# Numerical and Experimental Analysis of Labyrinth Seals with Rhomboidal Cells
## 4 Discussion and Conclusions  
This research presents a detailed analysis of the performance of an innovative labyrinth seal. In particular, three different patterns were proposed, alternative to the standard honeycomb, with a rhomboidal shape. The numerical models developed and validated in a previous study [14] were used to predict the performance of these innovative cells, and experimental tests were performed using the most promising alternative pattern. The parameter for the design of the rhomboidal cell was the ratio of the diagonals and, using the CFD approach, the three different rhomboidal patterns were evaluated, examining their discharge coefficient. All the cells analyzed turned out to be more dissipative than both the smooth labyrinth seals and the honeycomb ones, while maintaining the same pattern density as the honeycomb cells. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Temperature** & \multicolumn{2}{c}{**Shroud**} & \multicolumn{2}{c}{**Fin tip**} \\
**(K)** & **M9** & **M11** & **M8** & **T10** \\ \hline T\({}_{\text{HC,exp}}\) & 751.7 & 725.6 & 797.3 & 774.5 \\ T\({}_{\text{RB,exp}}\) & 749.1 & 722.0 & 795.2 & 774.2 \\ \(\Delta\)T & \(-2.9\) & \(-3.6\) & \(-2.1\) & \(-0.3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Temperature in the labyrinth seal region for test 4 and test 4 bis. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Test** & \begin{tabular}{c} **TFP** \\ **(K)** \\ \end{tabular} & \begin{tabular}{c} **pFF** \\ **(bar)** \\ \end{tabular} & \begin{tabular}{c} **\(\hat{m}_{\text{FP}}\)** \\ **(kg/s)** \\ \end{tabular} & \begin{tabular}{c} **TCoo** \\ **(K)** \\ \end{tabular} & \begin{tabular}{c} **pCoo** \\ **(bar)** \\ \end{tabular} & \begin{tabular}{c} **\(\hat{m}_{\text{Coo}}\)** \\ **(kg/s)** \\ \end{tabular} & \begin{tabular}{c} **TACC** \\ **(k)** \\ \end{tabular} & \
\begin{tabular}{c} **pFF** \\ **(bar)** \\ \end{tabular} &
\begin{tabular}{c} **\(\hat{m}_{\text{ACC}}\)** \\ **(kg/s)** \\ \end{tabular} \\ \hline
3 bis & 797 & 1.8 & 0.437 & 460 & 2 & 0.043 & 314 & 1.06 & 0.013 \\
4 bis & 807 & 1.8 & 0.441 & 567 & 2 & 0.044 & 321 & 1.05 & 0.013 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Experimental inlet conditions for the rhomboidal cells. The CFD analyses allowed highlighting that the rhomboidal cells perform better for two reasons. The first one is related to the effective clearance, since the geometrical increase parameter, \(\zeta_{\text{g}}\), for all the proposed rhomboidal patterns is around 1. In practical terms, this condition promotes the formation of stronger and, thus, more dissipative vortex structures within the seal cavities. In addition, the interaction between the carry-over and the cell wall near the fin tips introduces a second contribution related to friction, which allows the rhombi to perform even better than the smooth seal. Furthermore, the CFD analyses revealed that in the examined rhomboidal patterns, this dissipative effect increases with the ratio of the diagonals. The R3 cell is indeed the most efficient, with a 2.2% reduction in the discharge coefficient, compared to the smooth seal. Using the fluid model that reproduces the fluid dynamic behavior of the available Test Article, these numerical results were verified and confirmed for the R3 cell. The comparison of the leakage flows obtained using the honeycomb cell, and the rhomboidal cell showed that the latter is more efficient and that the fraction of the total mass flow rate that bypasses the blade is reduced by 29.4% (honeycomb) to 27% (rhombi). Using the coupled fluid-thermal model of the TA, the performances of the rhomboidal cells R3 were also evaluated from the thermal point of view. The obtained results showed that the differences with the hexagonal stator land are negligible, less than 1%. The following experimental tests confirmed the numerical forecast, with a good agreement. Conceptualization and methodology, E.C., A.D., M.P.M. and A.R. ; CFD analysis, A.D. and A.R. ; fluid-thermal coupled analysis, M.P.M. ; experimental validation, E.C.