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Abstract

To solve the problem of performance degradation caused by supersonic phenomena under high gas supply pressures in orifice-compensated aerostatic journal bearings (OCAJBs), a double-row gas supply micron-groove-orifice aerostatic journal bearing (MGOAJB) is designed, and the SST k-ω turbulence model is used to analyze the influences of the supersonic phenomena on the gas film flow field under different gas supply pressures, gas film thicknesses and orifice diameters. Compared with traditional OCAJBs, under high gas supply pressure, supersonic phenomena occur near the gas film inlet in both the MGOAJB and the OCAJB. The Mach number, maximum pressure, temperature and Reynolds number of the OCAJB are higher than those of the MGOAJB, but the MGOAJB has a more stable gas film pressure due to pressure equalization. As the gas film thickness of the MGOAJB increases, the Mach number, Reynolds number and temperature near the gas film inlet increase, but the pressure decreases, the recovery of the pressure and temperature becomes more obvious when the gas velocity drops to a subsonic speed. With increases in the MGOAJB orifice diameter, the pressure, temperature and Reynolds number increase, the Mach number decreases, and the recovery of the pressure and gas temperature becomes more obvious.

Keywords

Micron-groove-orifice aerostatic journal bearing high gas supply pressure supersonic phenomena Mach number gas film pressure

Introduction

As a kind of externally pressurized gas-lubricated bearing, orifice-compensated aerostatic bearings are new, highly technical products that can be used in extreme conditions such as very high or very low temperatures, ultrahigh speeds and ultraprecise applications,¹ and their performance largely depends on the gas supply pressure and restrictor structure.

Increasing the supply pressure can improve the performance, such as the load capacity and stiffness of orifice-compensated aerostatic bearings, but with the continuous increase in the supply pressure, the gas velocity accelerates from subsonic to supersonic, resulting in self-excited vibrations, shock waves, and even negative pressure and other phenomena that reduce the load capacity and stiffness.^2–4 As early as 1961, researchers began to study the supersonic phenomena of orifice-compensated aerostatic bearings. Mori et al.^5,6 analyzed the supersonic flow in a bearing gap and studied the pressure distribution in the gas film. When the gas supply pressure increases continuously, the pressure around the outlet of the orifices decreases rapidly, and the pressure drops sharply, which affects the load capacity. In 1984, Miyake et al.^7,8 was the first researcher to propose the design concept to use gas under high pressure to lubricate heavily loaded bearings. Miyake et al. designed a very stiff bearing with a high gas supply pressure to take advantage of the fact that low pressure disturbance downstream of a supersonic flow cannot affect the pressure distribution upstream. Referring to the relevant design theories of the contraction sections in wind tunnels, other researchers⁹ proposed a wall profile design suitable for the contraction section of thrust bearings lubricated with high-pressure gas so that the gas flow could theoretically undergo uniform flow in the gap, that is, the flow direction was parallel to the bearing race. In the classical Reynolds lubrication equation, it is assumed that the inertia force of the fluid in the gas film is ignored, but under high gas supply pressure, the average gas velocity in a gas film increases, and the influence of inertial forces should be considered.¹ The Reynolds equation of classical lubrication theory is no longer applicable. In the literature,¹⁰ the Navier–Stokes (N-S) governing equations, including the inertia term, were solved using commercial computational fluid dynamics (CFD) software to calculate the supersonic flow field in a thrust bearing under high pressure. The numerical results and the conventional Reynolds equation were compared with the experimental results, and relatively accurate results were obtained. In the literature,^11,12 the SST k-ω turbulent model is used to solve the supersonic flow field of thrust bearings under high gas supply pressure, and the calculation results are accurate. Although researchers have made many significant achievements in the supersonic flow field of orifice-compensated aerostatic bearings under high gas supply pressure, they are mainly based on thrust bearings, while journal bearings widely used in engineering are rarely studied.

Improving the restrictor structure of orifice-compensated aerostatic bearings is another way to improve their performance.^13,14 By connecting the orifices with the pressure-equalizing grooves, a new micron-groove-orifice (MGO) structure is formed, which remarkably improves the load capacity and stiffness of the aerostatic bearing. Researchers^15,16 improved the load capacity and stiffness of aerostatic bearings by setting different numbers and positions of pressure-equalizing grooves on the bearing surface. Researchers have designed radial groove-compensated¹⁷ and trapezoidal groove-compensated aerostatic thrust bearings¹⁸ to study the influence of the structural parameters on the static characteristics of aerostatic thrust bearings, and the static stiffness, bearing capacity and stability of the bearings were improved. Literature¹⁹ reveals that the adoption of the MGOs can substantially improve the load capacity and stiffness of the bearing and make the bearing maintain a uniform stress, which enhances the operating accuracy and life of the bearing. By setting different forms of latitudinal and longitudinal grooves, the homogenization effect of these grooves on pressure is utilized to reduce the velocity of the pressure drop and increase the pressure on the atmospheric film flow field.²⁰ In the literature,²¹ the resistance network method (RNM) is utilized to numerically solve the Reynolds equation required in the static performance of journal bearings with rectangular grooves and has good precision.

Combining the two aspects mentioned above, that is, when the gas supply pressure is increased, whether the gas film flow field of micron-groove-orifice aerostatic bearings will produce a supersonic phenomenon, what are the differences from orifice-compensated aerostatic bearings, whether it is more suitable for working under high gas supply pressure, and how the physical parameters (such as the Mach number, pressure, temperature, etc.) of the gas film flow field change with the gas supply pressure and structural parameters under the influence of supersonic phenomenon, etc., are all worthy of study. Therefore, in this paper, a double-row micron-groove-orifice aerostatic journal bearing (MGOAJB) is designed to study the influence of supersonic phenomena on the gas film flow field of aerostatic journal bearings under high gas supply pressure, enriching relevant theories and providing a basis for engineering applications.

Bearing structure of MGOAJB and OCAJB

Figure 1(a) shows a schematic diagram of the MGOAJB. Eight gas supply holes are provided around the externally pressurized gas bearing, and a double-row gas supply is adopted. At the outlet of the orifices on the inner surface of the bearing, axial microgrooves are arranged to form the MGO structure with the orifices; there are eight MGOs in the aerostatic bearing. For comparison, the structure of the OCAJB is shown in Figure 1(b). To prevent cylindrical surface compensation (cylindrical surface throttling has a lower load capacity and stiffness than orifice compensation), a circular stingy cavity is set at the end of the gas supply orifice (the gas film inlet) to ensure that the restrictive section is the orifice cross-section.¹

Figure 1.

Schematic diagram of the structure and size of (a) MGOAJB and (b) OCAJB.

where $α$ is the position angle of the gas supply hole relative to the bearing centerline, $e$ is the rotor eccentricity, $D$ is the bearing diameter, $B$ is the bearing width, $d$ is the orifice diameter, $h_{g}$ is the groove depth, $b$ is the groove width, $l$ is the groove length, and $h_{m}$ is the average gas film thickness. To avoid gas-hammer self-excited vibration caused by the increasing gas volume in the grooves of the MGOs (to improve the load capacity and stiffness),¹ the microgrooves should be as small as possible, and the width and depth of the grooves are usually $h_{g} = (5 ~ 10) h_{m}$ and $b = (2 ~ 3) d$ , respectively.^15,16

In general, to prevent cylindrical surface compensation (cylindrical surface compensation leads to a lower load capacity and stiffness than orifice compensation), a circular cavity is set at the end of the orifice to ensure that the restrictive section is the orifice cross-section.¹ $d_{1}$ and $h_{1}$ are the diameter and depth of the circular cavity of the OCAJB, respectively, and according to Liu et al.,¹ $d_{1} > d^{2} / 4 h_{m}$ and $h_{1} > d / 4 - h_{m}$ .

The main structure size parameters of the MGOAJB and OCAJB are shown in Table 1.

Table 1.

Main structure size parameters of the MGOAJB and OCAJB.

Structure size parameters	Selected value
Diameter $D$ (mm)	80
Width $B$ (mm)	80
Orifice diameter $d$ (mm)	0.2
Gas film thickness $h_{m}$ (μm)	15–40
Groove depth $h_{g}$ (μm)	75
Groove width $b$ (mm)	0.5
Groove length $l$ (mm)	50
Circular cavity diameter $d_{1}$ (mm)	0.8
Circular cavity depth $h_{1}$ (μm)	40

Mathematical model and simulation

Theoretical calculation of the gas film flow field under high gas supply pressure

In previous studies, the classical Reynolds lubrication equation was the main means to calculate the gas film flow field. However, the Reynolds equation is derived on the premise that the Reynolds number is low (laminar flow) and the effect of inertia force is ignored. When the gas supply pressure and gas film thickness increase, flows become more complex, a supersonic region may be generated, and the influence of inertia force cannot be ignored.^1,23 New governing equations must be established for the gas film flow field under high gas supply pressure. Researchers found that the shear stress transport (SST) k-ω turbulence model can effectively simulate the complex flow state near the gas film inlet under a high gas supply pressure. Compared with the laminar flow model, the results are more accurate and more consistent with the real gas film flow,^11,12 and thus, the SST k-ω turbulence model is adopted in this paper to calculate the pressure distribution of a gas film flow field. The tensor forms of its governing equations are as follows^11,23:

Continuity equation:

\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x_{i}} (ρ u_{i}) = 0

(1)

Momentum equation:

\frac{\partial}{\partial t} (ρ u_{i}) + \frac{\partial}{\partial x_{j}} (ρ u_{i} u_{j}) = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (μ \frac{\partial u_{i}}{\partial x_{j}} - ρ \bar{{u'}_{i} {u'}_{j}}) + S_{i}

(2)

Energy equation:

\frac{\partial}{\partial t} (ρ T) + \frac{\partial}{\partial x_{j}} (ρ u_{i} T) = \frac{\partial}{\partial x_{j}} (\frac{λ}{c_{p}} \frac{\partial T}{\partial x_{j}} - ρ \bar{T' {u'}_{j}}) + S_{T}

(3)

Gas state equation:

\frac{p}{ρ} = RT

(4)

where $u$ is the velocity vector of the flow; $u_{i}$ and $u_{j}$ are the velocity tensors corresponding to each of the coordinate axes; $ρ$ is the gas density; $p$ is the gas pressure; $R$ is the gas constant; $T$ is the gas temperature; $λ$ is the heat transfer coefficient; $c_{p}$ is the specific heat capacity; the superscript “—” represents the average with respect to time; $u'_{i}$ , $u'_{j}$ , and $T'$ indicate the pulsation value of the corresponding physical quantity; $S_{i}$ is the generalized source terms on each coordinate axis; and $S_{T}$ is the viscous dissipative term, the expression of which is shown in Sun et al.¹¹

The term related to the turbulence pulsation value $- ρ \bar{{u'}_{i} {u'}_{j}}$ in the time-mean form governing equations is the Reynolds stress term. Since it is a new unknown quantity, if the governing equations are solved, it is necessary to establish a new expression for the Reynolds stress term (to introduce the turbulence model equation) and connect the pulsation value of turbulent flow with the time-mean value, as shown in the following:

- ρ \bar{{u'}_{i} {u'}_{j}} = μ_{t} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{2}{3} (ρ k + μ_{t} \frac{\partial u_{i}}{\partial x_{i}}) δ_{ij}

(5)

where $μ_{t}$ is the turbulent viscosity; $k$ is the turbulent energy; and $δ_{ij}$ is the Kronecker delta.

k = \frac{1}{2} \bar{{u'}_{i} {u'}_{j}} = \frac{1}{2} (\bar{{u'}^{2}} + \bar{{v'}^{2}} + \bar{{w'}^{2}})

(6)

The turbulent viscosity $μ_{t}$ is defined as

μ_{t} = \frac{ρ a_{1} k}{max (a_{1} ω, Ω F_{2})}

(7)

where $ω$ is the turbulence specific dissipation, $ω = ε / k$ , $ε$ is the turbulence dissipation, $Ω$ is the absolute value of the eddy, $F_{2}$ is the blending function, and $a_{1}$ =0.31.

F_{2} = \tanh {[max (2 \frac{\sqrt{k}}{0.99 ω y}, \frac{500 μ}{ρ y^{2} ω})]}^{2}

(8)

The transport equation of turbulent energy $k$ and the turbulence specific dissipation $ω$ equation¹¹ of the SST k-ω turbulence model are as follows:

Transport equation of turbulent energy:

\frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial x_{i}} (ρ k u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + G_{k} - Y_{k}

(9)

Turbulence specific dissipation equation:

\frac{\partial}{\partial t} (ρ ω) + \frac{\partial}{\partial x_{i}} (ρ ω u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + σ_{ω} μ_{t}) \frac{\partial ω}{\partial x_{j}}] + G_{ω} - Y_{ω} + D_{ω}

(10)

where $G_{k}$ is the turbulent kinetic energy, $G_{ω}$ is the specific dissipation equation, $Y_{k}$ and $Y_{ω}$ are the divergence terms of $k$ and $ω$ , respectively, $D_{ω}$ is the orthogonal divergence term, and $σ_{k}$ and $σ_{ω}$ are the turbulent Prandtl numbers of $k$ and $ω$ , respectively.

According to Liu et al.,¹ when the gas supply pressure is large, the influence of the inertial force cannot be ignored, and the Reynolds number will increase accordingly. Therefore, it is necessary to calculate the Reynolds number to reflect the complexity of the gas flow.

Re = ρ \frac{u_{i} h_{m}}{μ}

(11)

Simulation

According to the principle of computational fluid dynamics (CFD) simulations, the gas in the restrictor and the gas film of the MGOAJB or OCAJB are taken as the research object. As shown in Figure 2, SolidWorks software is used to establish a three-dimensional model of the flow field in the MGOAJB. In general, the high-pressure gas flow enters the orifice after the pressure-stabilizing chamber in the aerostatic bearing spindle system, so the inlet of the pressure-stabilizing chamber is set as the pressure inlet of the high-pressure gas flow. The axial boundary of the gas film is set as the outlet of the gas film pressure. Because the double-row gas supply structure has symmetry in the bearing axial direction, the symmetry face is set along the axial direction, and half of the structure is simulated for simplification.

Figure 2.

3D model of the gas film flow field of the MGOAJB.

The mesh tool in Workbench software, the method for partition grid division, is used to mesh the 3D model of the bearing flow field (Figure 3). For the MGO or orifice structure, the “Hex Dominant Method” command is used to divide the grid, and the element size is selected according to the geometrical dimensions of the orifices and grooves to ensure that there are enough grid cells and that the calculation time will not be too long due to excessive grids. Through trial and error, 0.05 mm is found to be appropriate; for the gas film, the “Sweep Method” and “Edge Sizing” commands are used to divide the gas film into 750 and 2100 equal parts along the axial and circumferential directions, respectively, to ensure that the element size of the grid is the same order of magnitude as that of the MGO and orifice. The “Face Meshing” command is used to divide the gas film into five equal parts along the gas film thickness direction. Finally, the boundary conditions are set according to the method shown in Figure 2, and the mesh files are exported.

Figure 3.

Meshing of the gas film flow field of the MGOAJB.

FLUENT software is used to simulate the flow field. According to Sun et al.,^11,12 the SST k-ω model is more suitable for calculating the supersonic flow field and shock wave with an adverse pressure gradient, and thus, the SST k-ω model is used in this study to simulate the flow field under high gas supply pressure. The SIMPLEC algorithm is selected for iterative calculations, and the relaxation factor is adjusted to improve the convergence of the calculation. The simulation calculation flowchart is shown in Figure 4.

Figure 4.

Flowchart of the simulation calculation.

In previous applications of aerostatic bearings, it is necessary not only to ensure sufficient load capacity of bearings but also to avoid the adverse effects brought by supersonic phenomena of the flow field. The use range of the gas supply pressure was limited between 0.4 and 0.6 MPa. However, to study the supersonic phenomena of the flow field of MGOAJB, reveal the changes in physical parameters in the supersonic region, and determine the differences and improvements compared with the OCAJB in this paper, the gas supply pressure is increased as much as possible to exceed 0.4–0.6 MPa limit. Considering existing experimental equipment with published research results, 0.9, 1.1, 1.3, and 1.5 MPa are adopted as the gas supply pressures at the pressure inlet in this paper,^11,12 the inlet temperature is 20°C, and the pressure outlet of the gas film is connected with the external atmosphere.

Simulation results

As shown in Figure 5, the gas Mach number, pressure and streamline are shown from left to right. It is found that the supersonic region of the aerostatic bearings gas film flow field is mainly near the orifice outlet (gas film inlet), which is consistent with Liu et al.¹ and Sun et al.^11,22 It can also be seen that when high-pressure gas ( $p_{0}$ = 1.5 MPa) enters the gas film through the orifice, the gas Mach number of the OCAJB ( $Ma$ = 1.8) near the gas film inlet is significantly higher than that of the MGOAJB ( $Ma$ = 1.4). This indicates that after the high-pressure gas enters the gas film through the MGO structure, the gas velocity is slower than that of the OCAJB, which is more conducive to maintaining higher gas film pressure. As shown in Figure 6, by comparing with the experimental data (the gas film pressure at the corresponding position can be obtained by the pressure sensors located near the orifice outlet), the simulation results and the experimental data have the same variation trend and a small numerical error. Therefore, the above conclusions are verified, and the SST k-ω turbulence model can effectively calculate the gas film flow field of aerostatic bearings under high gas supply pressures.

Figure 5.

Cloud image of the gas film flow field in the supersonic region under high gas supply pressures ( $h_{m}$ = 40 μm, $p_{0}$ = 1.50 MPa, $d$ = 0.20 mm): (a) MGOAJB and (b) OCAJB.

Figure 6.

A comparison of the simulation results and the experimental data ( $h_{m}$ = 40 μm, $p_{0}$ = 1.50 MPa, $d$ = 0.20 mm): (a) MGOAJB and (b) OCAJB.

The gas flowing through the gas film enters from the subsonic flow into the supersonic flow and then rapidly descends to the subsonic flow. This is because the gas flow from the orifice to the turning point of the gas film is similar to the flow along the outer folded wall, and the change in flow direction and the curvature of the stream line produce expansion waves of gas, which make the gas flow speed increase while the pressure, density and temperature sharply decrease.^22,23 Then, the supersonic flow is compressed by the wall of the gap between the gas film and generates a compression wave, which merges into a weak oblique shock wave or pseudoshock^5,23 and causes the Mach number to change from supersonic flow to subsonic flow; on the other hand, the gas pressure begins to recover.

As shown in Figures 7 and 8, when the gas film thickness is reduced to $h_{m}$ = 20 μm, the maximum Mach number near the gas film inlet of the MGOAJB is less than 1.0 ( $Ma$ = 0.76), while that of the OCAJB is still greater than 1.0 ( $Ma$ = 1.09), resulting in supersonic phenomena. The gas film pressure of the MGOAJB is higher than that of the OCAJB and can maintain a higher value under the action of MGO-structure pressure equalization because after the high-pressure gas flows out through the orifice, the pressure equalizing groove in the MGOAJB performs secondary compensation to slow down the violent turning and the collision between gas molecules when the gas enters the gas film. Thus, the gas flow becomes relatively smooth, and the gas flow velocity is reduced. Although the OCAJB has circular cavities, their main function is to ensure that the restrictive section is the orifice cross-section to avoid the instability caused by cylindrical surface compensation,¹ and the pressure equalizing effect is very limited. Therefore, after the high-pressure gas flows out of the orifice, it almost directly enters the gas film, the flow turning and molecular collision are violent, and the gas is choked in the circular cavities,^3,23 which easily leads a dramatic change in the gas flow velocity and the gas film pressure. This indicates that compared with the OCAJB, the MGOAJB can effectively reduce the gas flow velocity and weaken the supersonic phenomenon under high gas supply pressure. When the gas film thickness is reduced (e.g. $h_{m}$ = 20 μm), the MGOAJB can even avoid the supersonic phenomenon and possible adverse effects compared with the OCAJB and obtain greater gas film pressure, thus improving the performance of aerostatic bearings.

Figure 7.

Cloud image of the gas film flow field in the supersonic region under high gas supply pressures ( $h_{m}$ = 20 μm, $p_{0}$ = 1.50 MPa, $d$ = 0.20 mm): (a) MGOAJB and (b) OCAJB.

Figure 8.

Cloud image of the gas film flow field in the supersonic region under high gas supply pressures ( $h_{m}$ = 20 μm, $p_{0}$ = 1.50 MPa, $d$ = 0.20 mm): (a) Mach number of gas flow $Ma$ and (b) gas film pressure $p$ .

Results and discussion

Gas supply pressure

As shown in Figure 9, when high-pressure gas ( $p_{0}$ = 0.9–1.5 MPa) enters the gas film inlet, the gas flow Mach number, gas film pressure, temperature and Reynolds number of the aerostatic bearing change abruptly near the film inlet. The reason is that at the inlet of the gas film, the gas produces an expansion wave and turns in the direction, and the gas velocity accelerates sharply while the pressure drops sharply. Not far from the gas film inlet, the supersonic flow is compressed by the wall of the gap between the gas film and generates a compression wave, and the gas velocity rapidly decreases to subsonic.²³ As shown in Figure 9(a), in the MGOAJB, the flow Mach number $Ma$ increases rapidly from 0.55 to more than 1.0 near the gas film inlet and reaches a maximum of 1.19–1.40 at a distance of $r$ = 0.25 mm from the inlet (the higher the gas supply pressure is, the greater the Mach number), and then the Mach number drops rapidly to subsonic speed and gradually becomes stable at a distance of $r$ = 1.50 mm from the gas film inlet. In the OCAJB, the flow Mach number reaches a maximum of 1.58–1.87 at a distance of $r$ = 0.20 mm. Under the same gas supply pressure, due to the lack of a structure like the MGO to smooth the gas flow, the gas flow velocity is significantly higher than that of the MGOAJB (32.7%–33.6%), and the supersonic phenomenon is more obvious. Then, the gas flow velocity rapidly decreases to subsonic speed and continues to decrease with increasing $r$ . There is no obvious stable state similar to that of the MGOAJB.

Figure 9.

Influence of the gas supply pressure on the supersonic phenomena in the MGOAJB and OCAJB ( $h_{m}$ = 40 μm, $d$ = 0.20 mm): (a) Mach number of gas flow $Ma$ , (b) gas film pressure $p$ , (c) gas temperature $T$ , and (d) gas Reynolds number $Re$ .

As shown in Figure 9(b), high-pressure gas ( $p_{0}$ = 0.9–1.5 MPa) enters the gas film inlet, as the gas speeds up rapidly, the gas film pressure drops rapidly, part of the pressure reduction in this section is converted into gas kinetic energy to increase the gas flow speed, and the remainder is related to local pressure loss due to the abrupt change of the restrictive section.^11,23 When the gas flow enters the supersonic state and reaches the maximum Mach number, the gas film pressures of the MGOAJB and OCAJB decreased to minimum gas pressures of 0.25, 0.28, 0.33, and 0.35 MPa and 0.15, 0.14, 0.13, and 0.12 MPa, respectively, in the supersonic region (Figure 10(a)). Under the same gas supply pressure, because the flow turns and the molecules collide more violently, the local pressure loss is greater, and the gas pressure of the OCAJB drops more obviously. Further, the higher the gas supply pressure is, the more serious the pressure drop is. When the gas flow begins to descend rapidly from the maximum Mach number, the gas flow velocity decreases rapidly from supersonic to subsonic, and the gas film pressures of the MGOAJB and OCAJB increase by 0.046, 0.057, 0.067, and 0.076 MPa and 0.16, 0.21, 0.28, and 0.32 MPa, respectively (Figure 10(b)), while the maximum gas film pressures are 0.30, 0.34, 0.40, and 0.43 MPa and 0.31, 0.35, 0.41, and 0.45 MPa, respectively (Figure 10(c)). Under the same gas supply pressure, the pressure recovery and the maximum gas film pressure of the OCAJB are higher than those of the MGOAJB (247.8%–321.1% and 3.3%–4.6%), and the higher the gas supply pressure is, the more obvious the difference, which indicates that the gas film pressure of the OCAJB is more affected by the supersonic phenomenon. Although the maximum gas film pressure in the OCAJB is slightly higher than that in the MGOAJB, as the distance from the gas film inlet $r$ increases, the gas film pressure continues to decrease (0.13–0.17 MPa). However, the MGOAJB can still maintain a certain gas film pressure (0.28–0.42 MPa) under the action of MGO-structure pressure equalization, thus improving the performance of bearings.

Figure 10.

Change in the gas film pressure in the supersonic region: (a) minimum gas film pressure in the supersonic region, (b) recovery of gas film pressure, and (c) maximum gas film pressure.

As shown in Figure 9(c), when the high-pressure gas enters the gas film, the gas temperature $T$ decreases rapidly to approximately 20°C as the gas velocity increases to the supersonic state. When the gas flow velocity decreases rapidly from supersonic to subsonic, the gas temperature begins to rise, the maximum gas temperatures of the MGOAJB and OCAJB appear at $r$ = 0.5–1.5 mm away from the gas film inlet, and the maximum gas temperature of the MGOAJB is slightly away from the gas film inlet. The maximum gas temperatures of the two bearings under different gas supply pressures ( $p_{0}$ = 0.9, 1.1, 1.3, and 1.5 MPa) are 43.3°C, 45.2°C, 47.0°C, and 50.6°C and 45.9°C, 48.6°C, 50.7°C, and 52.5°C, respectively. Meanwhile, the maximum temperature in the OCAJB is slightly higher than that in the MGOAJB because kinetic energy is lost due to the collision of gas molecules when the gas velocity goes from supersonic to subsonic, and this loss is converted into heat energy, which increases the temperature of the gas.¹² In the OCAJB, the turning and molecular collisions are more intense after the flow enters the gas film, so the temperature of the gas is higher. However, the change in the temperature is less dramatic than that in the pressure, compared with the gas film pressure and the flow Mach number, and the maximum temperature occurs at a position slightly farther from the gas film inlet (approximately $r$ = 1.0–1.5 mm). In addition, the higher the gas supply pressure is, the farther the maximum temperature is from the gas film inlet. To prevent accuracy reduction or even shaft holding caused by the temperature rise, the cooling system should be appropriate in the shaft system.

The Reynolds number of the gas near the gas film inlet varies with the gas supply pressure, as shown in Figure 9(d). Because the flow turns and becomes more complicated, the Reynolds number at the gas film inlet ranges from 3200 to 3700 and decreases rapidly as $r$ increases under high gas supply pressures. As the gas continues to flow downstream, the gas flow gradually moves away from the gas film inlet, and the flow gradually tends to stabilize in the range of $r$ = 0–1.5 mm, the Reynolds number decreases by approximately 75%, and then the decline rate slows and levels off. The higher the gas supply pressure is, the higher the Reynolds number near the gas film inlet, and the more complicated the flow state, which is obvious in the range of $r$ = 0.45–4.0 mm (the maximum difference is 28.01%). However, at the gas film inlet, the Reynolds numbers caused by different gas supply pressures differ by a statistically nonsignificant amount (less than 12.16%); when $r$ = 5.0 mm, the Reynolds number drops to the same level ( $Re$ = 227–272). For the OCAJB, the Reynolds number at the gas film inlet is basically equal to that of the MGOAJB. However, when the flow becomes complicated due to the influence of the circular gas cavity, the Reynolds number increases at $r$ = 0.45–0.50 mm, and although it subsequently decreases, it is still higher than the value in the MGOAJB at the same position. For example, when $r$ = 5.0 mm, $Re$ = 703–968, which is 2.55 times that in the MGOAJB.

Gas film thickness

As shown in Figure 11(a), in MGOAJBs, when high-pressure gas ( $p_{0}$ = 1.50 MPa) enters the gas film through the orifices, the gas flow Mach number rises and reaches the maximum value at approximately $r$ = 0.20 mm from the gas film inlet due to the change in the restrictive section. Under different gas film thicknesses ( $h_{m}$ = 20, 30, 40, and 50 μm), the maximum gas flow Mach numbers of the MGOAJB gas film flow field are 0.79, 1.05, 1.40, and 1.61. When $h_{m}$ = 30–50 μm, the maximum gas flow Mach number is more than 1.0 ( $Ma$ = 1.05–1.61), which indicates that with the increase in the gas film thickness, the gas flow velocity gradually accelerates. This result is due to the increase in the thickness of the gas film, the gas film outlet also becomes larger, which causes the high-pressure gas entering the gas film to be lost too quickly, resulting in an increase in the gas velocity and a decrease in the gas film pressure. However, when the gas drops rapidly from the maximum Mach number to subsonic speed, the gas flow Mach number is relatively stable (especially between $r$ = 1.5–5.0 mm), and there is no significant difference for different gas film thicknesses.

Figure 11.

Influence of the gas film thickness on the supersonic phenomena in the MGOAJB ( $p_{0}$ = 1.50 MPa, $d$ = 0.20 mm): (a) Mach number of the gas flow $Ma$ , (b) gas film pressure $p$ , (c) gas temperature $T$ , and (d) gas Reynolds number $Re$ .

As shown in Figure 11(b), high-pressure gas ( $p_{0}$ = 1.50 MPa) enters the gas film through the MGO structure. Under different gas film thicknesses ( $h_{m}$ = 20, 30, 40, and 50 μm), the gas film pressure drops to 0.81, 0.52, 0.35, and 0.26 MPa when the gas flow enters the supersonic state and reaches the maximum Mach number. When the gas rapidly descends to subsonic speed, the gas film pressure returns to the maximum pressure of 0.86, 0.58, 0.43, and 0.35 MPa, and the increase is 0.051, 0.064, 0.076, and 0.088 MPa, respectively. Then, the gas film pressure is maintained under the action of MGO-structure pressure equalization. When the gas film thickness increases, the loss of high-pressure gas in the gas film is intensified, the gas velocity increases, sometimes to the point of a supersonic phenomenon, the gas film pressure drops more obviously in the supersonic region (0.81–0.26 MPa), and the gas film pressure rebound is more obvious (0.051–0.088 MPa) when the gas descends to subsonic speeds, but the maximum pressure in the film still decreases (0.86–0.35 MPa) as the gas film thickness increases. When $h_{m}$ = 20 μm, the maximum Mach number in the gas film is less than 1.0, the supersonic phenomenon does not occur, and the gas film pressure is significantly higher than $h_{m}$ = 30, 40, and 50 μm (47.4%–147.4%). The larger the film thickness is, the faster the gas flow velocity, the more obvious the supersonic phenomenon, and the more serious the gas film pressure loss.

As shown in Figure 11(c), under high supply pressure, the temperature of the gas in the gas film gap drops to approximately 21.1°C in the supersonic region. When the gas rapidly descends to subsonic speed, the temperature increases. With the increase in the gas film thickness, the gas flow and velocity increase, the motion and collision between gas molecules become more intense, and the kinetic energy loss increases, leading to a rise in the gas temperature,¹² with maximum temperatures of 43.8°C, 47.6°C, 50.6°C, and 53.5°C. When $h_{m}$ = 20 μm, the maximum temperature (43.8°C) occurs at $r$ = 1.16 mm from the gas film inlet. Because the gas changes more dramatically, the heat cannot be transferred in time,¹ and as the gas film thickness increases, the maximum temperature gradually moves away from the gas film inlet. When $h_{m}$ = 50 μm, the maximum temperature (50.6°C) occurs at $r$ = 1.70 mm from the gas film inlet.

As shown in Figure 11(d), under high gas supply pressure, the Reynolds number rapidly decreases from the gas film inlet with increasing $r$ , and the influence of the gas film thickness on the Reynolds number is mainly reflected at the gas film inlet. A thicker gas film has less on an obstructive effect on the flow and corresponds to a faster gas flow velocity and a larger Reynolds number at the gas film inlet ( $Re$ = 3055–5011). However, when $r$ = 1.0–2.0 mm, the Reynolds number difference caused by the gas film thickness gradually decreases and is basically constant when $r$ = 5.0 mm ( $Re$ = 208–288).

In summary, although the MGOAJB has a lower gas flow velocity than the OCAJB under high gas supply pressure and the supersonic phenomenon is weaker than that of the OCAJB, it is necessary to make an appropriate choice for the gas film thickness in the design process of the MGOAJB under high gas supply pressure. If the gas film is too thick, the gas flow velocity is faster, the supersonic phenomenon is more obvious, the pressure loss is greater in the gas film and the gas temperature rises, leading to the load capacity and accuracy decline.

Diameter of the orifice

As shown in Figure 12(a), in MGOAJBs, when high-pressure gas ( $p_{0}$ = 1.50 MPa) enters the gas film, the gas velocity rises rapidly and reaches supersonic speeds ( $Ma$ ≥ 1.0) due to the change in the restricted section. The maximum Mach numbers are 1.82, 1.66, 1.40, 1.27, and 1.18 under different orifice diameters ( $d$ = 0.10, 0.15, 0.20, 0.25, and 0.30 mm). When the diameter of the orifice is larger, the throttling effect on the gas flow decreases, according to the law of mass conservation, and under the condition of a constant gas supply, the orifice diameter increases, and the gas velocity decreases. When $d$ = 0.10 mm, the maximum Mach number is 1.82 at a distance of $r$ = 0.05 mm from the gas film inlet. When $d$ =0.30 mm, the maximum Mach number is 1.18 at $r$ = 0.30 mm from the gas film inlet, and with the increase in the diameter of the orifice, the position of the maximum Mach number gradually moves away from the gas film inlet ( $r$ = 0.05–0.30 mm). Then, the gas flow decreases rapidly to subsonic speed; however, the flow Mach number in the subsonic region increases slightly with increasing orifice diameter.

Figure 12.

Influence of the orifice diameter on the supersonic phenomena in the MGOAJB ( $h_{m}$ = 40 μm, $p_{0}$ = 1.50 MPa): (a) Mach number of the gas flow $Ma$ , (b) gas film pressure $p$ , (c) gas temperature $T$ , and (d) gas Reynolds number $Re$ .

As shown in Figure 12(b), high-pressure gas ( $p_{0}$ = 1.50 MPa) enters the gas film through the orifice; as the gas velocity increases rapidly to supersonic speed, the gas film pressure drops to the lowest point rapidly. Under different orifice diameters ( $d$ = 0.10, 0.15, 0.20, 0.25, and 0.30 mm), the gas film pressure drops to 0.13, 0.21, 0.35, 0.53, and 0.62 MPa, respectively, in the supersonic region. When the gas flow rapidly descends from supersonic to subsonic, the gas film pressure returns to 0.17, 0.26, 0.43, 0.61, and 0.72 MPa, and the increase is 0.045, 0.053, 0.076, 0.083, and 0.096 MPa, respectively. With the increase in diameter of the orifice, the position of the maximum gas film pressure gradually moves away from the gas film inlet ( $r$ = 0.05–0.30 mm). Then, the gas film pressure is maintained under the action of MGO-structure pressure equalization. When the orifice diameter increases, the loss of gas film pressure decreases in the supersonic region, the recovery of gas film pressure is more obvious when the gas flow descends from supersonic to subsonic, and the maximum gas film pressure increases. This is because when the orifice diameter increases, the throttling effect is weakened, and the gas velocity decreases, leading to an increase in the gas film pressure in the supersonic region. However, with the increase in orifice diameter, the size difference between the orifice diameter and the gas film thickness increases, and the turning of high-pressure gas into the gas film becomes more intense; therefore, the gas film pressure recovery is more obvious in the subsonic region, and the maximum gas film pressure is higher.

As shown in Figure 12(c), under high supply pressure ( $p_{0}$ = 1.50 MPa), the temperature of the gas entering the gas film drops to 20.1°C when the gas velocity increases rapidly to supersonic speed and subsequently rises when the gas descends from supersonic to subsonic speed. As the orifice diameter increases, the high-pressure gas more intensely turns into the gas film, and the kinetic energy loss increases, leading to an increase in the gas temperature. The maximum temperatures are 44.5°C, 47.1°C, 50.6°C, 53.0°C, and 55.6°C. The maximum temperature occurs at a distance of $r$ = 1.38–1.50 mm from the gas film inlet.

As shown in Figure 12(d), under high gas supply pressure, the Reynolds number rapidly decreases from the gas film inlet with increasing $r$ ; with increasing diameter of the orifice, the Reynolds number of the gas near the gas film inlet gradually increases. As mentioned above, when the orifice diameter increases, the size difference between it and the gas film thickness increases, and the turning of high-pressure gas becomes more intense after entering the gas film; thus, the flow state becomes more complicated, and the difference is most obvious (369.4%) at $r$ = 1.01 mm.

In summary, it is necessary to make an appropriate choice of orifice diameter in the design process of the MGOAJB for operation under high supply pressure. If the diameter of the orifice is too small, the gas flow velocity is too fast, and the gas film pressure is greatly reduced. However, if the diameter of the orifice is too large, the dimensional difference from the gas film thickness becomes more significant, which results in the film pressure changing more dramatically when the gas flow changes from supersonic to subsonic speed. The diameter of the orifice is typically controlled between 0.15 and 0.25 mm.

Conclusions

A double-row gas supply micron-groove-orifice aerostatic journal bearing is designed, and the SST k-ω turbulence model is used to analyze the influences of the supersonic phenomena on the gas film flow field under different gas supply pressures, gas film thicknesses and orifice diameters. Under high gas supply pressure, supersonic phenomena occur near the gas film inlet in both the MGOAJB and OCAJB. After the high-pressure gas enters the gas film, the flow Mach number rapidly increases to supersonic and then sharply decreases to subsonic, while the pressure and temperature rapidly decrease and subsequently recover to the maximum value, and the Reynolds number reflects the complexity of the gas flow. The main conclusions are as follows:

With increasing gas supply pressure, the Mach number, gas film pressure, temperature and Reynolds number increase in the gas film under the influence of the supersonic phenomenon. Under the same gas supply pressure, the Mach number, Reynolds number and temperature of the gas flow in the OCAJB are higher than those of the MGOAJB, and the supersonic phenomenon is more obvious. Although the maximum gas film pressure of the OCAJB is slightly higher than that of the MGOAJB when the gas velocity drops to subsonic speed, the gas film pressure in the MGOAJB is more stable and higher than that in the OCAJB due to pressure equalization by the MGO structure.

With increasing gas film thickness of the MGOAJB, the Reynolds number of the gas increases, especially at the inlet of the gas film. The Mach number and temperature increase, but the gas film pressure decreases. The recovery of the gas film pressure and gas temperature becomes more obvious when the gas velocity drops to subsonic speed, as does the supersonic phenomenon in the gas film of the MGOAJB.

With increases in the MGOAJB orifice diameter, the gas film pressure, temperature and Reynolds number gradually increase, but the Mach number decreases, and the recovery of the gas film pressure and gas temperature becomes more obvious when the gas velocity drops to subsonic speed.

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Natural Science Foundation of Heilongjiang Province, Grant No. E2018001.

ORCID iDs

Shusen Li

Wei Cui

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Research on the influence of supersonic phenomena on the gas film flow field of micron-groove-orifice aerostatic journal bearings under high gas supply pressure

Abstract

Keywords

Introduction

Bearing structure of MGOAJB and OCAJB

Mathematical model and simulation

Theoretical calculation of the gas film flow field under high gas supply pressure

Simulation

Simulation results

Results and discussion

Gas supply pressure

Gas film thickness

Diameter of the orifice

Conclusions

Footnotes

Appendix

Declaration of conflicting interests

Funding

ORCID iDs

References