# CFD Analysis of Journal Bearing with a Heterogeneous Rough/Smooth Surface

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Governing Equations

_{i}(u

_{j}) is the average velocity of the lubricant along the coordinates x

_{i}(x

_{j}), that is the coordinate X, Y, or Z; p is the hydrodynamic pressure; µ is the viscosity; u′

_{i}and u′

_{j}are the fluctuation velocities; $\rho \overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ is the Reynolds stress. In the present study, the standard turbulent kinetic energy k and turbulent dissipation rate ε

_{d}models [29] are employed to solve the Reynolds stress.

_{A}is expressed as follows [29]:

_{i}and l are turbulence velocity and length scales, respectively and ${a}_{o}$ is the speed of the sound which is set to 1480 m/s. In Equation (3), a is a model constant. Furthermore, Equation (3) can be reduced in terms of k and ε

_{d}as follows:

#### 2.2. Cavitation Modeling

_{v}represents vapor volume fraction and ρ

_{v}refers to vapor density. R

_{g}and R

_{c}account for the mass transfer between the liquid and vapor phases in cavitation. For the Zwart–Gelber–Belamri model, assuming that all the bubbles have the same size in a system, the final form of the cavitation is as follows [29,30]:

_{evap}= evaporation coefficient = 50, F

_{cond}= condensation coefficient = 0.01, R

_{B}= bubble radius = 10

^{−6}m, α

_{nuc}= nucleation site volume fraction = 5 × 10

^{−4}, ρ = liquid density and p

_{sat}= saturation pressure.

#### 2.3. Roughness Modeling

_{s}is used to cover the surface uniformly. For modeling the surface roughness, the modified law-of-the-wall for mean velocity is employed. This equation can be expressed as follows [29]:

_{s}, while the height is assumed constant per surface [29].

_{s}is the equivalent sand-grain roughness height and is not equal to the geometric roughness height of the surface. Therefore, it is necessary to use the conversion factor to convert the geometric roughness height of the surface into an equivalent sand-grain roughness. In this work, the R

_{a}as shown in Figure 1b, is chosen as a parameter to represent the roughness height K

_{s}(Figure 1a). The R

_{a}represents the arithmetic average of the roughness profile, and in reality it is measured by the profilometer.

_{a}value will be an input to specify the roughness level of the heterogeneous rough/smooth bearing. According to the experiment performed by Adams et al. [31], the correlation between K

_{s}and R

_{a}can be defined as follows:

_{s}and the roughness height K

_{s}. Here, because the k-ε

_{d}turbulence model is used and the uniform sand-grain is assumed, the default roughness constant (C

_{s}= 0.5) is employed as suggested by ANSYS FLUENT [29].

## 3. Simulation Method

#### 3.1. Model

_{s}to model the roughness. Here, the film thickness of the lubricant will follow the surface profile as the input in the CFD program.

_{r}is always much larger than the critical one, Re

_{c}. For example, for the case of ε = 0.8, the Re

_{r}is 1575 which is much larger than the Re

_{c}of 0.32. From the physical framework, it means that turbulence may occur in the fluid film and thus, from the numerical framework, such turbulence phenomena must be modeled during the lubrication analysis to achieve accurate results.

#### 3.2. Meshing

#### 3.3. Assumption and Boundary Condition

#### 3.4. Solution Setup

## 4. Results and Discussion

#### 4.1. Validation

^{3}), as reflected in Figure 5. It can be found that the obtained values from the CFD code developed here are very close to the published ones both from the numerical and experimental results. Their deviations are less than 4% as indicated in Figure 5b, suggesting validation of the developed CFD code.

#### 4.2. At Varied Eccentricity Ratio

_{a}of 25 μm. As a note, the surface with a value R

_{a}of 25 μm is categorized as “rough” surface (R

_{a}= 12.5–100 μm) [33]. As demonstrated by Tauviqirrahman et al. [17], the surface class of “rough” has the strongest effect on the tribological performance [17] in comparison to other classes such as precision (R

_{a}= 0.1–0.2 μm), fine (R

_{a}= 0.4–0.8 μm), and medium (R

_{a}= 1.6–6.3 μm).

_{a}= 0 and denoted as S pattern) and the bearing with the heterogeneous rough/smooth area (i.e., 1 L, 2 L, 3 L patterns) is reflected in Figure 6. From Figure 6, several characteristics can be seen. First, for all cases, an increase in the eccentricity ratio will decrease the Sommerfeld number S. The decrease in the Sommerfeld number occurs significantly when the eccentricity ratio ε is greater than 0.2. Secondly, when ε = 0 to 0.6, the heterogeneous rough/smooth bearing pattern with two-rough zones (2 L) gives the lowest Sommerfeld Number value when compared with the conventional smooth bearing (S) pattern and other heterogeneous rough/smooth bearing patterns. However, for eccentricity ratios of 0.7 and 0.8, heterogeneous rough/smooth bearing with one-rough zone (1 L) gives the lowest Sommerfeld Number. In the other words, although not superior to all eccentricity ratios studied here, the 2 L pattern gives the best performance in reducing the Sommerfeld number, which means that the enhanced load-carrying capacity can be achieved. From the results depicted in Figure 6, it can also be observed that the heterogeneous rough/smooth bearing, irrespective of the rough patterns, can generate the load-carrying capacity for all values of eccentricity ratio including for the concentric position. As is known, for the conventional bearing, no load-carrying capacity is produced when the concentric condition is applied due to the absence of the hydrodynamic pressure. This finding is interesting, and hence, the heterogeneous rough/smooth bearing can be compared to the heterogeneous slip/no-slip pattern. Based on the reference [19,22], even though there is no wedge effect in the case of concentric journal bearing, the heterogeneous slip/no-slip pattern could produce a relatively high load-carrying capacity. It indicates that the behavior of a “rough” surface can be correlated to the wettability of the surface (in particular the surface with hydrophobic coating) inducing the slip boundary. This is understandable because as discussed by Patankar [34] and Jung and Bhushan [35], the wettability of a surface is a function of its roughness.

_{sat}, and the lubricant would rupture. In the present study, the saturation pressure P

_{sat}used is 2340 Pa (as shown in Table 1), and the pressure at the inlet and outlet boundaries are taken as the ambient pressure, i.e., zero pressure. As a consequence, for each value of local pressures in the computational domain, FLUENT will reduce them with the environmental pressure p

_{atm}of 1 atm ($\approx $101,325 Pa). Therefore, when the cavitation occurs, the local pressure will be set to the saturation pressure (2340 Pa). By FLUENT, these values are converted to the negative value, i.e., −98,985 Pa ($\approx $−0.1 MPa), as depicted in Figure 8, to show that the cavitation exists. Based on Figure 8b and Figure 9, when the eccentricity ratio is 0.8, the width of the cavitation zone does not change very much. It ranges from 50–70° depending on the bearing pattern. For example, in the case of 2 L pattern, the cavitation occurs at the circumferential angle θ of around 190°–236°.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{o}$ | Local speed of the sound |

B | Bearing width |

c | Radial clearance |

C_{s} | Roughness constant |

D | Bearing diameter |

E | Empirical constant |

f_{r} | Roughness function |

F_{evap} | Evaporation coefficient |

F_{cond} | Condensation coefficient |

h_{min} | Minimum film thickness |

h_{max} | Maximum film thickness |

K_{s} | Roughness height |

k | Turbulent kinetic energy |

l | Length scale |

L_{θ} | Circumferential length of the bearing |

n | Rotational speed |

p | Hydrodynamic pressure |

P_{A} | Acoustic power level |

p_{sat} | Saturation pressure |

W | Load-carrying capacity |

r | Shaft radius |

R | Bearing radius |

R_{a} | Arithmetic average of the roughness profile |

R_{B} | Bubble radius |

Re_{c} | Critical Reynolds number |

Re_{r} | Real Reynolds number |

R_{g}, R_{c} | Mass transfer between the liquid and vapor phase |

u_{p} | Mean velocity of the fluid at the near-wall node P |

u* | dimensionless velocity |

y_{p} | Distance from point P to the wall |

α_{nuc} | Nucleation site volume fraction |

α_{v} | Vapor volume fraction |

ε | Eccentricity ratio |

ε_{d} | Turbulent dissipation rate |

$\kappa $ | von Karman constant |

$\theta $ | Circumferential angle |

µ | Lubricant viscosity |

µ_{v} | Vapor viscosity |

ρ | Lubricant density |

ρ_{v} | Vapor density |

$\Phi $ | Attitude angle |

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**Figure 2.**Three types of heterogeneous rough/smooth bearing with different artificial roughness zones, (

**a**) one-rough zone (1 L); (

**b**) two-rough zones (2 L); (

**c**) three-rough zones (3 L).

**Figure 4.**Boundary condition of the computational domain: 1—moving wall, 2—stationary wall, 3–inlet, 4—outlet.

**Figure 7.**Effect of the arrangement of the rough zone on the load-carrying capacity under several eccentricity ratios, (

**a**) lubrication performance ratio of load-carrying capacity, (

**b**) improvement of the load-carrying capacity (compared with conventional bearing).

**Figure 8.**Hydrodynamic pressure distributions of the heterogeneous rough/smooth bearings for (

**a**) ε = 0, and (

**b**) ε = 0.8. The results are evaluated at the mid-plane of the bearing.

**Figure 9.**Comparison of the contour of vapor volume fraction between (

**a**) the conventional (smooth) bearing, (

**b**) the heterogeneous slip/no-slip bearing with 1L pattern, (

**c**) the heterogeneous slip/no-slip bearing with 2 L pattern, (

**d**) the heterogeneous slip/no-slip bearing with 3 L pattern.

**Figure 10.**Effect of the arrangement of the rough zone on friction force under several eccentricity ratios, (

**a**) lubrication performance ratio of friction force, (

**b**) deterioration of the friction force (compared with conventional bearing).

**Figure 11.**Effect of the arrangement of the rough zone on average acoustic power level under several eccentricity ratios, (

**a**) lubrication performance ratio of average acoustic power level, (

**b**) improvement of the average acoustic power level (compared with conventional bearing).

Parameter | Symbol | Value | Unit |
---|---|---|---|

Bearing radius | R | 50 | mm |

Width-diameter ratio | B/D | 0.8 | [[–] |

Radial clearance | $c$ | 0.152 | mm |

Eccentricity ratio | $\epsilon $ | 0; 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8 | [[–] |

Attitude angle | $\Phi $ | 54 | Deg |

Fluid density | $\rho $ | 998.2 | kg/m^{3} |

Fluid viscosity | $\mu $ | 0.001005 | Pa.s |

Rotational speed | $n$ | 2000 | rpm |

Saturation pressure | p_{sat} | 2340 | Pa |

Vapor density | ${\rho}_{v}$ | 0.5542 | kg/m^{3} |

Vapor viscosity | ${\mu}_{v}$ | 1.34 × 10^{−5} | Pa.s |

Roughness level | R_{a} | 25 | μm |

Mesh Criteria | Value |
---|---|

Edge sizing 1 | 400 division |

Edge sizing 2 | 60 division |

Face Meshing | 12-layers of division |

Method | Sweep |

Element number | 288,000 |

For case ε = 0 | |

Maximum skewness | 9.137 × 10^{−2} |

Minimum skewness | 7.194 × 10^{−3} |

Average skewness | 5.703 × 10^{−2} |

For case ε > 0 | |

Maximum skewness | 0.155 |

Minimum skewness | 5.019 × 10^{−3} |

Average skewness | 5.604 × 10^{−2} |

Boundary Condition | Setup |
---|---|

Inlet | Pressure inlet (0 Pa) |

Outlet | Pressure outlet (0 Pa) |

Stationary wall | No-slip |

Moving wall | No slip, n = 2000 rpm |

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## Share and Cite

**MDPI and ACS Style**

Tauviqirrahman, M.; Jamari, J.; Wicaksono, A.A.; Muchammad, M.; Susilowati, S.; Ngatilah, Y.; Pujiastuti, C.
CFD Analysis of Journal Bearing with a Heterogeneous Rough/Smooth Surface. *Lubricants* **2021**, *9*, 88.
https://doi.org/10.3390/lubricants9090088

**AMA Style**

Tauviqirrahman M, Jamari J, Wicaksono AA, Muchammad M, Susilowati S, Ngatilah Y, Pujiastuti C.
CFD Analysis of Journal Bearing with a Heterogeneous Rough/Smooth Surface. *Lubricants*. 2021; 9(9):88.
https://doi.org/10.3390/lubricants9090088

**Chicago/Turabian Style**

Tauviqirrahman, Mohammad, J. Jamari, Arjuno Aryo Wicaksono, M. Muchammad, S. Susilowati, Yustina Ngatilah, and Caecilia Pujiastuti.
2021. "CFD Analysis of Journal Bearing with a Heterogeneous Rough/Smooth Surface" *Lubricants* 9, no. 9: 88.
https://doi.org/10.3390/lubricants9090088