# Study on the Leakage and Inter-Stage Pressure Drop Characteristics of Two-Stage Finger Seal
## 3 Analysis Method of Two-Stage Finger Seal Based on FEM/CFD Coupling  
### Analysis Scheme  
Similar to the 1 SFS, the leakage flow of 2 SFS is mainly comprised of the main leakage flow (leakage flow through two 1 SFS/rotor radial clearances) and the side leakage flow (leakage flow through the two finger stick bundles (set of all the finger sticks)). For main leakage flow, only the radial clearances caused by the hysteresis effect are considered in the analysis, and the finite element method (FEM) is used to calculate the radial hysteresis clearances. For side leakage flow, the porous medium approach is adopted. In the previous study, the authors have equivalently regarded the finger stick bundle of 1 SFS as a porous medium [19]. However, as shown in Figure 1, the circumferential width of the finger stick stem (the region of finger stick corresponding to the foot upper circle to the finger base circle) is significantly smaller than the circumferential width of finger foot, the flow resistance of the fluid flowing through the micro-voids in the rough contact interface between adjacent finger feet is much greater than that between adjacent finger stick stems. Therefore, the finger stick bundle of 1 SFS is equivalent to two porous media in this paper: A porous medium representing the finger foot bundle (the region of finger stick bundle corresponding to the finger foot) and a porous medium representing the finger stick stem bundle (the region of finger stick bundle corresponding to the finger stick stem). When the finite element method is used to calculate the radial clearance of each 1 SFS caused by the hysteresis effect, it is necessary to take the axial pressure difference of each 1 SFS as the input parameters. In addition, for any 1 SFS in 2 SFS, the contact state between adjacent finger sticks is related to its axial pressure difference. Therefore, when the porous medium models are used to simulate the side leakage flow of 2 SFS, the resistance coefficients (including viscous resistance coefficients and inertial resistance coefficients) of porous media are a function of axial pressure difference of 1 SFS. However, for 2 SFS, the axial pressure difference of each 1 SFS is related to the structural and operating parameters, there is no obvious explicit relationship between the axial pressure difference of each 1 SFS and the total axial pressure difference of 2 SFS, so the axial pressure difference of each 1 SFS cannot be directly obtained by the total axial pressure difference of 2 SFS. To solve the problem, the FEM/CFD coupling iterative algorithm is proposed to study the leakage and inter-stage pressure drop characteristics of 2 SFS in this paper. The calculation flow is as follows. 1. Specify the initial axial pressure difference of each 1 SFS according to the total axial pressure difference of 2 SFS (it is generally assumed that each 1 SFS bears the same axial pressure difference). 2. Calculate the radial hysteresis clearance of each 1 SFS and the resistance coefficients of each porous medium as input parameters of the CFD model of 2 SFS. 3. Establish the CFD model of 2 SFS including main leakage flow and side leakage flow, then calculate the leakage of 2 SFS and the axial pressure difference of each 1 SFS.