## 2 Numerical Calculation Model of Four-Tooth Three-Stage Brush Seal
### The Verification of the Accuracy of the Numerical Method  
In this paper, based on the interstage pressure drop test data of multistage brush seals in the literature [7], the accuracy of the established numerical calculation method in solving the interstage pressure drop of multistage brush seals is verified. One study [7] has processed a three-stage brush seal test piece, constructed a multi-stage brush seal static experimental set up, measured the inter-stage pressure values of the three-stage brush seals at pressure ratios of 2.0 and 3.0, and pointed out that the experimental pressure ratios of the first-, second-, and third-stage brush bundles were 21-27%, 27-32%, and 41-52%. In this section, the study's test piece geometry is modeled as shown in Figure 5, and the numerical calculation method of this paper is applied to calculate the interstage pressure drop distributions for pressure differences of 2.0 and 3.0 under the same operating conditions and boundary conditions.  
As shown in Figure 6, the percentage of pressure drop of the first, second, and third brush bundles is 21.3%, 31.8%, and 46.9% at a pressure ratio of 2.0, and the percentage of  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Grids/10\({}^{6}\)** & **Flow Values of Imports and Exports/(g/s)** & **Calculation Accuracy/\%** \\ \hline
5.24 & 72.80064 & 89.19 \\
5.71 & 75.00450 & 91.89 \\
6.36 & 76.816553 & 94.11 \\
6.83 & 78.968718 & 96.74 \\
6.95 & 78.968718 & 96.74 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Grid-independent verification.  
Figure 4: Detail of mesh division of four-tooth three-stage brush seal combination.  
Figure 5: A structural model of the three-stage brush seal test piece from the literature.  
pressure drop of the first, second, and third brush bundles is 24.2%, 30.2%, and 45.6% at a pressure ratio of 3.0, the interstage pressure drop has the characteristic of increasing by stages. The numerical calculation results are in good agreement with the test results, which ensures the credibility of the numerical calculation method developed in this paper for the calculation of the pressure drop of multi-stage brush seals.  
To further verify the accuracy of the numerical calculation method established in this paper in terms of solving the sealing characteristics of multi-stage brush seals, this paper takes the four-tooth three-stage brush combination seal shown in Figure 2 as a test piece and carries out the three-stage brush seal differential pressure and leakage characteristics test (the test was completed in the high-temperature, high-speed sealing test bench of Shenyang University of Aeronautics and Astronautics). As shown in Figure 7, the test bench consists of an air supply system, power system, oil supply system, heating system, cooling system, and test system. The test bench has a rotational speed of 0-20,000 r/min, an inlet air temperature of 20-300 \({}^{\circ}\)C, a maximum test piece diameter of 630 mm, a maximum linear velocity of 500 m/s, and a maximum supply pressure of 1.0 MPa, which enables the measurement of the brush seal leakage flow rate under the working conditions of 250 \({}^{\circ}\)C and 1.0 MPa. The comparison of the leakage rate between the test data and the numerical calculation under different working condition parameters (differential pressure 0.1-1.0 MPa, 25 \({}^{\circ}\)C) is shown in Figure 8, and the error between the calculation results and the test results is \(\leq\)10%, which proves the accuracy of the numerical model of fractal geometries and porous media in the calculation of the sealing performance.  
Figure 6: A comparison of the numerical calculations and experimental values of the structure of the three-stage brush seal test piece from the literature [7].  
Figure 7: Differential Pressure Leakage Characteristics Test Rig for Four-Tooth Three-Stage Brush-Type Combination Seals.