# CFD Research for Air Bearing with Gradient-Depth Recesses
## 3 CFD Simulation and Structure Optimization
### Numerical Solution of AFVM for Gradient-Depth Recesses  
In order to solve the problem of non-convergence in the finite volume numerical solution of the compound throttling global flow field, the UDF (user-defined functions) in CFD method is proposed to capture and subdivide the mesh near the wall. As shown in Formulas (11) and (12), the dimensionless velocity \(u^{+}\) is modified as a mixed function of the linear law, which is associated with laminar flow, and of the logarithmic law, which is associated with turbulent flow. Then, the corrected \(u^{+}\) is incorporated into Formula (13) to obtain a new first grid height \(y_{min}\), as shown in Formula (14). \[u^{+}=e^{\intercal}u^{+}_{lam}+e^{\intercal}u^{+}_{turb} \tag{13}\]  
Figure 4: Structure, parameters, and computational domain of the guideway without air tube. Figure 5: The model of air film. (**a**) Unified-depth recesses. (**b**) Gradient-depth recesses [3] (\(\surd\) means matched and \(\times\) means not). where \(\Gamma=-\frac{C_{a}\left(y^{+}\right)^{4}}{1+C_{b}y^{+}}\), \(C_{a}=0.01\), \(C_{b}=5\);  
\[y_{min}=y^{+}_{min}u/\rho u\sqrt{\frac{0.058 Re^{-\frac{1}{3}}}{2}} \tag{14}\]  
where \(u^{+}=\frac{u}{u_{*}}\), \(y^{+}=\frac{u_{*}}{\mu}y\). The structure of the guideway without an air tube is divided into hexahedral meshes according to the actual size parameters, and its initial boundary conditions (BCs) are shown in Figure 6.  
BC1: Pressure inlet boundary. The pressure inlet boundary of the calculation domain is the outer edge of the throttle nozzle, with a total pressure of 4 atm (about 0.4 Mpa, \(\sigma=4\)) for gas supply, a static pressure of 1 atm, a hydraulic diameter of the throttle nozzle aperture, a flow direction set perpendicular to the inlet boundary, and an initial turbulent re-flux intensity set at 9%. BC2: Pressure outlet boundary. The pressure outlet boundary of the calculation domain is the outer edge of the gap between the air flotation shaft and the air flotation sleeve, with a total pressure of 1 atm and a hydraulic diameter of 2 times the air film gap. The flow direction is set perpendicular to the outlet boundary, and the initial turbulence re-flux intensity is set to 4%. The corresponding initial turbulent kinetic energy is set to \(k=2.96\times 10^{3}\)\(\text{m}^{2}/\text{s}^{2}\). BC3: Smooth wall boundary. The smooth wall boundary of the computational domain is the upper boundary of the gas film gap, which is also used to calculate the mechanical properties of the guide rail on the air flotation working surface. The heat exchange is set to isothermal flow, and there is wall slip, so a strengthened wall treatment is adopted. BC4: Microstructure wall boundary. The boundary of the microstructure wall in the computational domain is the lower boundary of the gas film gap, which includes the shallow cavity diameter, the width of the pressure equalization groove, and the depth parameters of these microstructures. Other settings are the same as BC3. BC5: Symmetrical boundary of the air duct. The velocity and acceleration perpendicular to the boundary direction are zero, and the normal gradient in other directions on the boundary is zero. BC6: Centrally symmetric boundary. Other settings are the same as BC5. Figure 6: Boundary conditions for initial mesh of guideway with recesses. BC7: Adjacent area interface boundary. Two adjacent areas that share interface boundaries are divided into different grids, but it is necessary to ensure that some of the grid nodes overlap, such as the difference points. BC8: Inner face boundary in the same area. The two parts of the same area that share inner face boundaries are divided into the same mesh method, so it is possible to achieve overlapping of all mesh nodes, such as point-to-point. The result of the adaptive method for the boundary layer is shown in Figure 7.  
In this method, the initial thickness of the layer is strictly divided according to \(2\eta_{min}\).