## 2 Materials and Methods
### CFD Setup  
The configuration of the CFD model, using STAR-CCM+ 14.02.010, starts by setting fluid properties, selecting air that is modeled as an ideal gas. This assumption is valid due to the low variation of static temperature in the studied cases through the rotor-less turbine. The simulation is set as a steady-state case, and the turbulence is modeled with an RANS \(k\)-\(\omega\) SST model. This model has been extensively used in turbomachinery by authors such as [PERSON] et al. or [PERSON] et al., due to its behavior when adverse pressure gradients and flow separation are present. Additionally, the inlet turbulence intensity was set with a value of 4%. A coupled implicit solver was set with a controlled CFL number to ensure stability. Furthermore, as the CFD case used for the validation is adiabatic, every wall in the domain possesses adiabatic conditions. In every wall, the option of non-slip boundary condition is enabled to consider wall friction. Besides the boundary condition of each solid wall, at the inlet region, where the duct begins, mass flow inlet condition was  
Figure 1: Printed nozzles with different vanes position.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Scaled Turbine** & **Real Turbine** \\  & **(ST)** & **(RT)** \\ \hline Blade number & \(11\) & \(11\) \\ Chord length & \(52.5\) mm & \(17.5\) mm \\ Vane height & \(24\) mm & \(8\) mm \\ Tip height & \(1.5\) and \(0.5\) mm & \(0.2\) mm \\ Tip-to-vane ratio & \(0.063\) and \(0.021\) & \(0.025\) \\ Blade angle-\(\alpha\) & \(51^{\circ}\) & \(65^{\circ}\) & \(73^{\circ}\) \\ VGT opening & \(80\%\) & \(60\%\) & \(30\%\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometry specifications.  
set, adding a total inlet temperature that fits with the test bench capacities. On the other hand, at the outlet section, static pressure was imposed with atmospheric conditions since the experimental setup was designed that way.  
For turbomachinery applications, it is crucial to obtain information on how the fluid behaves within the so-called nozzle vanes or blades when referring to the impeller. The regions composing the inner volume of the turbine, that is, volute, stator, and the place where the rotor would have been normally placed (but avoided in this design), are meshed with a unstructured polyhedral one, as shown in Figure 2a. Apart from that, inlet and outlet regions, built by extruding the corresponding sections at the intake and the outlet of the turbine, include an extruded mesh. The growth of notorious gradients in the boundary layer or separation bubbles on the stator vanes involve using a prism layer meshing strategy in the near-wall region (as shown in the vane-surrounding region in Figure 2b). A logarithmic function is implemented with the effect of having a higher cell density near the walls. A constant growth rate is applied in the normal direction to the wall until the chosen base cell size is reached in the free stream region. The size of the first cell at the wall must be selected according to the required level of accuracy for viscous effects, being thinner than the viscous sublayer, according to [PERSON] et al. [36]. Another value to be considered is the dimensionless wall distance \(y^{+}\) that, when using the \(k\)-\(\omega\) SST model, is recommended to have values \(y^{+}\) < 2, according to [PERSON] et al. [37]. However, this value can be enlarged if the complexity of the flow requires it.  
A mesh convergence analysis was carried out to obtain the most appropriate cell number, the least without interacting with the solution. A variation in the base size and the total number of cells in the prism layer meshing was performed. As previously stated, we were looking for a low Reynolds mesh.