To accurately predict the leakage flow and resistance characteristics of brush seals, the multiblock structured mesh and the mesh motion technique are applied to the three-dimensional (3D) staggered tube bundle model of brush seals. The multiblock structured mesh can easily add nodes and set boundary layers in the interbristle gap between adjacent bristles, which can ensure good mesh quality (orthogonal angle and expansion ratio). The mesh motion technique realizes the overall axial compactness of the bristle pack. The effects of pressure ratio, sealing clearance c, and bristle pack compactness on the leakage flow and resistance characteristics are investigated. To analyze the aerodynamic resistance of the brush seals, Euler number (Eu) is applied in this study. The numerical results are in good agreement with the experimental data. Thus, the accuracy of the presented numerical method is validated. For the contacting brush seal, has a significant effect on the leakage flow rate reduction. For the clearance brush seal, has little effect on the leakage flow rate reduction. The leakage flow passing through the sealing clearance keeps almost constant. As for aerodynamic resistance, the presence of the sealing clearance can effectively convert the pressure energy of the leakage flow into the kinetic energy. As a result, the leakage flow velocity exiting the bristle pack of the clearance brush seal is 1.5 to 2.0 times larger than that of the contacting brush seal. Although the existence of the sealing clearance obviously increases the leakage flow rate, it effectively reduces the aerodynamic forces acting on the bristles. The developed numerical approach based on the three-dimensional staggered tube bundle model and multiblock structured mesh can serve as a technical method for analysis of the sealing mechanisms of brush seals.