# Numerical and Experimental Analysis of Labyrinth Seals with Rhomboidal Cells
## 2 Numerical Analysis
### CFD Analysis on Schramm Geometry
\begin{table}
\begin{tabular}{c c} \hline Outlet static pressure & \(p_{0}\) = 1.01325 bar \\ \hline Pressure ratio & 1.1 \\ \hline Total inlet temperature & \(T_{0}\) = 300 K (Isothermal) \\ \hline Solution scheme & Upwind–First Order \\ \hline Turbulence model & k-\(\omega\) \\ \hline Y+ & Automatic wall treatment \\ \hline Analysis & Steady state \\ \hline Fluid & Air ideal gas (Sutherland) \\ \hline Turbulence intensity & 5\% \\ \hline \end{tabular}
\end{table} 
Table 4: Setup settings. Figure 3: CFD geometry, boundary conditions [19]. composed of multiple bodies having non-conformal grids at the interface. The pressure ratio is calculated between the inlet total pressure and the static outlet pressure. The calculations were performed with a Dell Precision T7600 computer, with two Intel Xeon CPU E5-2620 processors. Since each model has a different lateral width, comparing the numerical values of the leakage mass flow rates is not possible. In the present work, the discharge coefficient will be used to compare the performance of the several patterns analyzed:  
\[C_{D}=\frac{\dot{m}_{CFD}}{\dot{m}_{ideal}}\, \tag{2}\]  
where \(\dot{m}_{CFD}\) is the mass flow rate recorded during the numerical simulations, while \(\dot{m}_{ideal}\) is obtained with the following formula, proposed by [PERSON] [23]:  
\[\dot{m}_{ideal}=\frac{p_{0}\times A}{\sqrt{T_{0}}}\sqrt{\frac{2\kappa}{R( \kappa-1)}\left[\left(\frac{1}{\beta}\right)^{2/\kappa}-\left(\frac{1}{\beta} \right)^{(\kappa+1)/\kappa}\right]}\, \tag{3}\]  
Air (ideal gas) was used in the numerical model, meaning that \(R=287\,\mathrm{J/kgK}\) and \(\kappa=1.4\). Area \(A\) in Equation (3) is based on the nominal clearance value. Here, an important observation has to be made before showing the CFD results. When a pattern is utilized for the abradable stator land, the effective gap that can be measured between the fin tip and the closest cell wall can be significantly greater than the nominal clearance value s. In particular, a local value of effective clearance \(s_{eff}(y)\) can be defined. This value depends on the cell-fin tip misalignment d and on the function \(q(y)\), which represents the contour of the cell in the tangential direction [19]:  
\[s_{eff}=\sqrt{s^{2}+\left(q(y)-d-\frac{b}{2}\right)^{2}}\, \tag{4}\]  
Figure 4: Mesh rhomboidal cell detail. \begin{table}
\begin{tabular}{c c} \hline
**Pattern** & **\# Nodes** \\ \hline HC & 600,000 \\ R1 & 780,000 \\ R2 & 1,225,000 \\ R3 & 1,700,000 \\ \hline \end{tabular}
\end{table} 
Table 5: Mesh details for the Honeycomb cell (HC) and for the three rhomboidal cells (R1, R2, R3). where \(d\) has been set equal to zero, while \(q(y)\) is a periodical function defined by intervals. Figure 5 shows the points used in Equation (5) below:  
\[ \begin{gathered} from\ A\ to\ B\ q(y)=0\\ from\ B\ to\ C\ q\ (y)=\left(y-\frac{S_{R}}{2}\frac{1}{\sin\partial} \right)\times\sin\delta\\ from\ C\ to\ D\ q(y)=\left(D_{m}-b\right)/2-\left(y-\left(\frac{D_{M}}{2}+ \frac{S_{R}}{2}\frac{1}{\sin\partial}\right)\right)\times\sin\delta\\ from\ D\ to\ E\ \times\ q(y)=0\end{gathered}  \tag{5}\]  
By applying the procedure described above for the honeycomb cells [19] to Equations (4) and (5), it is possible to calculate the clearance values \(s_{mean}\) and \(s_{seal}\). The first represents the local average clearance value and refers to the single tooth, while the second is the average value among all the teeth of the labyrinth seal. [PERSON] et al. [20] took into account, for the first time, the contribution to leakage flow, due to the difference between effective and nominal clearance, introducing a geometrical increase parameter \(\zeta_{g}\) for honeycomb labyrinth seals.