# Numerical and Experimental Analysis of Labyrinth Seals with Rhomboidal Cells
## 2 Numerical Analysis
### CFD Analysis on Schramm Geometry
\(A\) similar definition, based on the averaged clearance \(s_{seal}\), is proposed in the present work:  
\[\zeta_{g}=\frac{s_{seal}}{s}\, \tag{6}\]  
The leakage mass flow rates, recorded in the numerical simulations, are used to calculate the discharge coefficients shown in Figure 6. Table 6 reports the percentage variation calculated with the formula reported in Equation (7) and using the smooth labyrinth seal as a reference value. \[\Delta C_{D}\%=\frac{C_{D,x}-C_{D,\ Smooth}}{C_{D,\ x}}\times 100\, \tag{7}\]  
Figure 5: Nominal and effective clearance. As known from the literature [20], stepped honeycomb labyrinth seals generally exhibit greater leakage flows than the equivalent smooth seal, when the cell diameter is larger than the clearance. The advantages of using the honeycomb pattern is mostly related to a lower density of the abradable material and better convective heat transfer within the cells [24]. The rhomboidal cells not only have these advantages, but also promote a stronger dissipation, leading to a leakage flow, which is even lower than the smooth seal. Moreover, this mechanism resulted in being more effective when rhombi with higher diagonal ratios were simulated. The reduction of the discharge coefficient for rhomboidal cells can be explained considering two different contributions. The first one is related to the shape of the cells that, unlike honeycomb cells, do not have edges perpendicular to the direction of flow. The absence of these perpendicular edges acts on the carry-over, promoting strong axial and vertical velocity gradients. Consequently, complex vorticous structures arise in the downstream cavity which are more dissipative not only than in the honeycomb cells, where this phenomenon occurs to a limited extent, but also than in smooth labyrinth seals. The second effect can be explained with a more detailed analysis of the geometrical increase parameter, shown in Figure 7. All the analyzed patterns exhibit a geometrical increase parameter approaching the value \(\zeta_{g}=1\) (i.e., the smooth configuration). In particular, the averaged seal clearance calculated for the rhomboidal cells is significantly smaller than the value recorded for the honeycomb labyrinth seal. Moreover, as the diagonal ratio increases, \(\zeta_{g}\) slightly decreases. \begin{table}
\begin{tabular}{c c} \hline \hline  & \(\mathbf{\Delta C_{d}}\) **\% (Reference = Smooth)** \\ \hline HC & \(+3.68\) \\ \hline R1 & \(-0.64\) \\ \hline R2 & \(-1.64\) \\ \hline R3 & \(-2.23\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: C\({}_{d}\) percentage variation. Figure 6: Discharge coefficient for the analyzed patterns. As pointed out by previous studies [19; 21], a smooth forward-stepped labyrinth seal shows two cavity counter-rotating vortices, which strongly dissipate the kinetic energy. This phenomenon, clearly visible in the vector velocity plot shown in Figure 8a, is due to the carry-over flowing directly towards the seal step. Here, the airflow impacts the step and is deflected by ninety degrees. The presence of any empty cell in the same typology of labyrinth seal may change this vorticous structure. When \(\zeta_{g}\gg 1\) the carry-over is already deflected by the cell wall. Although the interaction between cell wall and carry-over generates friction, which partially dissipates its kinetic energy, the flow deflection is enough to avoid the impact on the seal step. Consequently, the cavity vortices show a different structure. The condition \(\zeta_{g}\approx 1\) indicates that the effective seal clearance is practically equal to the nominal value. In this case, the structures observed in the seal cavities should be similar to those occurring in the smooth labyrinth seals. Indeed, the labyrinth seal with the rhomboidal cells shows stronger downstream vortices compared to the honeycomb seal, due to its smaller \(\zeta_{g}\) (compare Figure 8b,c). However, the rhomboidal cells are even more dissipative than the smooth configuration, as they add the friction contribution that occurs near the fin tips.