle, i.e., the plane (0,\(x\),\(y\)) in Figure 3;
* a 3D GIS to define the proper refinement level along the micro-nozzle depth direction (the \(z\)-direction in Figure 3). In both studies, the discretization error between two consecutive refinement levels was estimated by computing the grid convergence index (GCI), as suggested by [38]. In particular, the 2D GIS was successfully performed and discussed in the previous work [4], to which the authors refer for further details. The summary of the 2D GIS is reported in Table 3: the 2D GIS led to the choice of the intermediate refinement level corresponding to an error of about \(\text{GCI}_{12}=0.35\%\). Once we defined the refinement level in the plane (0,_x,y_), the 3D mesh was built by extrusion along the micro-nozzle depth direction, i.e., the \(z\)-axis in Figure 3. Therefore, the 3D GIS was performed to define the proper refinement level into the micro-nozzle along depth-wise direction (\(z\)-axis in Figure 3), which ensured the best trade-off between accuracy and computational cost. In this regard, the 3D computational domain composed of the micro-nozzle geometry and an outer discharge domain extending \(5W_{exit}\) upward, \(10W_{exit}\) downstream, and \(50H_{n}\) sideways, where \(W_{exit}\) refers to the width of the micro-nozzle exit, and \(H_{n}\) is the micro-nozzle depth. The 3D refinement of the mesh involved the micro-nozzle region and the thin region of the outer domain downstream of the micro-nozzle exit invested by the plume. In particular, three uniform grid steps have been considered in the depth-wise direction, i.e., 20 \(\upmu\)m, 15 \(\upmu\)m, and 10 \(\upmu\)m, which corresponded to overall 3D mesh sizes of 493,520 cells (coarse mesh level), 516,360 cells (intermediate mesh level), and 607,720 cells (fine mesh level). Due to the small thickness of the computational domain affected by the mesh refinement, the overall mesh sizes do not provide indication of the refinement level. Figure 3: 3D computational domain [4]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test Case** & **Refinement** & **Cell Number** & **Grid Spacing 1 [\(\upmu\)m]** & **Spacing Factor** & **GCI** & **Parameter,** & **GCI [\%]** \\  & **Level** & & **[\(\upmu\)m]** & & & \(\theta_{xy}\)2 [\(\upmu\)m]** & \\ \hline SIM1-2D & Fine & 35,690 & 48.89 & 1 & 28.79 & \(\text{GCI}_{12}=0.035\) \\ SIM2-2D & Intermediate & 23,931 & 83.81 & 1.71 & 28.68 & \(\text{GCI}_{23}=0.514\) \\ SIM3-2D & Coarse & 17,732 & 135.38 & 2.77 & 27.06 & - \\ \hline \hline \end{tabular}  
* referred to the exit section. 2 momentum thickness \(\theta_{xy}\) at the exit section on the sidewall.  
\end{table} 
Table 3: 2D grid independence study in the symmetry plane (0,_x,y_) [4]. Therefore, the grid spacing was defined by the grid steps along the \(z\)-axis direction, leading to spacing factors of 1, 1.5, and 2 from fine to coarse refinement, as summarized in Table 4.  
The compressible momentum thickness at the exit section of the sidewall in the symmetry plane (0,\(x\),\(y\)) was used as a grid convergence index (CGI) parameter, allowing us to estimate the discretization errors. Both fine-to-intermediate error \(GCI_{12}\) and intermediate-to-coarse refinement error \(GCI_{23}\) were computed, and the minimum among them was assumed the mesh refinement error. As reported in Table 4, the refinement error was about \(GCI_{23}\) = 5.26% during the coarse-to-intermediate refinement step, and it decreased to \(GCI_{12}\) = 0.67% when the mesh was refined from the intermediate level to the fine one.