# Reynolds Model versus JFO Theory in Steadily Loaded Journal Bearings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

#### 2.1. Reynolds Model

#### 2.2. JFO Theory

_{c}. The value of θ can be regarded as a fractional film content or the oil film height through the striations [10]. The pressure–density relation is

_{c}is the cavitation pressure. The definition of g suggested by Elrod [13] is expressed as

## 3. Bearing Condition and Parameters

_{a}is the ambient pressure, R is the journal radius, and C is the radius clearance.

_{max}, the film thickness profile is expressed by

_{max}. Due to the use of the Cartesian coordinate system, the relation between ϕ and x is

_{max}. In practical bearings, the position of h

_{max}depends on the attitude angle that varies with load. The precise oil supply location is hard to maintain. However, this oil supply method was largely applied in theoretical analysis [3,36,37] because it presents the characteristics of a plane oil film. There are four boundaries around the bearing geometry. All of them are treated as the pressure boundary condition with p

_{a}= 0 Pa. The bearing is considered as a submerged bearing.

_{s}is the rotational speed in rev/s. The radial and tangential loads, W

_{r}and W

_{t}, are, respectively calculated by

## 4. Cavitation Analysis

^{5}Pa and −1.0 × 10

^{5}Pa were analyzed for the two methods.

_{c}for the Reynolds model and the JFO theory. The pressure obtained using the Reynolds model is the same to that obtained using the JFO theory at the condition of cavitation pressure equaling ambient pressure (the black dotted line is covered by the black solid line). If the cavitation pressure is lower than the ambient pressure, the two methods provide different results. The oil film rupture boundary predicted using the Reynolds model is the same to that predicted using the JFO theory, but the oil film reformation boundary predicted using the Reynolds model is ahead of that predicted using the JFO theory. It implies that, the cavitated area is underestimated using the Reynolds model, and the difference between the two methods increases with the reducing cavitation pressure.

_{c}= −0.5 × 10

^{5}Pa for the Reynolds model and the JFO theory. The pressure distribution in the liquid region is basically the same for the two methods, while the pressure distribution in the cavitated region is obviously different. The difference is mainly reflected by the different shapes of cavitated areas, as shown in Figure 5b. The cavitation patterns predicted using the two methods look like half ellipses. The major axis of the ellipse predicted using the Reynolds model is shorter than that predicted using the JFO theory. This difference is due to the fact that the density effect in the cavitated region is not reflected in the Reynolds model. In other words, mass conservation is not always satisfied with the Reynolds model. The quantitative effect of cavitation pressure on the bearing characteristics is presented below.

_{c}= 0 Pa, as represented by the black lines. The resulting Sommerfeld number agrees well with the result provided by Khonsari and Booser [3] under different L/D and ε. Moreover, the Sommerfeld number predicted using the JFO theory is the same as that predicted using the Reynolds model (the black dotted lines are covered by the black solid lines), since there is no film reformation. Secondly, let us observe the blue lines that represent the bearings with p

_{c}= −1.0 × 10

^{5}Pa. (1) The Sommerfeld number of the bearing with p

_{c}= −1.0 × 10

^{5}Pa is smaller than that of the bearing with p

_{c}= 0 Pa for most cases. It implies that the load-carrying capacity is enhanced by the reducing cavitation pressure, which is in accordance with the comment on cavitation by Dowson and Taylor [20]. (2) The difference in the Sommerfeld number between the different cavitation pressures gradually decreases with the increasing ε and L/D. This can be explained by the extent of the hydrodynamic effect. With the increase in ε and L/D, the convergence clearance becomes narrower, and more oil is kept in the clearance; hence, the hydrodynamic effect is enhanced. The positive pressure in the convergence clearance region is increased, while the negative pressure in the cavitated region is kept in the constant cavitation pressure. This behavior weakens the cavitation effect on the load-carrying capacity. (3) The Sommerfeld number predicted using the JFO theory is basically the same to that predicted using the Reynolds model for most cases. The slight difference appears in the condition of medium ε and large L/D, where the Sommerfeld number predicted using the JFO theory is slightly smaller than that predicted using the Reynolds model. It implies that the Reynolds model underestimates the load-carrying capacity. This is contrarily compared to the case of microtextured thrust bearings, in which the load-carrying capacity is overestimated using the Reynolds model. In journal bearings, the cavitated area can increase the load because the integral calculation is along the circular surface and the cavitated region is on the opposite side of the positive pressure region. Finally, the bearing with p

_{c}= −0.5 × 10

^{5}Pa is presented by the red lines. The variation behavior of the Sommerfeld number is between the two situations mentioned above.

_{c}= 0 Pa for different L/D and ε (the black dotted lines are covered by the black solid lines), and agree well with the results provided by Khonsari and Booser [3]. On the other hand, (1) the effect of cavitation pressure on the attitude angle is larger than that on the Sommerfeld number. The attitude angle increases with the reducing cavitation pressure. This is due to the increase in the load-carrying capacity, making the bearing self-adjust to a large attitude angle to reduce the load. Moreover, the difference in the attitude angle between different cavitation pressures increases with the decrease in L/D. In extreme cases, the attitude angle reaches the maximum of 90°, which appears at the bearing with a small L/D and ε. In fact, the attitude angle of 90° is the full Sommerfeld solution. This is due to the weak hydrodynamic effect that cannot make cavitation occur. In other words, the negative pressure peak is higher than the predetermined cavitation pressure. (2) The attitude angle predicted using the Reynolds model is basically the same to that predicted using the JFO theory. This difference is not obvious.

^{2}indicates the Petroff multiplier [37]. The Petroff multiplier increases with the increasing eccentricity ratio. Hence, as the eccentricity ratio increases, the rate of decrease in the friction coefficient is slightly slower than that of the Sommerfeld number. Similar to the variation of the Sommerfeld number, the same behavior for the friction coefficient is obtained as (1) the friction coefficient decreases with the decreasing cavitation pressure for most cases; (2) the difference in the friction coefficient between different cavitation pressures gradually decreases with the increasing ε and L/D, since the hydrodynamic effect enhances; and (3) the friction coefficient is overestimated by the Reynolds model but the difference is small. In addition, the resulting friction coefficient is calculated by Equation (15), where the effect of θ on the friction coefficient is not considered. The viscous shear stress variation due to the cavitation effect needs to be further investigated.

_{L}is expressed by

_{L}= Q

_{3}+ Q

_{4}

_{c}= 0 Pa (the black dotted lines are covered by the black solid lines), and they agree well with the results provided by Khonsari and Booser [3]. The side leakage decreases with the decreasing cavitation pressure for most cases. This is because the negative gradient means the oil is sucked back into the bearing clearance. In other words, the oil flows away from the positive pressure region and reverses from the negative pressure region. On the other hand, the side leakage remains zero at the low ε and L/D. It is explained by the full Sommerfeld solution: the symmetrical pressure distribution results in an equal outflow and reflux. In addition, the side leakage is overestimated by the Reynolds model, because the pressure gradient in the divergence region is weakened by the underestimated cavitated area, causing the reduction in the inverse flow.

_{net}is calculated by

_{net}= Q

_{1}− Q

_{2}− Q

_{3}− Q

_{4}

## 5. Conclusions

- (1)
- The low cavitation pressure leads to a decrease in the Sommerfeld number, friction coefficient, and side leakage, and an increase in the attitude angle. The cavitation effect is weakened with the increased L/D, eccentricity ratio, and bearing number due to the improvement of the hydrodynamic effect.
- (2)
- The cavitated area is underestimated by the Reynolds model due to the inaccurate oil film reformation boundary, leading to the underestimation of the load-carrying capacity and the overestimation of the friction coefficient and side leakage.
- (3)
- To sum up, the load-carrying capacity is improved by the decrease in the cavitation pressure, and the effect is significant in lightly loaded cavitated bearings. In the non-cavitated case and the cavitated case with intermediate and heavy loads, the difference between the Reynolds model and the JFO theory can be effectively ignored, but the accuracy of the leakage predicted using the Reynolds model should be carefully evaluated.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Effect of bearing number on attitude angle for Reynolds model and JFO theory: (

**a**) Λ = 100; (

**b**) Λ = 1000.

**Figure A2.**Effect of bearing number on friction coefficient for Reynolds model and JFO theory: (

**a**) Λ = 100; (

**b**) Λ = 1000.

**Figure A3.**Effect of bearing number on side leakage for Reynolds model and JFO theory: (

**a**) Λ = 100; (

**b**) Λ = 1000.

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**Figure 2.**Central pressure variation for the journal bearing. (D = 100 mm, L = 133 mm, C/R = 0.00291, ε = 0.60, n = 459.3 r/min, μ = 0.0127 Pa·s, p

_{c}= −72.1 kPa, β = 1 × 10

^{10}Pa, P = (p − p

_{a}) (C/R)

^{2}/(μω)).

**Figure 4.**Central pressure variation with different p

_{c}for Reynolds model and JFO theory at the condition of L/D = 1, ε = 0.5, and Λ = 10.

**Figure 5.**Pressure and cavitation distribution for p

_{c}= −0.5 × 10

^{5}Pa at the condition of L/D = 1, ε = 0.5, and Λ = 10: (

**a**) Pressure distribution; (

**b**) Cavitation distribution.

**Figure 6.**Effect of cavitation pressure on Sommerfeld number for Reynolds model and JFO theory at Λ = 10.

**Figure 7.**Effect of cavitation pressure on attitude angle for Reynolds model and JFO theory at Λ = 10.

**Figure 8.**Effect of cavitation pressure on friction coefficient for Reynolds and JFO theory at Λ = 10.

**Figure 9.**Effect of cavitation pressure on side leakage for Reynolds model and JFO theory at Λ = 10.

**Figure 10.**Effect of cavitation pressure on net leakage for Reynolds model and JFO theory at Λ = 10.

**Figure 11.**Effect of bearing number on Sommerfeld number for Reynolds model and JFO theory: (

**a**) Λ = 100; (

**b**) Λ = 1000.

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**MDPI and ACS Style**

Xu, W.; Zhao, S.; Xu, Y.; Li, K.
Reynolds Model versus JFO Theory in Steadily Loaded Journal Bearings. *Lubricants* **2021**, *9*, 111.
https://doi.org/10.3390/lubricants9110111

**AMA Style**

Xu W, Zhao S, Xu Y, Li K.
Reynolds Model versus JFO Theory in Steadily Loaded Journal Bearings. *Lubricants*. 2021; 9(11):111.
https://doi.org/10.3390/lubricants9110111

**Chicago/Turabian Style**

Xu, Wanjun, Shanhui Zhao, Yaoyao Xu, and Kang Li.
2021. "Reynolds Model versus JFO Theory in Steadily Loaded Journal Bearings" *Lubricants* 9, no. 11: 111.
https://doi.org/10.3390/lubricants9110111