## 6 Algorithm Validation  
To validate the accuracy of the MATLAB algorithm, CFD simulations were used to determine the thermodynamic properties and turbine performance in comparison to the numerical predictions. The CFD was conducted utilising ANSYS CFX application with turbo mode and the following setup used:  
* The fluid is modelled as Air (Ideal Gas)
* K-\(\omega\) SST turbulence model was chosen. * Interfaces between stator and rotor, and rotor and outlet are selected as stage (mixing-plane). * Interfaces between stator blades and interfaces between rotor blades are set as periodic. * The meshes were setup using ANSYS Turbo Grid consisting of a total of 225,000 nodes. * Stator blade average Y + 12.2053. * Rotor blade average Y + 9.8685. Two CFD cases were run, with Case 1 using total pressure and total temperature at the inlet and a mass flow rate boundary condition at the outlet. This is the most realistic setup for simulating operating conditions, however it did not respond well to choked flow and would often result in numerical instability. The mass flow rate inlet and static pressure outlet boundary conditions were to be used during the design optimisation and turbine map generation. This is labelled as Case 2 in Figure 12.  
Using the preliminary design boundary conditions and design parameters the tip gap was varied between k = 0% and k = 5%. The results can be seen in Figure 12a,b. It is worth noting that as the MATLAB algorithm \"designs\" a turbine based on the input parameters, and k is changing during the sweep, the blade angles would also be changing. For the CFD cases only the tip gap is changed, leaving the rest of the blade geometry identical, meaning that the MATLAB code will certainly be overpredicting the efficiency compared to the CFD in cases other than k = 2.5%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**(a) Stator** & **Hub** & **Mean** & **Tip** & **(b) Rotor** & **Hub** & **Mean** & **Tip** \\ \hline Axial Chord (mm) & 6.43 & 6.43 & 6.43 & Axial Chord (mm) & 9.36 & 6.93 & 4.63 \\ Stagger Angle (\({}^{\circ}\)) & 48.96 & 48.96 & 48.96 & Stagger Angle (\({}^{\circ}\)) & 41.64 & 56.45 & 64.24 \\ Chord (mm) & 9.79 & 9.79 & 9.79 & Chord (mm) & 12.52 & 12.54 & 10.65 \\ LE Wedge (\({}^{\circ}\)) & 26 & 26 & 26 & LE Wedge (\({}^{\circ}\)) & 24 & 24 & 24 \\ TE Wedge (\({}^{\circ}\)) & 4 & 4 & 4 & TE Wedge (\({}^{\circ}\)) & 2 & 3 & 2 \\ LE Thickness (mm) & 0.35 & 0.35 & 0.35 & LE Thickness (mm) & 0.62 & 0.45 & 0.38 \\ TE Thickness (mm) & 0.17 & 0.17 & 0.17 & TE Thickness (mm) & 0.22 & 0.22 & 0.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Preliminary stator **(a)** and blade profile geometry **(b)** used for bgi generation. Figure 11: 3-D full view of preliminary design generated. It is illustrated that where k \(\leq\) 2%, the total to static efficiency is predicted with less than 1% difference to the CFD result. The gradients begin diverging after this, although this may be due to the overprediction outlined above. When looking at CFD Case 1, modelling free flow conditions, the gradient of the prediction is well matched to the results. Due to the relationship between the pressure ratio and the turbine power production, CFD Case 1 shows in the blade number sensitivity study that the power generation is heavily based on the throat area. At \(Z_{rotor}\) = 9 the power and efficiency of Case 1 results show the fluid is not being accelerated enough to expand the flow and create sufficient torque. By selecting \(Z_{rotor}\) = 17 for the design it can be said with confidence that the turbine is producing enough power, rather than being misled by the power output of CFD Case 2. CFD Case 2, by enforcing static pressure at the outlet, inflates the power value compared to what would be experienced in operation. It is also seen that significant over-acceleration of the flow with higher blade numbers causes significant efficiency losses related to the supersonic flow regime.