## 3 Results and Discussion
### Blow-Down Prediction  
[PERSON] et al. [53] proposed a formula for calculating the blow-down effect. However, this formula is not universally applicable to brush seals with clearances and could not be employed under the conditions used to analyze the [PERSON] et al. [27] geometry. Therefore, we developed a new blow-down formula specifically for the [PERSON] et al. [27] geometry, based on the concept of effective clearance [48] utilized by [PERSON] et al. [53]. Effective clearance is useful for accurately predicting and modeling the performance of brush seals and assists in controlling leakage flow in complex fluid dynamics environments such as turbines [54]. The blow-down formula proposed by [PERSON] et al. [53] is challenging to use because it is tailored to a specific experimental setup involving a particular brush seal configuration and clearance geometry. The lack of a clear specification for the effective clearance further complicates the application of this formula. The formula accounts for factors such as bristle pack dynamics, pressure distribution, and axial compression, which differ significantly from the conditions employed by [PERSON] et al. [27], where different pressure ratios and clearance conditions result in varying blow-down behaviors. To address these issues, a new calculation formula was developed to determine the blow-down, based on the effective clearance suggested by [PERSON] et al. [48]. This effective clearance is calculated using Equations (6) and (7), where \(\gamma\) and \(R\) are set to 1.4 and 287 J/kg\(\cdot\)K, respectively, based on the properties of air. \(Q\) is a flow function that accounts for the compressibility and flow characteristics of the fluid, as expressed in Equation (7). \[h_{eff}=\dot{m}\;\sqrt{T_{u}}/\left(\pi d_{1}p_{u}Q\right), \tag{6}\]  
\[Q=\left\{ \begin{aligned} \sqrt{\frac{2\gamma}{R\left(\gamma-1 \right)}\left[\left(\frac{p_{d}}{p_{u}}\right)^{\frac{2}{\gamma}}-\left(\frac {p_{d}}{p_{u}}\right)^{\frac{2+1}{\gamma}}\right]}\;,\;\;\text{if}\;\frac{p_{ u}}{p_{d}}\leq\left(\frac{\gamma+1}{2}\right)^{\frac{2}{\gamma-1}}\\ \sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{\frac{2+1} {\gamma-1}}}\;,\;\;\text{if}\;\frac{p_{u}}{p_{d}}\geq\left(\frac{\gamma+1}{2} \right)^{\frac{\gamma}{\gamma-1}}\end{aligned} \right. \tag{7}\]  
Following the methodology of [PERSON] et al. [53], the effective clearance was calculated for both the non-blow-down state and the contact state. Additionally, the increase in blow-down with increasing pressure ratio was accounted for. Since the final calculated blow-down is in millimeters, the blow-down clearance can be obtained by subtracting the  
Figure 11: Velocity vectors around the fence height for contact [25] and clearance (0.27 mm) [27] brush seals under a pressure ratio of 2.0.  
blow-down value from the initial clearance. The blow-down calculation formula used for the [PERSON] et al. [27] geometry is shown in Equation (8):  
\[Blow-down=\frac{h_{eff}1-h_{eff}2}{\left(\frac{CL}{d}\right)^{0.21}}-\frac{CL}{ \left(\frac{CL}{d}\right)^{\frac{PS}{2}}}. \tag{8}\]  
In Equation (8), \(h_{eff}1\) represents the effective clearance calculated using the mass flow rate considering the initial clearance, without blow-down, and \(h_{eff}2\) represents the effective clearance calculated using the mass flow rate in the contact state, without clearance. The formula is designed to account for variations in blow-down depending on the clearance level. Using Equation (8), the blow-down clearance and leakage flow for the [PERSON] et al. [27] geometry were determined and were compared with the values reported by [PERSON] et al. [52], as shown in Figures 12 and 13. Based on the comparison, the maximum relative error was 4.8%, which occurred at a pressure ratio of 1.5 and a clearance of 0.75 mm. Therefore, Equation (8) can accurately predict the blow-down behavior for the [PERSON] et al. [27] geometry. Figure 12: Comparison of predicted blow-down clearance (Equation (8)) with simulation results reported by [PERSON] et al.