## 2 Numerical Calculation Model of Four-Tooth Three-Stage Brush Seal
### Numerical Modeling
#### 2.2.1 Fractal Geometry Porous Media Model
From Equations (1) and (2), the total pressure drop loss can be derived as the sum of viscous damping dissipation per unit length and inertial damping dissipation per unit length:  
\[ \begin{array}{c}\frac{\Delta p_{1}}{l}+\frac{\Delta p_{2}}{l}=\frac{32\mu}{ L^{1-D_{T}}}\frac{3+D_{T}-D_{f}}{2-D_{f}}\frac{1-\mathcal{O}}{\mathcal{O}} \frac{v_{s}}{\lambda_{max}^{\gamma(1+D_{T})}}\\ +\left(\frac{3}{2}+\frac{1}{\beta^{4}}-\frac{5}{2\beta^{2}}\right)\frac{\rho v _{s}^{2}}{2\mathcal{O}^{2}}\frac{1}{l}\Big{[}D_{T}L_{0}^{(D_{T}-1)}\overline{ \lambda}^{(1-D_{T})}\Big{]}^{2}\end{array}  \tag{3}\]  
(2)  
Numerical modeling of multi-stage brush seals for fractal geometry porous media  
The governing equations of the fractal geometry porous media model were derived in detail above, leading to the specific physical mechanisms of the viscous and inertial damping dissipation coefficients, and next, we will use the fractal geometry porous media in the numerical model of the brush seal. Brush bundles consist of numerous fine filaments densely packed together. When airflow passes through the brush bundle, it undergoes significant density variations. Therefore, the gas flowing through both upstream and downstream regions is treated as an ideal compressible fluid:  
\[p=\rho RT \tag{4}\]  
where \(P\) denotes the gas pressure (Pa), \(\rho\) represents the gas density (kg/m\({}^{3}\)), \(R\) is the gas constant, and \(T\) indicates the temperature (K). Given that the brush bundle region in brush seals exhibits anisotropic porous media flow characteristics, its RANS flow governing equations are:  
\[\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{5}\]  
\[\frac{\partial\left(\rho u_{j}u_{i}\right)}{\partial x_{j}}=-\frac{\partial p}{ \partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{6}\]  
where \(x_{i}\) denotes the flow direction; \(U_{i}\) represents the velocity in the \(x_{i}\)-direction (m/s); \(P\) indicates the gas pressure (Pa); \(\tau_{ij}\) is the stress tensor (N/m\({}^{2}\)), \(\tau_{ij}=\mu_{eff}\left[\left(\partial u_{i}/\partial x_{j}\right)+\partial u _{j}/\partial x_{i}\right]-\left(2/3\right)\delta_{ij}\mu_{eff}\left(\partial u _{1}/\partial x_{1}\right)\), \(\mu_{eff}=\mu+\mu_{i}\); and \(\mu,\mu_{t}\) are molecular viscosity and turbulent viscosity (kg/(m-s)), respectively. For fluid regions outside the brush bundle, turbulence modeling is required to establish additional Reynolds stresses for solving the RANS equations. This study employs the k-\(\omega\) SST model. Given the anisotropic structure of the brush bundle and the additional resistance induced by laminar flow within it, a fractal porous media model is adopted. This model specifically enhances the original formulation 39 for the brush bundle region, by incorporating non-Darcian high-speed seepage effects through the addition of viscous and inertial resistance terms derived from the fractal geometric porous media model:  
\[F_{i}=-A_{i}\mu u_{i}-B_{i}\rho|u_{i}|u_{i} \tag{7}\]  
i.e.,  
\[\frac{\partial\left(\rho u_{j}u_{i}\right)}{\partial x_{j}}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial\tau_{ij}}{\partial x_{j}}-A_{i}\mu u_{i}-B_{i }\rho|u_{i}|u_{i} \tag{8}\]  
where  
\[\left\{ \begin{matrix}A_{i}=\left(a_{z}\begin{matrix}0\\ 0&a_{n}\cos\theta+a_{s}\sin\theta\end{matrix}\right)\\ B_{i}=\left(\begin{matrix}b_{z}&0\\ 0&b_{n}\cos\theta+b_{s}\sin\theta\end{matrix}\right)\end{matrix} \right. \tag{9}\]  
where  
\[\left\{ \begin{matrix}a_{z}=a_{n}=\frac{32\mu}{L^{1-D_{T}}}\frac{3+D_{T}-D_{f} }{2-D_{f}}\frac{1-\rho}{\Delta}\frac{1}{\Delta^{(1+D_{T})}}\\ a_{s}=0.4\frac{\lambda_{\min}}{\lambda_{\max}}\frac{\rho(4-D_{f})}{\lambda_{ \max}}\frac{1}{\rho(4-D_{f})}a_{\max}\\ b_{z}=\text{b}_{n}=\left(\begin{matrix}3&\frac{1}{2}+\frac{1}{\beta^{4}}&- \frac{5}{2\beta^{2}}\end{matrix}\right)\frac{\rho}{2\sigma^{2}}\frac{1}{2} \left[D_{T}L_{0}^{(D_{T}-1)}\overline{\lambda}^{(1-D_{T})}\right]^{2}\\ b_{s}=0\end{matrix} \right.