Effects of Grooved Surfaces and Lubrication Media on the Performance of Hybrid Gas Journal Bearings

Abstract

Gas bearings are attractive for sustainable, high-speed, and cryogenic applications, where gases replace liquid lubricants. This study numerically analyzed hybrid gas journal bearings lubricated with hydrogen, nitrogen, air, and helium, and quantifies the impact of circumferential micro-grooves. The compressible Reynolds equation was solved by the finite element method with constant-flow valve restrictors, while Gauss–Seidel iterations were used for convergence. The model was verified against published theoretical and experimental data with maximum deviations below 6%, and mesh independence is confirmed. The parametric results show that the gas type and texturing jointly controlled static and dynamic performance. Helium (highest viscosity) yielded the largest minimum film thickness, whereas hydrogen (lowest viscosity) attained higher peak pressures at a lower film thickness for a given load. Grooves redistributed pressure and reduced both the maximum pressure and the minimum film thickness, but they also lowered the frictional torque. Quantitatively, the hydrogen-lubricated grooved bearing reduced the frictional torque by up to 50% compared with the non-grooved air-lubricated bearing at the same load. Relative to air, hydrogen increased stiffness and damping by up to 10% and 50%, respectively, and raised the stability threshold speed by 110%. Conversely, grooves decreased the stiffness, damping, and stability threshold speed compared with non-grooved surfaces, revealing a trade-off between friction reduction and dynamic stability. These findings provide design guidance for selecting gas media and surface texturing to tailor hybrid gas journal bearings to application-specific requirements.

1. Introduction

Nowadays, a wide variety of lubricants are being developed for tribological systems to obtain the desired lubricating performance [1,2,3,4,5]. Hydrogen is a green energy source as it is clean, non-toxic, and has no greenhouse effect [3,6,7]. Hydrogen energy is used in various applications, such as solid oxide batteries, molten carbonate fuels, hydrogen-powered vehicles, and hydrogen power production [8,9,10]. Furthermore, space transfer power systems and launching vehicles require long-life liquid hydrogen turbopumps [11,12,13]. Moreover, hydrogen turbo-expanders are crucial devices used for liquefying hydrogen and directly depend on the behavior of hydrogen gas-lubricated bearings [14]. Generally, the proper functioning of gas bearings is important to obtain the desired performance of turbo-expanders. Yang et al. [15] investigated the behavior of orifice-compensating double-row hole-entry hybrid aerostatic bearings. They used the finite difference method (FDM) for solving the governing Reynolds equation and analyzed the influences of the numbers and location of hole entry on bearing performance indicators. The authors reported that six holes provided better bearing stability compared with three holes. Leonard and Rowe [16] provided the mathematical correlations for obtaining the dynamic parameters of hydrostatic aerostatic bearings. They presented analytical expressions to calculate the dynamic forces and derived dynamic coefficients that can be used for bearings with different pockets. Xiao et al. [17] presented a numerical model to investigate the performance of hybrid aerostatic journal bearings. They revealed that the aerodynamic effect enhances the static performance of the bearing system when the bearing operates at high speeds with large eccentricities. Lo et al. [18] applied the FDM to analyze the behavior of hole-entry aerostatic bearings and provided the design parameters valid for non-circular journal, slider bearings, and thrust bearings. Liu et al. [19] numerically explored the behavior of externally pressurized gas bearings. Thus, the authors determined the distribution of gas film pressure and bearing load-carrying capacity considering both aerostatic and aerodynamic effects. Li et al. [20] used flow difference as the feedback in the Gauss–Seidel iterative technique and examined the effects on load-carrying capacity and stiffness coefficients. Their findings confirmed that this approach was insensitive to initial conditions and useful for decreasing iteration times. Furthermore, Gao et al. [21] used the finite element method (FEM) for aerostatic journal bearings to examine the combined influence of aerodynamic and aerostatic effects on aerostatic bearings. Cui et al. [22] examined the effects of manufacturing errors on the performance of aerostatic bearings using the FEM. They reported that the deflection angle and manufacturing errors substantially affect the distribution of gas film pressure. Bouchehit [23] investigated the use of various gases, including hydrogen, air, helium, and pentafluoropropane, as lubricants in journal bearings. They reported that hydrogen-lubricated bearings can achieve lower frictional torque compared with other gases, making it a promising ecological alternative to conventional liquid lubrication. Recently, Liu et al. [24] computed the static characteristics of organic working fluid journal bearings, taking into account microscale effects to better understand their performance in organic Rankine cycle systems. They analyzed the influence of key design parameters and found that the eccentricity ratio and gas film thickness were the most influential factors affecting load-carrying capacity and attitude angle, while surface roughness significantly enhanced load capacity.

Nowadays, the application of textured surfaces in a tribological system is an efficient method to enhance their performance. Surface texturing is typically understood to be deterministic and well-organized, whereas roughness is a random process and inherent in nature [25,26,27,28,29,30]. Textured surfaces can be created on bearing surfaces as discrete (e.g., micro-dimples) or continuous patterns (e.g., micro-grooves). These textured surfaces also act as lubricant reservoirs, entrapping foreign particles and preventing the deterioration of the bearing surface [31,32,33,34]. The advancement of several precision manufacturing technologies in recent years, including laser surface texturing [35,36], chemical etching [37,38], vibro-machining texturing [39], novel dressing techniques [40,41], etc., has made it possible to create textured bearing surfaces with higher-order accuracy down to the micro level. Bonneau et al. [42] used the FEM to consider the effects of grooved surfaces on gas thrust bearings and face seals. The pressure distribution and leakage flow are discussed in their study. They also reported that the FEM is an efficient numerical technique for designing gas bearings and face seals compared with the finite difference method. Wang et al. [43] used the FDM to analyze a flexible rotor supported by gas journal bearings with grooves. They presented the relationship of the rotor mass with the bearing number, considering the dynamic behavior of the system. Chen et al. [44] scrutinized the effects of circumferential grooves on the behavior of aerostatic thrust bearings. They revealed that grooved surfaces could be beneficial to obtain an enhanced load-carrying capacity. Du et al. [45] investigated the influence of grooved surfaces on an externally pressurized gas journal bearing using the FEM. They concluded that pressure-equalizing grooves opening in axial and circumferential directions enhance loading capacity. Luan et al. [46] examined the effects of herringbone grooves on gas foil thrust bearings using the FDM. They obtained the optimum design parameters for a gas foil thrust bearing with grooves to obtain a larger load-carrying capacity. Recently, Zhang [2] reported that micro-groove textures can significantly enhance the lubrication performance of water-lubricated bearings by improving load-carrying capacity and reducing friction.

To summarize, the published literature related to grooved hybrid gas bearings is rather limited and is primarily confined to externally pressurized gas thrust bearings. The earlier-mentioned studies reveal that the application of grooved surfaces significantly influences the performance of tribological systems. Furthermore, hydrogen gas has been a new area of attention for tribologists due to its characteristics, such as being non-toxic and clean, and having non-greenhouse effects. It has also been noticed that different lubricant gases significantly affect the performance of gas bearings. The analysis of hybrid gas bearings has not yet been performed, considering the influence of grooves and different gases. Consequently, a more comprehensive numerical model is required to compute the influence of the application of grooved surfaces and different gases on the performance of a hybrid gas journal bearing. The present study deals with the comparison of different lubrication media in a hybrid gas journal bearing, along with the influence of grooved surfaces. Hydrogen, nitrogen, air, and helium gases were considered to analyze the behavior of different gases on the performance characteristics of hybrid gas bearings compensated with constant flow valve restrictors. The FEM was used to obtain bearing performance indicators, i.e., the minimum gas film thickness, pressure distribution, frictional torque, stiffness, and dynamic coefficients.

2. Theory and Methods

2.1. Problem Formulation

The bearing was mathematically modeled to consider the influence of grooved surfaces on hybrid gas journal bearings, such as those being used in turbomachinery. Three-dimensional representations of the non-grooved and grooved hole-entry gas bearings are depicted in Figure 1a,b, respectively. The developed surface of the bearing with a grooved surface is shown in Figure 1c. The gas was supplied in the bearing clearance domain through two rows of supply holes by constant flow valve restrictors. The latter were used to prevent the interruption of flow in the bearing and maintain a constant flow rate [47]. The micro-grooves were applied on the bearing surface in the circumferential direction in front of the supply holes.

2.2. Reynolds Equation

The Reynolds equation to determine the film pressure distribution for gas journal bearings for steady-state conditions is given as [48,49,50]

$${\displaystyle \frac{\partial }{\partial x}}\left({\rho h}^3{\displaystyle \frac{\partial p}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\rho h}^3{\displaystyle \frac{\partial p}{\partial y}}\right)=6\mu U{\displaystyle \frac{\partial \left(\rho h\right)}{\partial x}} ,$$

(1)

Assuming isothermal and isoviscous conditions, and using the perfect gas equation, the above equation is converted into

$${\displaystyle \frac{\partial }{\partial x}}\left({ph}^3{\displaystyle \frac{\partial p}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({ph}^3{\displaystyle \frac{\partial p}{\partial y}}\right)=6\mu U{\displaystyle \frac{\partial \left(ph\right)}{\partial x}} ,$$

(2)

with the pressure p, film height h, viscosity μ, velocity U, and the spatial coordinates x and y. The following parameters are used to convert the above equation into a dimensionless form:

$$\overline{h}={\displaystyle \frac{h}{c}}, \alpha ={\displaystyle \frac{x}{r}}, \beta ={\displaystyle \frac{y}{r}} \overline{p}={\displaystyle \frac{p}{p_{\mathrm{s}}}}, \mathrm{\Lambda }={\displaystyle \frac{6\mu Ur}{c^2{p}_{\mathrm{s}}}}.$$

Thereby, c refers to the radial clearance, r is the bearing radius, and p_s is the supply pressure.

Thus, the dimensionless Reynolds equation for gas journal bearings is given as

$${\displaystyle \frac{\partial }{\partial \alpha }}\left({\overline{p}\overline{h}}^3{\displaystyle \frac{\partial \overline{p}}{\partial \alpha }}\right)+{\displaystyle \frac{\partial }{\partial \beta }}\left({\overline{p}\overline{h}}^3{\displaystyle \frac{\partial \overline{p}}{\partial \beta }}\right)=\mathrm{\Lambda }{\displaystyle \frac{\partial \left(\overline{p}h\right)}{\partial \alpha }} .$$

(3)

2.3. Gas Film Thickness Equation

The gas film thickness equation for the gas journal bearings with grooves can be defined as [30,48]

$$\overline{h}=1-{\overline{X}}_{\mathrm{j}}\cos \alpha -{\overline{Z}}_{\mathrm{j}}\sin \alpha +{\overline{h}}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d}}.$$

(4)

Here, ${\overline{h}}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d}}$ refers to the gas film thickness for rectangular-shaped grooves and may be given as

$${\overline{h}}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d}}={\overline{d}}_{\mathrm{g}}; \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{g}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e} \mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n},$$

(5)

$${\overline{h}}_{\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d}}=0; \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{n}\mathrm{o}\mathrm{n}-\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{d} \mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n}.$$

(6)

2.4. FEM Formulation

The domain in the bearing clearance region was discretized using four-node isoparametric quadrilateral elements. The gas film pressure across an element was approximated by the Lagrangian function:

$$\overline{p}=\sum _{j=1}^{{\eta }^e}{\overline{p}}_{\mathrm{j}}{N}_{\mathrm{j}},$$

(7)

where $N_{\mathrm{j}}$ denotes the nodal shape function, and ${\overline{p}}_{\mathrm{j}}$ denotes the jth node pressure of an element. Galerkin’s orthogonality criterion was used to minimize the residual. The weak solution of Equation (4) was computed using the Galerkin weighted residual method [49]:

$$\iint \left({\displaystyle \frac{\partial }{\partial \alpha }}\left({\overline{h}}^3 \overline{p}{\displaystyle \frac{\partial \overline{p}}{\partial \alpha }}\right)+{\displaystyle \frac{\partial }{\partial \beta }}\left({\overline{h}}^3 \overline{p}{\displaystyle \frac{\partial \overline{p}}{\partial \beta }}\right)-\mathrm{\Lambda }{\displaystyle \frac{\partial \left(\overline{p}\overline{h}\right)}{\partial \alpha }}\right)N_{i}d\alpha d\beta =0 .$$

(8)

The elemental matrices were assembled using the general assembly procedure, and the film pressure distribution in the bearing gap was computed.

2.5. Boundary Conditions

The boundary conditions applied were as follows:

(1)

The gas pressure at the outer nodes of the bearings was the same as the ambient pressure.

(2)

The lubricating gas flow rate at the inlet of the supply hole was equal to the flow through the constant-flow valve restrictor.

(3)

The film pressure distribution was periodic in the circumferential direction.

2.6. Solution Procedure

The flow chart of the numerical solution method applied in the present work is illustrated in Figure 2. The effects of grooved surfaces and the behavior of different gases on the hybrid gas journal bearings were numerically simulated by the FEM, whereby the iterative Gauss–Seidel approach was applied to obtain the value of the gas film pressure distribution in the bearing domain. The new journal center position was established for a given external load by obtaining the gas film pressure distribution in the bearing clearance space. These iterations were repeated until the following convergence criterion was obtained:

$$\left|{\displaystyle \frac{{\left[{\left({∆\overline{X}}_{\mathrm{j}}^i\right)}^2+{\left({∆\overline{Z}}_{\mathrm{j}}^i\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{{\left[{\left({\overline{X}}_{\mathrm{j}}^i\right)}^2+{\left({\overline{Z}}_{\mathrm{j}}^i\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\right|\times 100\le 0.001$$

(9)

The performance parameters of the hybrid gas journal bearings are obtained once the above convergence criterion is satisfied. The geometrical and operational parameters used in this work have been selected from the published literature [27,30,47,48] and are summarized in

Table 1. Geometric and operating parameters for the hybrid gas journal bearings with different lubricating gases.

Parameters	Values
Bearing radius (R)	20 mm
Land width ratio $\left({\overline{a}}_b\right)$	0.25
Clearance (c)	20 μm
Supply pressure $\left(p_{\mathrm{s}}\right)$	4 MPa
Concentric design pressure ratio $\left({\beta }^{*}\right)$	0.5
Speed	80,000 min−1
No. of supply holes in a row	12
No. of grooves	12
No. of supply hole rows	2
Gases	Hydrogen, nitrogen, air, and helium
Viscosity of lubricating gases $\left(\mu \right)$	Hydrogen: 0.880; nitrogen: 1.730; air: 1.846; and helium: 1.960; ×10−5 Pa-s
Groove width $\left({\overline{w}}_{\mathrm{g}}\right)$	0.4
Groove depth $\left({\overline{d}}_{\mathrm{g}}\right)$	1
Groove reference length $\left({\overline{l}}_{\mathrm{g}\mathrm{r}}\right)$	0.25

. The simulation was implemented in a source code in MATHWORKS MATLAB 2024a. A mesh sensitivity test was carried out to select the appropriate mesh size for a hydrogen-grooved gas bearing, as depicted in Figure 3. The accuracy of the solution as well as the computational time increased with the refinement of the meshing. However, there was no significant improvement in the accuracy of the solution beyond a certain level of refinement (beyond 14,000 elements), while the computational time increased significantly. The optimum mesh size to numerically simulate the performance of the bearing with sufficient accuracy and a reasonable computational cost was found to be 209 by 67 nodes, which was also chosen for the present study.

2.7. Bearing Performance Parameters

The expressions mentioned in this section were used to obtain the effects of grooved surfaces on hybrid gas journal bearings with different lubricating gases. As such, the gas film reaction $\left({\overline{F}}_{\mathrm{o}}\right)$ was obtained by the integration of the gas film pressure distribution $\left(\overline{p}\right)$ in the axial and circumferential directions:

$${\overline{F}}_{\mathrm{o}}={\left[{{\overline{F}}_{\mathrm{x}}}^2+{{\overline{F}}_{\mathrm{z}}}^2\right]}^{{\displaystyle \frac{1}{2}}}={\left[{\left\{\underset{A}{\iint } \overline{p} cos\alpha d\alpha d\beta \right\}}^2+{\left\{\underset{A}{\iint } \overline{p} sin\alpha d\alpha d\beta \right\}}^2\right]}^{{\displaystyle \frac{1}{2}}} ,$$

(10)

where, ${\overline{F}}_{\mathrm{x}}$ and ${\overline{F}}_{\mathrm{z}}$ are the gas film reaction components in the x- and z-directions. Moreover, the frictional torque $\left({\overline{T}}_{\mathrm{f}}\right)$ was computed by integrating the shear stress acting on the surface of the bearing:

$${\overline{T}}_{\mathrm{f}}=\underset{A}{\iint } \left({\displaystyle \frac{\overline{h}}{2}}{\displaystyle \frac{\partial \overline{p}}{\partial \alpha }}+{\displaystyle \frac{\overline{\mu }\Omega }{6\overline{h}}}\right)d\alpha d\beta .$$

(11)

The dynamic performance characteristics of a journal bearing are obtained by introducing small perturbations into the position and velocity of the journal center about its equilibrium position. When the journal executes a small harmonic motion with certain velocity components, it results in incremental fluid film reactions. These incremental film reactions are linearized to define the fluid film stiffness $\left({\overline{S}}_{\mathrm{i}\mathrm{j}}\right)$ and damping coefficients ${(\overline{C}}_{\mathrm{i}\mathrm{j}})$ that govern the rotor–bearing dynamic behavior. The stiffness coefficients of the bearing system are evaluated by partially differentiating the fluid film reactions with respect to the journal center displacements in the respective directions [47]. The damping coefficients of the bearing system are computed by partially differentiating the fluid film reactions with respect to the journal center velocity [47]. Furthermore, the stiffness and damping coefficients are expressed as

$${\overline{S}}_{\mathrm{i}\mathrm{j}}=-{\displaystyle \frac{\partial {\overline{F}}_{\mathrm{i}}}{\partial {\overline{q}}_{\mathrm{j}}}} \left(i,j=x,z \mathrm{a}\mathrm{n}\mathrm{d} {\overline{q}}_{\mathrm{j}}={\overline{X}}_{\mathrm{j}},{\overline{Z}}_{\mathrm{j}}\right),$$

(12)

$${\overline{C}}_{\mathrm{i}\mathrm{j}}=-{\displaystyle \frac{\partial {\overline{F}}_{\mathrm{i}}}{\partial {\overline{\dot{q}}}_{\mathrm{j}}}} \left(i,j=x,y,z \mathrm{a}\mathrm{n}\mathrm{d} {\overline{\dot{q}}}_{\mathrm{j}}={\overline{\dot{X}}}_{\mathrm{j}},{\overline{\dot{Z}}}_{\mathrm{j}}\right).$$

(13)

Finally, the dynamic coefficients can also be defined in matrix form as

$$\left[\begin{array}{cc}{\overline{S}}_{\mathrm{x}\mathrm{x}}& {\overline{S}}_{\mathrm{x}\mathrm{z}}\\ {\overline{S}}_{\mathrm{z}\mathrm{x}}& {\overline{S}}_{\mathrm{z}\mathrm{z}}\end{array}\right]=-\left[\begin{array}{cc}{\displaystyle \frac{\partial {\overline{F}}_{\mathrm{x}}}{\partial {\overline{X}}_{\mathrm{j}}}}& {\displaystyle \frac{\partial {\overline{F}}_{\mathrm{x}}}{\partial {\overline{Z}}_{\mathrm{j}}}}\\ {\displaystyle \frac{\partial {\overline{F}}_{\mathrm{z}}}{\partial {\overline{X}}_{\mathrm{j}}}}& {\displaystyle \frac{\partial {\overline{F}}_z}{\partial {\overline{Z}}_{\mathrm{j}}}}\end{array}\right] ; \left[\begin{array}{cc}{\overline{C}}_{\mathrm{x}\mathrm{x}}& {\overline{C}}_{\mathrm{x}\mathrm{z}}\\ {\overline{C}}_{\mathrm{z}\mathrm{x}}& {\overline{C}}_{\mathrm{z}\mathrm{z}}\end{array}\right]=-\left[\begin{array}{cc}{\displaystyle \frac{\partial {\overline{F}}_{\mathrm{x}}}{\partial {\overline{\dot{X}}}_{\mathrm{j}}}}& {\displaystyle \frac{\partial {\overline{F}}_{\mathrm{x}}}{\partial {\overline{\dot{Z}}}_{\mathrm{j}}}}\\ {\displaystyle \frac{\partial {\overline{F}}_{\mathrm{z}}}{\partial {\overline{\dot{X}}}_{\mathrm{j}}}}& {\displaystyle \frac{\partial {\overline{F}}_{\mathrm{z}}}{\partial {\overline{\dot{Z}}}_{\mathrm{j}}}}\end{array}\right] .$$

(14)

Thereby, $i$, ${\overline{q}}_{\mathrm{j}}$, and ${\overline{\dot{q}}}_{\mathrm{j}}$ indicate the directions of the force, displacement of the journal center, and velocity components:

$$\begin{array}{cc}{\overline{S}}_{\mathrm{x}\mathrm{x}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{X}}_{\mathrm{j}}}}\cos \alpha d\alpha d\beta , & {\overline{C}}_{\mathrm{x}\mathrm{x}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{\dot{X}}}_{\mathrm{j}}}}\cos \alpha d\alpha d\beta ,\\ {\overline{S}}_{\mathrm{x}\mathrm{z}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{Z}}_{\mathrm{j}}}}\cos \alpha d\alpha d\beta ,& {\overline{C}}_{\mathrm{x}\mathrm{z}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{\dot{Z}}}_{\mathrm{j}}}}\cos \alpha d\alpha d\beta ,\\ {\overline{S}}_{\mathrm{z}\mathrm{x}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{X}}_{\mathrm{j}}}}\sin \alpha d\alpha d\beta ,& {\overline{C}}_{\mathrm{z}\mathrm{x}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{\dot{X}}}_{\mathrm{j}}}}\sin \alpha d\alpha d\beta ,\\ {\overline{S}}_{\mathrm{z}\mathrm{z}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{Z}}_{\mathrm{j}}}}\sin \alpha d\alpha d\beta ,& {\overline{C}}_{\mathrm{z}\mathrm{z}}=-\underset{A}{\iint } {\displaystyle \frac{\partial \overline{p}}{\partial {\overline{\dot{Z}}}_{\mathrm{j}}}}\sin \alpha d\alpha d\beta .\end{array}$$

(15)

The stability threshold speed $\left({\overline{\omega }}_{th}\right)$ is a key parameter for assessing the stability of a journal bearing. Disturbances arising from unbalanced fluid film forces can induce whirl motion of the journal center around its equilibrium position. For a given initial disturbance, the motion of the instantaneous journal center can be determined by integrating the governing equations of motion, expressed as [47,48,51]

$$\left[\begin{array}{cc}{\overline{\mathrm{M}}}_{\mathrm{j}}& 0\\ 0& {\overline{\mathrm{M}}}_{\mathrm{j}}\end{array}\right]\left\{\begin{array}{c}{\overline{\ddot{\mathrm{X}}}}_{\mathrm{j}}\\ {\overline{\ddot{\mathrm{Z}}}}_{\mathrm{j}}\end{array}\right\}+\left[\begin{array}{cc}{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{x}}& {\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{z}}\\ {\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{x}}& {\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{z}}\end{array}\right]\left\{\begin{array}{c}{\overline{\dot{\mathrm{X}}}}_{\mathrm{j}}\\ {\overline{\dot{\mathrm{Z}}}}_{\mathrm{j}}\end{array}\right\}+\left[\begin{array}{cc}{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{x}}& {\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{z}}\\ {\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{x}}& {\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{z}}\end{array}\right]\left\{\begin{array}{c}{\overline{\mathrm{X}}}_{\mathrm{j}}\\ {\overline{\mathrm{Z}}}_{\mathrm{j}}\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\} .$$

(16)

The balancing forces for the disturbed journal center, relative to its equilibrium position, depend on the instantaneous position and velocity of the journal center. The expression of the stability threshold speed $\left({\overline{\omega }}_{th}\right)$ using Routh’s stability criterion is given as [47,48,51]

$${\overline{\omega }}_{th}=\sqrt{\left({\displaystyle \frac{{\overline{M}}_c}{{\overline{F}}_o}}\right)} ,$$

(17)

where ${\overline{F}}_o$ represents the fluid film reaction under static conditions $\left({\overline{\dot{X}}}_j={\overline{\dot{Z}}}_j=0\right),$ and ${\overline{M}}_c$ is the critical journal mass, which is calculated as

$${\overline{M}}_c={\displaystyle \frac{{\overline{A}}_1}{{\overline{A}}_2-{\overline{A}}_3}} ,$$

(18)

where

$${\overline{\mathrm{A}}}_1=\left[{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{x}}{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{z}}-{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{x}}{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{z}}\right],$$

(19)

$${\overline{\mathrm{A}}}_2={\displaystyle \frac{\left[{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{x}}{\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{z}}-{\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{x}}{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{z}}\right]\left[{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{x}}+{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{z}}\right]}{\left[{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{x}}{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{z}}+{\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{z}}{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{x}}-{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{z}}{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{x}}-{\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{x}}{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{z}}\right]}} ,$$

(20)

$${\overline{\mathrm{A}}}_3={\displaystyle \frac{\left[{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{x}}{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{x}}+{\overline{\mathrm{S}}}_{\mathrm{x}\mathrm{z}}{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{z}}+{\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{x}}{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{x}}+{\overline{\mathrm{S}}}_{\mathrm{z}\mathrm{z}}{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{z}}\right]}{\left[{\overline{\mathrm{C}}}_{\mathrm{x}\mathrm{x}}+{\overline{\mathrm{C}}}_{\mathrm{z}\mathrm{z}}\right]}} .$$

(21)

The journal bearing system is stable when the stability threshold speed is greater than the operating journal speed.

3. Results and Discussion

3.1. Model Verification

Initially, the developed model was verified step-by-step against previously published theoretical and experimental works. In the first stage, our model was compared for gas journal bearings with the results from Yang et al.’s study [50] for different values of eccentricity ratios, as shown in

Table 2. Comparison of load-carrying capacity in reference work [50] with our present approach.

Λ	$\mathit{\epsilon }$	${\overline{\mathit{W}}}_0$ (Reference Work)	${\overline{\mathit{W}}}_0$ (Present Work)
0.6	0.1	0.0884	0.0879
0.6	0.2	0.1804	0.1798
0.6	0.4	0.3976	0.3957
0.6	0.6	0.7332	0.7273
0.6	0.8	1.6104	1.5778
3	0.1	0.34	0.3378
3	0.2	0.6948	0.6926
3	0.4	1.5208	1.5173
3	0.6	2.8456	2.8193
3	0.8	6.1456	6.0176
12	0.1	0.5148	0.5123
12	0.2	1.0728	1.0703
12	0.4	2.54	2.5325
12	0.6	5.0828	5.0471
12	0.8	11.336	11.014

. The geometric and operating parameters were chosen based on reference studies. The verification of the two results showed good agreement, with a maximum deviation of 2.84%. In the second stage, our model was verified for the grooved surfaces with the experimental results published by Hirs [30]. This small deviation between the two results could be attributed to experimental uncertainties, such as measurement errors, environmental conditions, manufacturing tolerances, etc. Similarly, as in the previous stage, the operating and geometric parameters were selected as in the reference work. The variations in the results computed from the present study with the reference work are summarized in Figure 4. The comparison showed a good agreement, with a maximum deviation of 6.25%. The small deviation in the results of the present model from the reference work could be attributed to variations in mesh sizes, solution schemes, and convergence criteria. These validations ensure that the current model is adequate to numerically investigate bearing performance.

3.2. Effects of Different Gases and Grooved Surfaces on Performance Characteristics

In this section, the influences of different lubricating gases (hydrogen, air, nitrogen, and helium) and grooved surfaces on different performance parameters of hybrid gas journal bearings are presented and discussed. Gas bearings are most commonly air-lubricated bearings. Therefore, the percentage difference in the performance parameters of hydrogen gas-lubricated journal bearings was compared with that of air-lubricated bearings.

3.2.1. Minimum Gas Film Thickness

The effects of different gases and grooved surfaces on the minimum gas film thickness against external loading are shown in Figure 5. The minimum gas film decreased with an increase in the external loading, which is a well-known phenomenon. It was also observed that different gases substantially influenced the value of the minimum gas film thickness. Such behavior was observed due to the variations in the viscosities for different types of lubricating gases. The more viscous lubricating gas, i.e., helium gas, provided a larger value of the minimum gas film thickness than other gases. In contrast, less viscous lubricating gases, i.e., hydrogen gas, provided a lower level of the minimum gas film thickness. Furthermore, it was found that grooved surfaces reduced the value of the minimum gas film thickness. This behavior was found due to the redistribution of gas pressure in the bearing domain due to the application of grooved surfaces.

3.2.2. Maximum Gas Film Pressure

Representative contour plots of the gas film pressure distribution in the bearing clearance domain for both grooved and non-grooved surfaces for different lubricating gases for one selected load case are illustrated in Figure 6. It may be noticed that the hydrogen gas-lubricated bearing (Figure 6a) provided a higher value of maximum pressure compared with other gases, e.g., helium (Figure 6b). This behavior occurred due to the lower viscosity of hydrogen gas, which provided a higher value of maximum pressure at a smaller minimum film thickness in order to support the external load. In contrast, helium gas offered a lower value of maximum gas pressure. Furthermore, it may be noticed that the introduction of grooved surfaces reduced the maximum gas film pressure (Figure 6c–f). This behavior occurred due to the redistribution of the gas film pressure by fabricating the grooved surfaces.

3.2.3. Frictional Torque

The effects of different gases and grooved surfaces on the frictional torque depending on external loading are summarized in Figure 7. Generally, frictional torque values increased with an increase in the external load [52,53,54]. It can also be observed that the bearing lubricated with hydrogen gas provided a lower value of frictional loss. This behavior occurred due to the lower viscosity of hydrogen gas compared with other gases. A similar trend was reported in a previous study, which observed a lower frictional torque for hydrogen gas-lubricated bearings compared with those lubricated with other gases [23]. In contrast, the helium gas-lubricated bearing featured higher friction. Furthermore, it can be noticed that the application of grooved surfaces decreased the frictional torque. The grooves acted as reservoirs of lubricating gas, which increased the nominal gas film thickness. The increment in the nominal film thickness reduced the shear stress and, thus, lowered the frictional torque [2,4]. The grooved bearing operating with hydrogen gas showed the lowest values of frictional torque, representing, for example, a 51% reduction compared with the air-lubricated bearing with a non-grooved surface.

3.2.4. Stiffness Coefficients

The variations in the stiffness coefficients $\left({\overline{S}}_{\mathrm{x}\mathrm{x}} \mathrm{a}\mathrm{n}\mathrm{d} {\overline{S}}_{\mathrm{z}\mathrm{z}}\right),$ considering the effects of different gases and grooved surfaces, are shown in Figure 8a,b. It may be noticed in Figure 8a that the hydrogen gas-lubricated bearings offered higher values of the stiffness coefficients than other gases, whereas the bearing lubricated with helium gas provided a lower stiffness value. The percentage change in ${\overline{S}}_{xx}$ for the hydrogen gas-lubricated bearing at ${\overline{W}}_{\mathrm{o}}=0.5$ was 9.15% compared with the air-lubricated bearing. Similar trends were observed in the case of ${\overline{S}}_{\mathrm{z}\mathrm{z}}$ (see Figure 8b). It may also be noticed that the introduction of grooved surfaces reduced the stiffness coefficients. Such behavior can be attributed to variations in the pressure gradients caused by the application of grooved surfaces. A similar behavior was observed in a previously published study for textured journal bearings operating with a constant-flow valve restrictor [47].

3.2.5. Damping Coefficients

The effects of different lubricating gases and grooved surfaces on the damping coefficients $\left({\overline{C}}_{\mathrm{x}\mathrm{x}} \mathrm{a}\mathrm{n}\mathrm{d} {\overline{C}}_{\mathrm{z}\mathrm{z}}\right)$ are depicted in Figure 9a,b. In Figure 9a, it can be noticed that the hydrogen-lubricated bearing featured higher damping coefficients than other gases. The percentage variation in ${\overline{C}}_{\mathrm{x}\mathrm{x}}$ for the hydrogen gas-lubricated bearing at ${\overline{W}}_{\mathrm{o}}=0.5$ was 53% compared with the air-lubricated bearing. The lowest values of the damping coefficients were observed for the bearings lubricated with helium gas. A similar trend could be noticed for ${\overline{C}}_{\mathrm{z}\mathrm{z}}$ (Figure 9b), where the percentage increment for hydrogen ${\overline{W}}_{\mathrm{o}}=0.5$ was 51% compared with air. It may also be noticed that the introduction of the grooved surfaces reduced the values of the damping coefficients. Such behavior may have been observed due to the reduction in the bearing land area by the application of grooves, which resulted in a reduction in the damping capabilities.

3.2.6. Stability Threshold Speed

The effects of different lubricating gases and grooved surfaces on the stability threshold speed $\left({\overline{\omega }}_{\mathrm{t}\mathrm{h}}\right)$ are shown in Figure 10. It can be observed that the hydrogen-lubricated bearings exhibited the highest stability threshold speed compared with bearings lubricated with other gases. The percentage increment in ${\overline{\omega }}_{\mathrm{t}\mathrm{h}}$ for the hydrogen gas-lubricated bearing at ${\overline{W}}_{\mathrm{o}}=0.5$ was 109% compared with the air-lubricated bearing. The lowest value of the stability threshold speed was observed for the bearings lubricated with helium gas. It may also be noted that the introduction of grooved surfaces reduced the stability threshold speed. This behavior may be attributed to the reduction in the damping coefficients due to the application of grooved surfaces. A similar trend was observed in a previously published study for textured journal bearings operating with a constant-flow valve restrictor [47].

4. Conclusions

This study numerically investigated hybrid gas journal bearings lubricated with hydrogen, nitrogen, air, and helium, and assessed the influence of circumferential micro-grooves. The finite element model, verified against prior theoretical and experimental data, showed that the lubricant gas properties and surface texturing jointly shaped static and dynamic bearing behaviors. Across loads, hydrogen consistently lowered the frictional torque, while helium increased the minimum film thickness. Grooves redistributed pressure, reducing both the maximum pressure and frictional torque at the expense of stiffness, damping, and the stability margin. The derived conclusions can be summarized as follows:

• Modeling fidelity supports robust parameter studies: The validated FEM framework with constant-flow restrictors provides reliable trends for film thickness, pressure distribution, torque, stiffness/damping, and stability, enabling systematic design exploration. Mesh independence and convergence criteria ensure that observed effects stem from physical parameter variations rather than numerical artifacts.
• Gas selection dominates the performance envelope: Hydrogen-lubricated bearings yield the lowest frictional torque and higher dynamic coefficients. Helium favors a larger minimum film thickness. Lower viscosity reduces shear stress (lower torque) but requires higher peak pressures at a smaller film thickness to sustain the load. Higher viscosity increases hydrodynamic support and film thickness, moderating peak pressures.
• The pressure–film coupling sets the load support and peak pressure: For a given load, gases with lower viscosities develop higher local peak pressures at reduced minimum film thicknesses, and higher-viscosity gases distribute pressure more broadly with thicker films. This is because the compressible Reynolds balance links viscosity and film geometry to pressure gradients, and reduced viscous resistance intensifies pressure build-up where clearance is smallest.
• Surface texturing introduces a friction–stability trade-off: Circumferential grooves reduce frictional torque but also reduce stiffness, damping, and the stability threshold speed as grooves act as micro-reservoirs that increase the local nominal film thickness and lower shear, yet they diminish the effective load-bearing land and alter pressure gradients, reducing dynamic support.
• Design implication: Pairing low-viscosity gases with carefully parameterized grooves is advantageous where torque reduction and efficiency are paramount. Non-grooved designs with lower-viscosity gases are preferable when stiffness, damping, and stability margins are critical. Thereby, application-specific weighting of shear losses versus dynamic coefficients should guide the selection of gas properties and groove geometry.

Overall, bearing performance can be tuned by co-designing the lubricant gas and groove topology. Choose hydrogen and pronounced grooving for minimal torque, or favor helium/air and reduced grooving when dynamic rigidity and stability dominate the requirements.