# Numerical and Experimental Analysis of Labyrinth Seals with Rhomboidal Cells
## 2 Numerical Analysis  
### CFD Analysis on Schramm Geometry  
In order to investigate the leakage behavior of the labyrinth seal, a set of CFD simulations were performed. The present study was carried out on the basis of previous research, accomplished by validating a numerical model [19] on the literature data [7; 20]. The reference model is a three-fin stepped labyrinth seal. In this seal, the effects observed on the leakage flow are not only related to the type of cell used in the stator design but also strongly depend on the forward or backward stepped labyrinth seal configuration [21]. The already-validated numerical procedure was applied to a different stator land pattern, composed of rhomboidal cells and the obtained results were compared with the traditional honeycomb cells. The numerical model was implemented with the software Ansys(r) CFX 14.5(r). This code uses a coupled solver with an algebraic multi-grid approach and a combined algorithm of finite volume and element. The model geometry is shown in Figure 1 and described in Table 1.  
In addition to the honeycomb land (whose geometrical parameters are listed in Table 2 and illustrated in Figure 2), rhomboidal cells were analyzed as an alternative pattern. Three kinds of cells were obtained by changing the diagonal ratio, which is defined in Equation (1). \[C_{R}=\frac{D_{M}}{D_{m}}=\frac{1}{tan\delta}\, \tag{1}\]  
The design constraint imposed for the rhombi is to keep their minor diagonal, which is disposed in the flow direction, equal to the honeycomb cell diameter. The investigation of the rhomboidal pattern consisted of a CFD numerical sensitivity, performed by modifying the ratio of the diagonals to point out the effects on the leakage flow \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline t & Tooth pitch & 28 mm \\ s & Seal clearance & 1.204 mm \\ NT & Number of teeth & 3 \\ H & Step height & 3.92 mm \\ h & Tooth height & 12.88 mm \\ b & Fin tip thickness & 1.316 mm \\ r & Fin tip fillet & 4 mm \\ \(\alpha\) & Angle between tooth faces & 20 deg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Labyrinth seal geometrical parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline L & Diameter & 3.22 mm \\ H\({}_{\text{HC}}\) & Depth & 24.08 mm \\ S\({}_{\text{HC}}\) & Cell wall thickness & 0.07 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Honeycomb cell geometrical parameters. Figure 1: Schramm Model geometry [19]. Figure 2: Rhombus (**left**) and honeycomb (**right**) cells. of the labyrinth seal. The names of the different generated patterns, with their respective \(C_{R}\), \(\delta\), and cell area ratio (with respect to the honeycomb cell) are listed in Table 3.  
The CFD model used is static and isothermal. While [PERSON] [7] dealt with turbulence using the k-\(\epsilon\) model, paying special attention in setting y+ near the wall domain, in this paper, instead, the two-equation model k-\(\omega\) with automatic wall treatment was chosen, referring to the previous analyses performed by [PERSON] [21]. The reason for this choice is that this model proved to perform better with adverse pressure gradients or separating flows as it accurately replicates the vortex pattern that is generated in the sealing cavity [22]. The computational domain has a width of two cells, with the condition of translational periodicity applied to the lateral boundaries (Figure 3). The outlet duct has an opening boundary condition in order to allow recirculating flows during the simulations. The stator cells above the tip of the first fin are centered on the fin tip. The setup settings are shown in Table 4.  
The mesh is composed of hexahedrons. The number of nodes for each simulation is shown in Table 5, while Figure 4 shows the detail of the grid of the rhomboidal cell. In the CFD model, the fin tip region and the stator land are connected by means of the General Grid Interface (GGI) scheme, which is suitable when the analyzed model is