Research and Experimental Verification of the Static and Dynamic Pressure Characteristics of Aerospace Porous Media Gas Bearings

Abstract

Porous media gas bearings utilize gas as a lubricating medium to achieve non-contact support technology. Compared with traditional liquid-lubricated bearings or rolling bearings, they are more efficient and environmentally friendly. With the uniform gas film pressure of gas bearings, the rotating shaft can achieve mechanical motion with low friction, high rotational speed, and long service life. They have significant potential in improving energy efficiency and reducing carbon emissions, enabling oil-free lubrication. By eliminating the friction losses of traditional oil-lubricated bearings, porous media gas bearings can reduce the energy consumption of industrial rotating machinery by 15–25%, directly reducing fossil energy consumption, which is of great significance for promoting carbon neutrality goals. They have excellent prospects for future applications in the civil and military aviation fields. Based on the three-dimensional flow characteristics of the bearing’s fluid domain, this paper considers the influences of the transient flow field in the variable fluid domain of the gas film and the radial pressure gradient of the gas film, establishes a theoretical model and a three-dimensional simulation model for porous media gas bearings, and studies the static–dynamic pressure coupling mechanism of porous media gas bearings. Furthermore, through the trial production of bearings and performance tests, the static characteristics are verified, and the steady-state characteristics are studied through simulation, providing a basis for the application of gas bearings made from porous media materials in the civil and military aviation fields.

1. Introduction

In 2024, considering aspects such as global fossil fuel consumption, aviation carbon emissions, and the flight economy of commercial aircraft, the International Civil Aviation Organization (ICAO) released the “Aviation and Environment Report”, which indicated that pure electric and hybrid aircraft are the future development directions of aircraft [1]. Many aircraft manufacturers around the world have carried out relevant research and development work in order to gain a head start in the development of pure electric and hybrid aircraft. The hybrid aircraft ZA10 developed by Zunum Aero (Kirkland, Washington, USA) is shown in Figure 1. The E-Fan X large commercial hybrid aircraft developed by Airbus (Blagnac, France), as shown in Figure 2, has a passenger capacity of 100 people. It replaces one of the four turbofan engines with a 2 MW electric ducted fan. At present, among hybrid power solutions, the hybrid power solution of a gas turbine generator combined with a solid oxide fuel cell can achieve a cycle thermal efficiency of over 70%, making it an important choice for hybrid power. Among them, the rotor system of the gas turbine generator is mainly supported by oil-lubricated bearings. There may be situations in which the sealing device of the lubricating oil cavity of the rotor system fails, and the lubricating oil evaporates and leaks into the fuel cell, along with high-pressure air, which can affect the service life of the battery. When gas bearings are used, there is no situation in which lubricating oil contaminates the battery, and relevant problems can be solved easily. Replacing traditional oil-lubricated bearings with gas bearings can reduce the need for the lubricating oil system, reduce the system’s weight, and lower the risk of the lubricating oil catching fire. Therefore, the rotor system supported by gas bearings is one of the key development directions for the rotor systems of gas turbine generators used in civil aviation in the future.

In order to improve the operational stability of gas bearings, many scholars have carried out research on gas bearings using various throttling methods, such as orifice throttling and slot throttling [2,3,4]. The porous medium material has countless tiny pores distributed throughout its interior and on its surface. Gas can pass through the interconnected pores, thus forming a uniform pressure. This kind of material includes metal, ceramic, and graphite types. Its preparation process is completed by adding a curing agent and additives to the corresponding powder, followed by applying processes such as high-temperature sintering or pressing. The graphite porous medium part is shown in Figure 3a,b presents the water immersion test of the porous medium part. When the porous medium material with an external gas supply is immersed in water, it can be seen that bubbles are uniformly generated on the surface of the part. This indicates that the gas can pass through the porous medium layer to form a pressure film on the surface of the rotating shaft, thus playing the role of supporting the rotating shaft and improving the bearing capacity of the bearing [5,6].

On the other hand, when the rotating shaft bears a dynamic load, the squeezing action of the rotating shaft causes the gas to flow back into the porous medium material, resulting in friction loss due to the viscous resistance. The damping of the porous medium gas bearing mainly comes from the flow of gas within the porous medium layer and the laminar flow of the gas film layer. The tortuous pore channels in the porous medium layer cause viscous losses in the gas flow. There is a pressure gradient in the radial direction of the gas film layer, and viscous losses also occur in the stratified flow. Therefore, the porous medium gas bearing has excellent damping characteristics [7], which can improve the stability of the rotor system. The method of porous medium throttling involves adding an additional layer of porous medium material inside the bearing to fabricate a porous medium gas bearing. The gas can form a uniform pressure through the porous medium material, which endows the porous medium gas bearing with a higher bearing capacity and greater stiffness, and also avoids problems such as the instability of the air hammer [8].

Table 1. Performance comparison of porous medium gas bearings and other types of gas bearings.

Gas Bearing	Dimension $\mathit{D}\times \mathit{L}(\mathbf{mm})$	$\mathbf{Clearance}\mathsf{\mu }\mathbf{m}$	Dimensionless Bearing Capacity $\mathit{W}/(\mathit{D}\times \mathit{L}\times \mathbf{Pa})$	Rotational Speed (r/min)	Stiffness Coefficient ${\mathit{k}}_{\mathit{x}\mathit{x}}$$(\mathbf{MN}/\mathbf{m})$	Damping Coefficient ${\mathit{c}}_{\mathit{x}\mathit{x}}$$(\mathbf{KN}/\mathbf{m})$
metal mesh foil bearing [12]	31.4 × 30	20	0.16	40,000	1.2	0.4
double-layered foil bearing [13]	30 × 30	15	0.56	50,000	3.1	3.4
tilting pad bearing [14]	28 × 33	20	0.56	50,000	1.5	0.05
porous medium bearing [15]	40 × 40	20	0.625	25,000	20	-
porous medium bearing [16]	30 × 40	15	0.92	20,000	19	38

shows that the porous medium gas bearing has a higher dimensionless bearing capacity than the metal mesh foil bearing, the double-layered foil bearing, and the tilting pad gas bearing at higher rotational speeds (40,000 and 50,000 r/min) when the rotational speed is relatively low, at 20,000 and 25,000 r/min. Moreover, the stiffness and damping coefficients of the porous medium gas bearing are also increased by an order of magnitude. The advantages of the porous medium gas bearing in terms of bearing capacity, stiffness, and damping endow it with a good application prospect. Many researchers have employed various methods in the numerical simulation of gas bearings, such as the finite element method [9] and the finite volume method [10,11]. Currently, research on porous medium gas bearings abroad has been applied to precision rotating machinery, while domestic research is still catching up with the advanced foreign level. In particular, aspects such as the theoretical model of the porous medium gas bearing, experimental research, the coupling of the bearing–rotor system, and the variation laws of the steady-state characteristics of the rotor system all require continuous in-depth study. There are many factors that affect the performance of the porous medium gas bearing, and the involved mechanisms are profound. Its technological progress and widespread application face challenges. Therefore, studying the porous medium gas bearing can promote both the perfection of its theory and its engineering applications.

This paper applies the porous medium material to the aerospace gas bearing, establishes a three-dimensional model of the porous medium gas bearing, designs the structural form according to the characteristics of the porous medium material, studies the steady-state characteristics of the bearing based on the established simulation model of the porous medium gas bearing, builds a test bench, and verifies the application performance of the porous medium gas bearing.

2. Model and Verification of the Porous Medium Gas Bearing

2.1. Porous Medium Materials

Porous medium materials are solid structures that contain countless regular or irregular and connected or unconnected micropores. Its production methods generally involve adding curing agents and additives to powders of copper alloys, stainless steels, ceramics, graphite, etc., and then sintering the mixture at high temperatures or molding it by pressing [17,18]. Among them, isostatic-pressed graphite porous medium materials, which have characteristics such as compactness, uniformity, a low coefficient of thermal expansion, self-lubrication, corrosion resistance, and good machinability, have become a research hotspot and have certain applications in the aerospace field. Therefore, in this thesis, isostatic-pressed graphite porous medium materials are selected to produce the porous medium layer of the bearing. Porosity and permeability are two main macroscopic parameters of porous medium materials.

2.1.1. Porosity of Porous Medium Materials

The volume occupied by countless connected or unconnected pores within the porous medium material is called the total pore volume of the porous medium material. Fluids can flow in the connected pores, and the volume occupied by the connected pores is called the effective pore volume, while the volume occupied by the unconnected pores is the ineffective pore volume. Due to the complexity of the internal pore channel structure and quantity, it is very difficult to accurately describe them using geometric methods. Generally, a statistical method is adopted to characterize the microscopic pore structure based on the macroscopic porosity. Its definition is the ratio of the total pore volume inside the porous medium material to the total volume of the porous medium material, which is represented by $\gamma$. It is a dimensionless parameter [19].

$$\gamma ={\displaystyle \frac{V_e}{V_t}},$$

(1)

In the formula, $\gamma$ represents the porosity of the porous medium material, $V_e$ represents the total pore volume inside the porous medium material (m³), and $V_t$ represents the total volume of the porous medium material (m³).

2.1.2. Permeability of Porous Media Materials

Permeability is an important parameter that characterizes the pore conductivity performance of porous media materials. It represents the degree of difficulty for high-pressure gas to pass through the porous media and is affected by factors such as the pore structure and density. The larger the pores and the more interconnected pores there are, the smaller the viscous resistance of the porous media materials to the gas, and the greater the permeability of the materials. By measuring the changes in the gas pressure and flow rate at the inlet and outlet of the porous media materials through experiments, the permeability can be calculated according to Darcy’s law as follows [20,21]:

$$k_s={\displaystyle \frac{2\mu H{p}_2}{A(p_1^2-p_2^2)}}Q,$$

(2)

In this formula, k_s is the permeability, m²; $\mu$ is the gas dynamic viscosity coefficient in N·s/m²; H is the thickness of the porous medium layer, m; Q is the volumetric flow rate flowing through the porous medium material, m³; A is the cross-sectional area of the porous medium material perpendicular to the flow direction, m²; P₁ is the inlet pressure of the porous medium material, Pa; and P₂ is the outlet pressure of the porous medium material, Pa.

The isostatic pressing graphite porous medium material selected in this paper is shown in Figure 4. A pressure flow test is adopted to measure the pressure and flow rate variation curves of different porous medium materials to determine their permeability. The temperature and humidity of the test environment may have a slight impact on the measured pressure and flow rate. However, the focus of this paper is not on studying the influencing factors of the characteristics of the porous medium materials. Therefore, it is assumed that the permeability is not affected by the gas temperature and humidity. The test device is shown in Figure 5. It mainly uses a gas source to provide high-pressure air. The air is depressurized by using a control valve to adjust the pressure. An oil–water separator removes the moisture, and the air enters the test device. The porous medium material specimen is sealed and fixed inside the test device. A pressure sensor with a measuring range of 0.05–1.0 MPa and a sensitivity of ±0.2%, along with a flowmeter with a measuring range of 0–50 L/min and a sensitivity of ±1.5%, are used to measure the changes in pressure and flow rate passing through the porous medium material specimen. The density of the porous medium material specimen is 1.709 g/cm³, with dimensions of 34 × 34 mm in length and width and thicknesses of 3.0 mm and 4.5 mm. Four bolts are used to press the specimen tightly against the aluminum test device with sealant applied to the edges, ensuring that all high-pressure air flows out through the porous medium material specimen.

During the test, the ambient temperature was 22 °C. The curve of the gas flow rate through the specimens changes with the supply pressure ratio, as shown in Figure 6. It can be seen that as the supply pressure ratio increases from 1.0 to 7.0, the gas flow rates of both specimens increase. When the supply pressure ratio is less than 3.8, the gas flow rate increases non-linearly. When the supply pressure ratio is greater than 3.8, the gas flow rate increases nearly linearly. As the thickness of the porous medium material increases, the gas flow rate decreases, indicating that the flow resistance increases. Based on the data measured in the test, the permeability curve of the porous medium material, calculated using Darcy’s law, is shown in Figure 7.

As can be seen in Figure 6, within the range of the supply pressure ratio tested in the experiment, the permeability of the porous medium material does not change significantly with the thickness. The average value is 1.27 × 10⁻¹⁴ m². The results show that the permeabilities of porous medium materials with different thicknesses are basically equal. Therefore, in this paper, porous medium materials with densities of 1.617 g/cm³, 1.673 g/cm³, and 1.709 g/cm³ are selected for processing the porous medium layer of the bearing for research purposes. The corresponding average permeabilities are 1.12 × 10⁻¹² m², 1.17 × 10⁻¹³ m², and 1.27 × 10⁻¹⁴ m², respectively. This is also consistent with the common selection of materials with permeabilities ranging from 1.0 × 10⁻¹⁴ m² to 1.0 × 10⁻¹² m² for preparing porous medium gas bearings in existing studies [22]. The permeabilities of the porous medium materials obtained through the abovementioned experiments and calculations can be used as material physical property parameters when modeling the porous medium gas bearings, providing a guarantee for the accurate simulation of their characteristics.

2.2. Airflow of the Porous Medium Gas Bearing

The porous medium gas bearing is a hybrid hydrostatic and hydrodynamic bearing that relies on an external gas supply to provide the main supporting force. Meanwhile, when the rotating shaft rotates at a high speed, it can exhibit hydrodynamic pressure characteristics. The schematic diagrams of the structure and airflow of the porous medium gas bearing are shown in Figure 8. The bearing is mainly composed of a bearing sleeve and a porous medium layer. The gas flow within the porous medium layer includes radial flow, as well as axial and circumferential flows. The porous medium layer is pressed into the bearing sleeve through an interference fit, forming an air chamber between the porous medium layer and the bearing sleeve. The air chamber is connected to the high-pressure gas inlet on the bearing sleeve. External high-pressure gas enters the air chamber through the high-pressure gas inlet and flows through the porous medium layer to form an air film. This air film has a uniform pressure, which plays a role in supporting the rotating shaft.

2.3. Simulation Model of the Porous Medium Gas Bearing

Most of the models of porous medium gas bearings couple the flow control equations of the air film layer and the porous medium layer. The finite difference method is adopted, and a self-written program is used to solve for the flow field distribution in the fluid domain. Then, by integrating parameters such as pressure and flow rate, the steady-state characteristics, such as the load-carrying capacity and mass flow rate, are obtained. This often requires simplifying and approximatingf the model, ignoring the radial pressure gradient of the air film and the velocity slip boundary conditions. Based on the fluid lubrication theory and the Darcy–Forchheimer law within the porous medium material, and taking into account the three-dimensional flow and the velocity slip boundary, this paper establishes a three-dimensional simulation model of the porous medium gas bearing. The finite volume method is used to solve the complete Navier–Stokes equations, analyze the microscopic flow characteristics of the air film layer and the porous medium layer, and further study the steady-state characteristics of the porous medium gas bearing.

The porous medium gas bearing is mainly composed of a bushing and a porous medium layer. Radial air inlets (Zhengjia, China) are machined onto the bushing, and the inner cylindrical surface forms an air chamber with the porous medium layer. The porous medium gas bearing is fitted with a bearing seat for installation and positioning, and its schematic diagram is shown in Figure 9. High-pressure gas enters from the air inlet and then enters the air chamber through the air inlet holes of the bushing. After pressure equalization, it passes through the porous medium layer and enters the bearing clearance, forming a pressure film on the surface of the rotating shaft, which plays a role in supporting the rotating shaft. Finally, the gas flows out from the clearance at the axial end face of the bearing. A three-dimensional design software is used for the structural design of the porous medium gas bearing and the bearing seat. Based on digital manufacturing technology, the processing of the porous medium layer of the bearing, the bushing, etc. is implemented.

When the porous medium gas bearing is in operation, the fluid domain mainly consists of two parts, namely, the gas flow inside the porous medium layer and the gas flow within the bearing clearance. Figure 10 is a schematic diagram of the control volume of the porous medium gas bearing. Figure 10a is a partial schematic diagram of the porous medium gas bearing. The gray area (Region 1) represents the air film layer, and the blue area (Region 2) represents the porous medium layer. There is a gas flow along the three directions of the Cartesian coordinate system, namely, x, y, and z. Figure 10b is a schematic diagram of the flow of an infinitesimal element within the bearing clearance (air film layer). The gas flows from the porous medium layer into the air film layer along the y direction, then flows along the x and z directions. Since the inner cylindrical surface of the infinitesimal element of the air film is in contact with the rotating shaft in the radial direction, there is no gas outflow. Figure 10c is a schematic diagram of the flow of an infinitesimal element in the porous medium layer, where the gas flows along the three directions of x, y, and z. Figure 10d is an infinitesimal element at the interface between the porous medium layer and the air film layer. Along the y-axis direction, the gas can flow between the porous medium layer and the air film layer. At the same time, the gas flow in the air film layer is a slip flow, with velocity slip on the surface of the porous medium layer. Based on the abovementioned flow characteristics of the infinitesimal elements in the fluid domain, comprehensively considering the velocity slip boundary at the interface between the porous medium layer and the air film layer, and coupling the flow control equations of the porous medium layer and the air film layer, a model of the porous medium gas bearing is established, laying a foundation for further analyzing its steady-state and dynamic performance.

The pores in tight, graphite-based porous materials are generally in the nanoscale. With regard to the gas flow in such small pores, the collisions between gas molecules and solid walls are much more obvious than those in a conventional situation. Meanwhile, at the interface between porous media and the gas film, the gas from the micro-pores of porous media collides with the gas in the gas film. The presence of radial flow and radial pressure gradient in the internal flow of porous media gas bearings complicates the establishment and solution of the model. In this paper, nonlinear partial differential equations are employed to solve the model. The flow of the air film layer within the bearing clearance complies with the law of conservation of mass, the law of conservation of momentum, and the law of conservation of energy, which can be expressed using the Navier–Stokes equations [17,22,23].

The law of conservation of mass is expressed as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot (\rho \overrightarrow{U})=0,$$

(3)

In the formula, $\rho$ is the gas density, kg/m³; $t$ is the time, s; and $\overrightarrow{U}$ is the velocity vector $(u,v,w)$.

The momentum conservation laws in the x, y, and z directions are expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial (\rho u)}{\partial t}}+\nabla \cdot (\rho u\overrightarrow{U})=-{\displaystyle \frac{\partial p}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{xx}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yx}}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{zx}}{\partial z}}+F_x\\ {\displaystyle \frac{\partial (\rho v)}{\partial t}}+\nabla \cdot (\rho v\overrightarrow{U})=-{\displaystyle \frac{\partial p}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{xy}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yy}}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{zy}}{\partial z}}+F_y\\ {\displaystyle \frac{\partial (\rho w)}{\partial t}}+\nabla \cdot (\rho w\overrightarrow{U})=-{\displaystyle \frac{\partial p}{\partial z}}+{\displaystyle \frac{\partial {\tau }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yz}}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{zz}}{\partial z}}+F_z\end{array},$$

(4)

In the formula, $\rho$ is the pressure in Pa, $u,v,w$ are the velocity components in the three directions in m/s, $\tau$ is the shear stress in N/m², and $F_x,F_y,F_z$ are the body forces in the three directions in N.

For Newtonian fluids, the shear stress, $\tau$, is proportional to the strain rate of change and can be expressed as follows [24,25]:

$$\begin{array}{l}{\tau }_{xx}=\eta \left(\nabla \cdot \overrightarrow{U}\right)+2\mu {\displaystyle \frac{\partial u}{\partial x}};{\tau }_{xy}={\tau }_{yx}=\mu ({\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial u}{\partial y}})\\ {\tau }_{yy}=\eta \left(\nabla \cdot \overrightarrow{U}\right)+2\mu {\displaystyle \frac{\partial v}{\partial y}};{\tau }_{xz}={\tau }_{zx}=\mu ({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}})\\ {\tau }_{zz}=\eta \left(\nabla \cdot \overrightarrow{U}\right)+2\mu {\displaystyle \frac{\partial w}{\partial z}};{\tau }_{yz}={\tau }_{zy}=\mu ({\displaystyle \frac{\partial w}{\partial y}}+{\displaystyle \frac{\partial v}{\partial z}})\end{array},$$

(5)

Among them, $\mu$ is the dynamic viscosity coefficient, m²/s; $\eta$ is the second dynamic viscosity coefficient, m²/s; and $\eta =-2\mu /3$.

The energy conservation equation can be expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial t}}\left[\rho T\right]+\nabla \cdot \left[\rho T\overrightarrow{U}\right]\\ =\rho \dot{q}+{\displaystyle \frac{\partial }{\partial x}}\left(k_g{\displaystyle \frac{\partial T}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(k_g{\displaystyle \frac{\partial T}{\partial y}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k_g{\displaystyle \frac{\partial T}{\partial z}}\right)-{\displaystyle \frac{\partial (up)}{\partial x}}-{\displaystyle \frac{\partial (vp)}{\partial y}}-{\displaystyle \frac{\partial (wp)}{\partial z}}\\ +{\displaystyle \frac{\partial (u{\tau }_{xx})}{\partial x}}+{\displaystyle \frac{\partial (u{\tau }_{yx})}{\partial y}}+{\displaystyle \frac{\partial (u{\tau }_{zx})}{\partial z}}+{\displaystyle \frac{\partial (v{\tau }_{xy})}{\partial x}}+{\displaystyle \frac{\partial (v{\tau }_{yy})}{\partial y}}+{\displaystyle \frac{\partial (v{\tau }_{zy})}{\partial z}}\\ +{\displaystyle \frac{\partial (w{\tau }_{xz})}{\partial x}}+{\displaystyle \frac{\partial (w{\tau }_{yz})}{\partial y}}+{\displaystyle \frac{\partial (w{\tau }_{zz})}{\partial z}}+\overrightarrow{F}\overrightarrow{U}\end{array},$$

(6)

where T is the temperature in K; $\dot{q}$ represents the heat generated by heat conduction in J; C_p is the specific heat capacity at constant pressure in J/(kg·K); and $k_g$ is the effective thermal conductivity of the gas, W/m·K.

A phenomenon known as the velocity slip boundary condition could emerge as flow geometries are scaled down to micro-/nano-scales. Under these conditions, surface boundary conditions become increasingly dominant. The new features of the flow in small gaps, such as the gas film between the rotor and the PGB, can arise. The characteristics are determined by the Knudsen ($Kn$) number, $Kn=\lambda /h$. The $\lambda$ is the mean free molecular path, and $h$ is the characteristic fluid thickness [26]. For example, A. The flows with $Kn\le 0.001$ are regarded as no-slip flows; B. $0.001<Kn\le 0.1$ are slip flows; C. $0.1<Kn\le 10$ are flows in the transitional regime; and D. $Kn>10$ are considered free molecular flows [27,28,29]. Luo et al. [30,31] summarized and reviewed the development of thin film lubrication in the past 20 years and explored the molecular orientation and friction performance in the contact region with a gap size of tens of nanometers. The $Kn$ number researched in this paper is between 0.001 and 0.1, and the flow of the gas film is the slip flow. The Navier–Stokes equations can be applied to the slip flow unless the appropriate boundary conditions are provided. References [32,33] proposed some boundary conditions. Maxwell’s equation is used to estimate the slip coefficient of rough walls in classical theory, which is adopted to calculate the velocity slip boundary condition as follows:

$$\begin{array}{c}{u}_s=(1-l_k^2)U_o+{\displaystyle \frac{2-{\sigma }_t}{{\sigma }_t}}\lambda \left\{{\displaystyle \frac{\partial }{\partial y_k}}\left[(1-l_k^2)u-l_k{m}_{k}v-l_k{n}_{k}w\right]+(1-l_k^2){\displaystyle \frac{\partial v_k}{\partial y}}-l_k{m}_k{\displaystyle \frac{\partial v_k}{\partial y}}-l_k{n}_k{\displaystyle \frac{\partial v_k}{\partial y}}\right\}\\ v_s={\displaystyle \frac{2-{\sigma }_t}{{\sigma }_t}}\lambda \left\{{\displaystyle \frac{\partial }{\partial y_k}}\left[(1-m_k^2)v-l_k{m}_{k}u-m_k{n}_{k}w\right]+(1-m_k^2){\displaystyle \frac{\partial v_k}{\partial y}}-l_k{m}_k{\displaystyle \frac{\partial v_k}{\partial y}}-m_k{n}_k{\displaystyle \frac{\partial v_k}{\partial y}}\right\}\\ w_s={\displaystyle \frac{2-{\sigma }_t}{{\sigma }_t}}\lambda \left\{{\displaystyle \frac{\partial }{\partial y_k}}\left[(1-n_k^2)u-l_k{n}_{k}v-m_k{n}_{k}v\right]+(1-n_k^2){\displaystyle \frac{\partial v_k}{\partial z}}-l_k{m}_k{\displaystyle \frac{\partial v_k}{\partial x}}-m_k{n}_k{\displaystyle \frac{\partial v_k}{\partial y}}\right\}\end{array}$$

(7)

where $V_s=(u_s,v_s,w_s)$ is the slip velocity of gas on the rough wall, $U_0$ is the velocity of the moving wall, $u,v,w$ represent the velocity of the gas near the wall, $v_k=l_{k}u+m_{k}v+n_{k}w$,$\partial /\partial y_k=l_k\partial /\partial x+m_k\partial /\partial y+n_k\partial /\partial z$, and $L=(l_k,m_k,n_k)$ is the unit normal vector of the grid on the wall.

2.4. The Mesh Model and Mesh Independence of the Porous Medium Gas Bearing

Both the porous medium layer and the bushing are cylindrical, and their structures are symmetrical around the central section of the bearing. Based on the structure of the bearing, this paper establishes a full-scale simulation model of the porous media gas bearing. The fluid domain of the porous medium gas bearing is divided into a porous medium layer and an air film layer. Compared with the porous medium layer, the radial dimension of the air film layer is very small. The thickness of the air film is at the micrometer level, while the thickness of the porous medium layer is generally 2.0–4.0 mm. Both fluid domains are regular cylinders, and hexahedral meshes can be used. The mesh model ensures that the shape of the geometric model and its boundary characteristics are maintained, and at the same time, it improves the calculation speed. Additionally, the number of mesh layers is encrypted in the thickness direction of the air film layer to ensure the accuracy of the calculations. The mesh model and boundary conditions are shown in Figure 11.

The calculation accuracy and speed of the three-dimensional model are affected by the mesh scale. The smaller the mesh scale, the larger the number of meshes, and the more accurate the calculation will be [34,35]. However, the calculation speed will decrease. To improve the calculation speed as much as possible while ensuring accuracy, a variety of mesh schemes are adopted to verify the independence of the number of meshes for the bearing model. Three mesh schemes are selected from all the schemes for comparison, and finally, the calculation mesh model is determined. The total number of meshes for the three schemes are as follows: Scheme A has 1,064,100 meshes; Scheme B has 1,641,600 meshes; and Scheme C has 2,395,800 meshes. Taking Scheme C as a reference point, based on the load-carrying capacity and pressure distribution of the axial middle cross-section for different numbers of meshes, it is determined whether the number of meshes converges. The results are shown in

Table 2. Comparison of results for three mesh schemes.

Scheme	Total Number of Meshes	Load-Carrying Capacity (N)	Deviation
Scheme A	1,064,100	68.58	−5.21%
Scheme B	1,641,600	71.20	−1.58%
Scheme C	2,395,800	72.35	-

. It can be seen that the difference in the load-carrying capacity between Scheme C and Scheme B is very small, indicating that the number of meshes in Scheme B can meet the requirements for calculation accuracy. Figure 12 shows the pressure on the rotating shaft of the axial middle cross-section of the porous medium gas bearing for different numbers of meshes. The pressure curves of Scheme B and Scheme C basically coincide, while that of Scheme A is on the low side. Finally, Scheme B is selected as the mesh for the steady-state calculation.

During the mesh generation process of the air film layer, the enhanced wall function is used to handle the fluid domain near the wall surface [36]. Without the need to encrypt the wall mesh model, good calculation results can be obtained using relatively few computational resources. In terms of boundary settings, the outer cylindrical surface of the porous medium layer is set as a pressure inlet boundary, and the two axial end faces are set as walls to prevent gas leakage from the end faces. The inner cylindrical surface of the air film layer rotates in tandem with the rotating shaft and serves as a rotating wall surface. The axial length of the bearing is 57 mm, the diameter of the bearing is 25 mm, and the ratio of length to diameter is 2.28. It should be treated as a short bearing. The gaps at the two axial end faces of the bearing are set as pressure outlet boundaries, and the interface between the air film layer and the porous medium layer has a slip velocity. The boundary conditions of the simulation model are shown in

Table 3. Boundary conditions of the simulation model.

Location	Boundary Definition
Outer cylindrical surface of the porous layer	Pressure Inlet
Inner cylindrical surface of the porous layer	Slip
Axial end face of the porous layer	Wall
Outer cylindrical surface of the gas film layer	Slip
Inner cylindrical surface of the gas film layer	Wall
Axial end face of the gas film layer:	Pressure Outlet

below.

2.5. Solution of the Steady-State Characteristics of the Porous Medium Gas Bearing

The control equations of the porous medium gas bearing are nonlinear partial differential equations, making it very difficult to solve them directly using analytical methods. The fluid computational software FLUENT 19.1 not only has advantages in aspects such as the microscopic visualization of the flow field and the capture of complex flows near the boundary layer but it can also comprehensively evaluate the influence of the porosity of the porous medium layer, the viscous and inertial damping coefficients, etc., on the performance of the bearing. Therefore, in this paper, with the aid of the FLUENT software, the finite volume method is adopted to solve the simulation model of the porous medium gas bearing. The solution process of FLUENT is shown in Figure 13. In the pre-processing stage, the geometric model is discretized into a mesh model. Then, the quality of the mesh model is checked. The mesh model that meets the calculation requirements is imported into FLUENT. In the relevant modules, the geometric dimensions, gas state parameters, material physical properties, and calculation boundary conditions are set. Subsequently, iterative calculations are carried out until convergence is achieved.

3. Static Characteristics of the Porous Medium Gas Bearing

The static characteristics of the porous medium gas bearing mainly involve studying the situation in which the rotating shaft is stationary. By changing the supply gas pressure, the pressure change law within the air film layer is measured through a pressure sensor. Then, it is compared with the air film pressure from simulation results under the same working conditions to verify the accuracy of the parameter settings, such as the solver and physical property parameters of the porous medium material in the simulation model of the porous medium gas bearing. In order to measure pressure values at different positions in the axial and circumferential directions within the bearing clearance and obtain the air film pressure distribution, a pressure measurement test bench for the air film layer of the porous medium gas bearing has been designed, as shown in Figure 14.

As shown in Figure 14a, the pressure measurement device is composed of a base, a support, a bearing seat, a porous medium gas bearing, a pressure measuring shaft, etc. An air supply hole is included on the side of the bearing seat. External high-pressure air enters through the air supply hole, passes through the shaft sleeve and the porous medium layer, and then enters the bearing clearance. The pressure measuring shaft is installed inside the porous medium gas bearing. Figure 14b shows the pressure measuring shaft. There are four micro-holes—pressure collection holes—evenly distributed at the middle cross-section of the shaft, which are used to collect the pressure inside the air film. Four pressure collection channels, which communicate with the pressure collection holes, are processed axially on the shaft, and they are connected to the pressure sensor through a pressure-resistant tube to realize the collection of the pressure inside the air film layer. The NI data acquisition card collects the voltage value from the pressure sensor and converts it into a pressure value. The pressure sensor used is a Kistler high-precision transient pressure sensor, with a measurement range of 0–5 bar, a response frequency greater than 100 kHz, a linearity less than or equal to 0.2% FSO, and a sensitivity less than or equal to 0.1% FSO.

Figure 15 shows the pressure measurement test bench. The pressure measuring shaft is horizontally installed in the porous medium gas bearing, and the pressure collection holes of the pressure measuring shaft are aligned with the axial middle cross-section of the bearing, which can measure the pressure values at four positions: upper, lower, left, and right of the air film at the middle cross-section. By axially moving the pressure measuring shaft and changing the position of the pressure collection holes relative to the bearing, the collection of pressure values across the entire axial direction of the bearing is realized.

The overall view of the test device is shown in Figure 16. The power supply supplies power to the gas supply pressure gauge, flowmeter, and pressure sensors. The pressure-reducing valve adjusts the gas supply pressure, that is, the gas pressure entering the porous medium gas bearing. Four transient pressure sensors are connected to the pressure collection channels, and the NI data acquisition system is used to collect the pressure changes in real time and process and save the test data. The porous medium layer of the tested porous medium gas bearing is processed from isostatic pressing graphite porous medium material. The material density is 1.709 g/cm³, the permeability is 1.27 × 10⁻¹⁴ m², and the porosity is 0.1. The determination of parameters such as the bearing’s length, diameter, and clearance, as well as the thickness and permeability of the porous medium material, is based on determining the design input conditions according to the bearing usage requirements. The influence of different parameters on the bearing performance has been calculated, and the basic structural dimensions of the bearing have been determined through multi-parameter optimization. In the existing literature, there are many studies on the influence of bearing structural parameters on bearing performance. Therefore, this part of the content is not elaborated in detail in this paper, and the basic structural parameters of the studied bearing are directly provided. The geometric parameters of the pressure-measuring shaft and the porous medium gas bearing are shown in

Table 4. Main parameters of the porous medium gas bearing and the pressure-measuring shaft.

Component	Parameter	Value
Porous medium gas bearing	inner diameter/mm	12.5
	outer diameter/mm	19.5
	bearing length/mm	57
	thickness of porous medium layer/mm	3
	length of porous medium layer/mm	57
Pressure measuring shaft	outer diameter/mm	12.5
	length/mm	91.5
	diameter of pressure collection hole/mm	0.2
	diameter of axial pressure channel/mm	3

below.

The inner diameter of the porous medium gas bearing was measured to be 25.024 mm using a three-jaw inside micrometer, and the outer diameter of the pressure measuring shaft was measured to be 24.986 mm using an outside micrometer, resulting in a radial clearance of the bearing of 0.019 mm. According to this radial clearance and the geometric parameters of the bearing and the pressure-measuring shaft, a static simulation model of the porous medium gas bearing is established to obtain the air film pressure distribution under different gas supply pressures.

The pressure values and errors at different cross-sections are shown in Figure 17. Under different gas supply pressures, the pressure values inside the air film layer are all lower than the gas supply pressure, which indicates that there is a viscous loss when the gas flows through the porous medium layer, and the gas flow passing through the porous medium layer can generate a uniform pressure. The pressure distribution in the air film layer is characterized by higher pressure in the middle and lower pressures at both ends, approximately symmetrically distributed throughout the axis. This is consistent with the physical model of the porous medium gas bearing during its actual operation, in which gas enters from the outer cylindrical surface and exits from the two end faces of the air film. The error curve shows a trend indicating that the error is small in the middle and gradually increases at both ends. The reason for this is found to be related to problems such as more serious gas leakage when the pressure collection holes are closer to the end faces during the test process. However, the overall error is within an acceptable range, especially near the middle cross-section, where the error is less than 5%. To evaluate the measurement uncertainty for pressure in the gas film, the method presented by Bich [36] to determine the standard uncertainty has been adopted for sensors [36]. We repeated the test at least five times for each condition and took the average value to reduce the measurement uncertainties and avoid contingency. The errors of multi-measurement of pressure are influenced by a number of elemental error sources, such as unsteadiness in the “steady-state” phenomenon being measured, errors caused by imperfect installation of the transducer, etc. For each operating condition, the standard uncertainty is quite small. Therefore, the measurements were considered precise and reliable. This indicates that the established simulation model is reasonable and accurate in terms of the solver and the determination of the physical property parameters of the porous medium layer. At the same time, it has been proven that a gas bearing manufactured from the porous medium material can form an air film to achieve suspension of the rotating shaft.

4. Steady-State Characteristics of Porous Medium Gas Bearings

When a porous medium gas bearing is in operation, the coupled effect of static and dynamic pressure has a significant impact on the bearing’s performance. Taking the porous medium gas bearing in Table 3 as the research object, a porous medium gas bearing that is stationary and only affected by external gas supply is defined as a pure hydrostatic bearing. A porous medium gas bearing without an external gas supply and only affected by rotation is defined as a pure hydrodynamic bearing. The porous medium gas bearing studied in this paper, which is affected by both external gas supply and rotation, is referred to as a hybrid static–dynamic pressure porous medium gas bearing (simply referred to as a hybrid static–dynamic pressure bearing) in this section. By calculating the performance of the three types of bearings described above, a comparative study is conducted on the working mechanisms of static and dynamic pressure of porous medium gas bearings. The operating conditions for the three types of bearings are as follows: For the pure hydrostatic bearing, the rotational speed is 0.0 r/min, the gas supply pressure ratio is 6.0, and the permeability of the porous medium layer is 1.27 × 10⁻¹⁴ m². For the pure hydrodynamic bearing, the gas supply pressure ratio is 0.0, the rotational speeds are 40,000 r/min and 80,000 r/min, and the permeability of the porous medium layer is 0.0 m². For the hybrid static–dynamic pressure bearing, the gas supply pressure ratio is 6.0, the rotational speed ranges from 10,000 to 80,000 r/min, and the permeability of the porous medium layer is 1.27 × 10⁻¹⁴ m². The external ambient pressure under these working conditions is 0.1 MPa.

Figure 18 shows the dimensionless gas film pressure distribution curves along the axial mid-section of pure hydrostatic, pure hydrodynamic, and hybrid bearings at eccentricities of 0.4 and 0.8. In Figure 18a (eccentricity = 0.4), the pressure curve of the pure hydrodynamic bearing exhibits both positive and negative pressure regions, indicating that the rotational motion of the shaft generates hydrodynamic effects. The pressure difference between these regions provides the supporting force for the shaft. At high rotational speeds, the dimensionless pressure of the pure hydrodynamic bearing is lower than that of the pure hydrostatic and hybrid bearings, suggesting that external gas supply generally increases the gas film pressure and eliminates negative pressure regions in the circumferential direction. When the supply pressure ratio is 6.0, the peak pressure of the pure hydrostatic bearing is lower than that of the hybrid bearings at various rotational speeds, indicating that the rotational motion in hybrid bearings generates hydrodynamic effects that enhance the peak pressure. As the rotational speed of the hybrid bearing increases, the peak dimensionless pressure rises while the minimum pressure decreases, further demonstrating that hydrodynamic effects enlarge the circumferential pressure difference, which is beneficial for improving the bearing capacity. The dimensionless gas film pressures of pure hydrodynamic and hybrid bearings are not symmetrical across the x-axis, whereas that of the pure hydrostatic bearing is. Viscous resistance during shaft rotation results in a deflection angle, which increases with rotational speed in hybrid bearings. In Figure 18b (eccentricity = 0.8), the pressure distribution patterns of the three bearing types are similar to those at an eccentricity of 0.4, except that both the maximum and minimum pressures increase. Compared with an eccentricity of 0.4, the maximum gas film pressure of the pure hydrodynamic bearing increases by 35%. The larger eccentricity amplifies the wedge effect, leading to a more significant pressure difference between the convergent and divergent regions of the gas film. This increased circumferential pressure difference improves the bearing capacity.

Figure 19 presents the load capacity curves of three types of bearings at varying eccentricities. For the pure hydrostatic bearing, the calculation conditions are as follows: the supply pressure ratio is 6.0 (the pressure of the gas supplied into the porous medium divided by the external atmospheric pressure), and the rotating shaft speed is 0 r/min. For the pure hydrodynamic bearing, the calculation conditions are as follows: the external supply pressure ratio is 1.0, and the rotating shaft speeds are 40,000 r/min and 80,000 r/min. For the hybrid hydrostatic–hydrodynamic bearing, the calculation conditions are as follows: the external supply pressure ratio is 6.0, and the rotating shaft speeds are 40,000 r/min and 80,000 r/min. Eccentricity is defined as the ratio of the distance between the center of the journal and the center of the bearing bore to the radial clearance of the bearing. As the eccentricity increases, the load capacity of all three bearings rises, with the rate of increase accelerating. Compared with pure hydrostatic bearings, hybrid bearings exhibit higher load capacity at the same eccentricity, indicating that the hydrodynamic effect generated by rotational speed enhances the load-carrying capacity of the bearing. When compared with pure hydrodynamic bearings at the same rotational speed, hybrid bearings show higher load capacity, demonstrating that external gas supply also improves the load capacity of the bearing.

Figure 20 shows the principal stiffness variation curves for pure hydrostatic bearings, pure hydrodynamic bearings, and hydrostatic–hydrodynamic hybrid bearings. As the eccentricity increases, the principal stiffness of all three types of bearings rises, and the rate of increase accelerates, indicating that under large eccentricities; the principal stiffness of gas bearings exhibits strong nonlinearity. At the same supply pressure ratio, hybrid bearings and pure hydrostatic bearings have higher stiffness values than pure hydrodynamic bearings. At the same eccentricity, stiffness increases with rising rotational speed, demonstrating that the hydrodynamic effect of rotational speed improves the stiffness of porous media gas bearings. Higher stiffness increases the critical rotational speed of the shaft, enabling use under high-speed operating conditions. Compared with hybrid bearings, eccentricity has a more significant influence on pure hydrodynamic bearings: as eccentricity increases, the principal stiffness of pure hydrodynamic bearings rises at a greater rate, indicating that hybrid bearings exhibit better stiffness stability when eccentricity changes.

In the cross-stiffness variation curves of pure hydrodynamic bearings and hybrid bearings shown in Figure 21, the cross-stiffness of hybrid bearings exhibits slight fluctuations with increasing eccentricity at low rotational speeds of 10,000 r/min and 20,000 r/min. Starting from 30,000 r/min, as eccentricity increases, the fluctuations weaken, while the cross-stiffness curve first increases slowly and then rapidly. This is primarily due to the enhanced nonlinear variation of gas film force at larger eccentricities. Compared with pure hydrodynamic bearings at the same rotational speed, the cross-stiffness of hybrid gas bearings is more significantly affected by rotational speed, especially under large eccentricities, where cross-stiffness increases rapidly with rising speed. The main reason is that the external gas supply of hybrid bearings increases overall gas film pressure, and when superimposed with the influence of large eccentricity, the variation of gas film force becomes more drastic, leading to a rapid increase in cross-stiffness at large eccentricities.

5. Conclusions

By applying porous medium materials to air bearings and manufacturing porous medium gas bearings, the application of gas bearings in the aerospace field is fulfilled. Coupling the gas flow control equations within the porous medium layer and the air film layer, a simulation model for porous medium gas bearings is established, and simulation calculations and analyses are conducted. Experiments are used to verify the accuracy of the model calculations for static and steady-state characteristics, and the causes of errors are analyzed. The effects of parameters such as rotational speed, supply pressure ratio, and permeability on the steady state of the bearings are studied. The achievements are summarized as follows:

(1) A simulation model for porous medium gas bearings is established by coupling the flow control equations inside the porous medium layer with the gas flow control equations in the air film layer. An experimental device for measuring the pressure in the air film layer is designed and built, and experimental tests are carried out to investigate the pressure distribution in the air film layer of porous medium gas bearings.

(2) The simulation and experimental results show that under different supply pressures, the error in the air film pressure near the middle section of the bearing is less than 4%, and it slightly increases near the two ends. The static pressure test verifies the accuracy of the model solver and the physical property parameters of the porous medium materials. By referring to the research conclusions in the relevant literature, the accuracy of the model established in this paper for calculating the steady-state characteristics of the bearings is indirectly verified, with an error of less than 4%.

(3) On the basis of a comparative analysis of the performance of pure hydrostatic, pure hydrodynamic, and hybrid hydrostatic–hydrodynamic porous medium gas bearings, the hybrid mechanism is studied, i.e., the static pressure effect generated by external gas supply increases the air film pressure, eliminates negative pressure zones in the circumferential direction of the air film, and enhances bearing stability. The hydrodynamic effect produced by the high-speed rotation of the shaft increases the circumferential pressure difference in the air film, which is conducive to improving the bearing capacity. The eccentricity angle changes gently under different working conditions, indicating that the bearing has good operational stability.

(4) The mass flow rate decreases as the eccentricity ratio and rotational speed increase, and the decline is faster at high rotational speeds, indicating that the hydrodynamic effect at high rotational speeds has a rotational sealing effect. The main stiffness of the bearing increases nonlinearly as the eccentricity ratio increases, and it also increases as the supply pressure ratio and rotational speed increase. The rotational speed has a significant effect on cross-stiffness, and at large eccentricity ratios, the cross-stiffness increases rapidly as the rotational speed increases. Inside the porous medium gas bearing, the gas flow exhibits self-circulation, which makes the porous medium gas bearing adaptive and can reduce unstable oscillations of the rotor, laying a foundation for the application of porous medium gas bearings in the aerospace field.

In future research, based on the established model of the porous media gas bearing, we will adopt a self-developed embedded Six Degrees of Freedom approach to establish a transient strong coupling model of the rotor system supported by porous media gas bearings. Meanwhile, we will build a rotor system test bench to verify the accuracy of the rotor system model through experiments. Using the verified rotor system model, we will study the influences of parameter changes—such as rotational speed, supply pressure ratio, bearing clearance, residual unbalance, and dynamic load—on the dynamic characteristics of the rotor system supported by porous media gas bearings.