**1. Introduction**

High-speed rotating machinery such as compressors, pumps, gas turbines, and automobiles is used all over the world [1, 2]. Oil-lubricated journal bearings are used widely as a support element of high-speed rotating shafts for reducing friction, enhancing the rotating accuracy. On the journal bearing, many researchers and engineers have been interested in the gaseous phase generated in the bearing clearance, and they tried to predict the existence of the gaseous phase because the bearing characteristics are strongly affected by the gaseous-phase areas. For example, Hashimoto and Ochiai clarified the stability characteristics under starved

lubrication both theoretically and experimentally, and they propose a stabilization method utilizing the starved lubrication [3]. Furthermore, Naruse and Ochiai have experimentally studied the relation between gaseous-phase area and temperature distribution [4]. However, these observations have not been theoretically investigated [5], because the calculation method for the detailed gaseous-phase area has not been proposed. In addition, in actual bearing systems, if the lubricant is not able to supply the bearing sufficiently, it means there are starved lubrication condition and also a high possibility of serious erosion damage or serious seizure on the bearing surfaces. Therefore, it is believed that predicting the gaseous area in journal bearings is very important.

the flooded and starved lubrication conditions while using a CFD model. Further, effects of VOF, surface tension, and vapor pressure of the setting condition were studied, and these analytical results were compared with the experiment, and the applicability was verified. Then, the authors considered the influence of surface tension on journal bearing from the Weber number, *We*. Furthermore, thermal analysis results under two types of supply oil conditions are shown. The effect of supply oil on bearing temperature characteristics was discussed from the results of

*The Multiphase Flow CFD Analysis in Journal Bearings Considering Surface Tension and Oil…*

As mentioned above, the VOF method has some advantages of calculation cost, convergency, and easy to handle compared with other methods [21–24]. Therefore, the authors selected the VOF method, and actually ANSYS FLUENT 15.0 is used in

Instead of the Reynolds equation, we applied the Navier–Stokes equation considering the surface tension to the journal bearing analysis in this study. The mass

! þ *ρ g*

! ¼ 0 (1)

! � �*δint* (2)

! þ ∇*σ* þ *σγn*

! velocity vector, *p* fluid pressure, *μ* fluid viscosity,

! normal vector, *γ* curvature of the

� � (3)

*ρ* ¼ *Fρ*<sup>1</sup> þ ð Þ 1 � *F ρ*<sup>2</sup> (4)

*μ* ¼ *Fμ*<sup>1</sup> þ ð Þ 1 � *F μ*<sup>2</sup> (5)

! � � (6)

conservation equation and the momentum equation are shown as follows:

<sup>∇</sup> <sup>¼</sup> *<sup>∂</sup>*

*<sup>∂</sup><sup>x</sup>* , *<sup>∂</sup> ∂y* , *∂ ∂z*

In this VOF calculation model, the fluid density and fluid viscosity are expressed

where *F* is the volume fraction and subscription 1 means oil and 2 means gaseous

This CFD analysis can analyze the internal flow of an oil-filler port with the bearing clearance simultaneously because the inertia term is considered in the basic

h i <sup>¼</sup> *<sup>λ</sup>*∇*<sup>T</sup>* <sup>þ</sup> *Sh* <sup>þ</sup> *<sup>μ</sup>* !*u*�∇<sup>2</sup>*<sup>u</sup>*

On the other hand, the energy equation is shown as follows:

!∇ð Þ *T*

*ρcp u*

∇ � *u*

calculation and experimental results.

*DOI: http://dx.doi.org/10.5772/intechopen.92421*

**2. Theory and calculation model**

**2.1 Governing equations**

*ρ ∂u* ! *∂t*

*ρ g*

as follows:

phase.

equation.

**195**

þ ∇ *u* ! � *u* ! � � " # ¼ �∇*<sup>p</sup>* <sup>þ</sup> *<sup>μ</sup>*∇<sup>2</sup>*<sup>u</sup>*

! gravitational force, *σ* surface tension, *n*

boundary surface, and *δint* Dirac's delta function. Moreover, ∇is a differential operator, defined by

where *ρ* is the fluid density, *u*

this study. The used methodology is explained below.

Generally, to analyze the oil-lubricated journal bearing, the Reynolds equation is used, and the half-Sommerfeld condition or the Swift-Stieber condition has been applied [6, 7] in determining the gaseous areas for the Reynolds equation. However, in these models, it is assumed that the negative-pressure areas exist only in the gas phase and there is no oil in the bearing clearance area. Therefore, the flow-rate conservation does not hold in the calculations. As a more advanced method, Coyne and Elrod's condition [8, 9] is used. This model assumes oil-film rupture calculates the surface tension between the oil film and gaseous phase which is ignored in the half-Sommerfeld or the Swift-Stieber conditions. However, it is impossible to estimate the cavitation area of the entire journal bearing including the oil-filler port. Therefore, boundary condition models that consider the cavitation have been studied. For example, Ikeuchi and Mori have analyzed the oil-film cavitation areas while using the modified Reynolds equation [10, 11]. In this method, the two-phase flow is considered as an averaged single-phase flow of oil and gas. However, in the case of high eccentricity ratio and starved lubrication condition, they did not conduct the experimental verification with the cavitation area. On the other hand, considering the finger-type cavitation, analytical methods have been proposed by Boncompine et al. [12] and Hatakenaka et al. [13]. However, the estimations of the variation of the gaseous-phase area against the changing of amount of oil lubricant have not been able to for bearing engineers or researchers. Furthermore, it is reported by Hashimoto, Ochiai, and Sakai that the oil-filler port and supply oil quantity affect the journal bearing characteristics [14, 15]. However, the internal flow of the oil-filler port has not been mentioned specifically. Therefore, a different approach is required to analyze the journal bearing while considering the internal flow of an oil-filler port.

The computational fluid dynamics (CFD) treating a two-phase flow has been proposed [16] recent years. The volume of fluid (VOF) method has the advantage of convergency and calculation times comparing with other methods [17]. Moreover, the VOF method also has the merit of reproducibility of slag flow in journal bearings. Therefore, there are some studies treating a two-phase flow CFD analysis utilizing the VOF method. For example, Zhai et al. and Dhande et al. [18, 19] studied the effect of vapor pressure and rotational speed on the gaseous-phase area of a journal bearing.

However, the experimental verifications have not been done in this study. On the other hand, a combination of the Reynolds equation and CFD analysis considering the two-phase flows was reported by Egbers et al. [20]. Furthermore, the gaseous-phase area in the oil-filler port and the opposite load side that was obtained by the analytical method in this study have been compared with that obtained in the experimental results. However, the analytical results cannot precisely produce the gaseous-phase scale and shape, because the influence of the surface tension has not been calculated.

In this situation, the authors have tried to reproduce the gaseous-phase area on both the bearing surface and the oil-filler port in a small-bore journal bearing under *The Multiphase Flow CFD Analysis in Journal Bearings Considering Surface Tension and Oil… DOI: http://dx.doi.org/10.5772/intechopen.92421*

the flooded and starved lubrication conditions while using a CFD model. Further, effects of VOF, surface tension, and vapor pressure of the setting condition were studied, and these analytical results were compared with the experiment, and the applicability was verified. Then, the authors considered the influence of surface tension on journal bearing from the Weber number, *We*. Furthermore, thermal analysis results under two types of supply oil conditions are shown. The effect of supply oil on bearing temperature characteristics was discussed from the results of calculation and experimental results.

### **2. Theory and calculation model**

As mentioned above, the VOF method has some advantages of calculation cost, convergency, and easy to handle compared with other methods [21–24]. Therefore, the authors selected the VOF method, and actually ANSYS FLUENT 15.0 is used in this study. The used methodology is explained below.

#### **2.1 Governing equations**

lubrication both theoretically and experimentally, and they propose a stabilization method utilizing the starved lubrication [3]. Furthermore, Naruse and Ochiai have experimentally studied the relation between gaseous-phase area and temperature distribution [4]. However, these observations have not been theoretically investigated [5], because the calculation method for the detailed gaseous-phase area has not been proposed. In addition, in actual bearing systems, if the lubricant is not able to supply the bearing sufficiently, it means there are starved lubrication condition and also a high possibility of serious erosion damage or serious seizure on the bearing surfaces. Therefore, it is believed that predicting the gaseous area in journal

Generally, to analyze the oil-lubricated journal bearing, the Reynolds equation is used, and the half-Sommerfeld condition or the Swift-Stieber condition has been applied [6, 7] in determining the gaseous areas for the Reynolds equation. However, in these models, it is assumed that the negative-pressure areas exist only in the gas phase and there is no oil in the bearing clearance area. Therefore, the flow-rate conservation does not hold in the calculations. As a more advanced method, Coyne and Elrod's condition [8, 9] is used. This model assumes oil-film rupture calculates the surface tension between the oil film and gaseous phase which is ignored in the half-Sommerfeld or the Swift-Stieber conditions. However, it is impossible to estimate the cavitation area of the entire journal bearing including the oil-filler port. Therefore, boundary condition models that consider the cavitation have been studied. For example, Ikeuchi and Mori have analyzed the oil-film cavitation areas while using the modified Reynolds equation [10, 11]. In this method, the two-phase flow is considered as an averaged single-phase flow of oil and gas. However, in the case of high eccentricity ratio and starved lubrication condition, they did not conduct the experimental verification with the cavitation area. On the other hand, considering the finger-type cavitation, analytical methods have been proposed by

Boncompine et al. [12] and Hatakenaka et al. [13]. However, the estimations of the variation of the gaseous-phase area against the changing of amount of oil lubricant have not been able to for bearing engineers or researchers. Furthermore, it is reported by Hashimoto, Ochiai, and Sakai that the oil-filler port and supply oil quantity affect the journal bearing characteristics [14, 15]. However, the internal flow of the oil-filler port has not been mentioned specifically. Therefore, a different approach is required to analyze the journal bearing while considering the internal

The computational fluid dynamics (CFD) treating a two-phase flow has been proposed [16] recent years. The volume of fluid (VOF) method has the advantage of convergency and calculation times comparing with other methods [17]. Moreover, the VOF method also has the merit of reproducibility of slag flow in journal bearings. Therefore, there are some studies treating a two-phase flow CFD analysis utilizing the VOF method. For example, Zhai et al. and Dhande et al. [18, 19] studied the effect of vapor pressure and rotational speed on the gaseous-phase area

However, the experimental verifications have not been done in this study. On the other hand, a combination of the Reynolds equation and CFD analysis considering the two-phase flows was reported by Egbers et al. [20]. Furthermore, the gaseous-phase area in the oil-filler port and the opposite load side that was obtained by the analytical method in this study have been compared with that obtained in the experimental results. However, the analytical results cannot precisely produce the gaseous-phase scale and shape, because the influence of the surface tension has not

In this situation, the authors have tried to reproduce the gaseous-phase area on both the bearing surface and the oil-filler port in a small-bore journal bearing under

bearings is very important.

*Computational Fluid Dynamics Simulations*

flow of an oil-filler port.

of a journal bearing.

been calculated.

**194**

Instead of the Reynolds equation, we applied the Navier–Stokes equation considering the surface tension to the journal bearing analysis in this study. The mass conservation equation and the momentum equation are shown as follows:

$$\nabla \cdot \overrightarrow{u} = \mathbf{0} \tag{1}$$

$$\rho \left[ \frac{\partial \overrightarrow{u}}{\partial t} + \nabla \left( \overrightarrow{u} \cdot \overrightarrow{u} \right) \right] = -\nabla p + \mu \nabla^2 \overrightarrow{u} + \rho \overrightarrow{g} + \left( \nabla \sigma + \sigma \gamma \overrightarrow{n} \right) \delta\_{\text{int}} \tag{2}$$

where *ρ* is the fluid density, *u* ! velocity vector, *p* fluid pressure, *μ* fluid viscosity, *ρ g* ! gravitational force, *σ* surface tension, *n* ! normal vector, *γ* curvature of the boundary surface, and *δint* Dirac's delta function.

Moreover, ∇is a differential operator, defined by

$$\nabla = \left(\frac{\partial}{\partial \mathbf{x}}, \frac{\partial}{\partial \mathbf{y}}, \frac{\partial}{\partial \mathbf{z}}\right) \tag{3}$$

In this VOF calculation model, the fluid density and fluid viscosity are expressed as follows:

$$
\rho = F\rho\_1 + (\mathbf{1} - F)\rho\_2 \tag{4}
$$

$$
\mu = F\mu\_1 + (\mathbf{1} - F)\mu\_2 \tag{5}
$$

where *F* is the volume fraction and subscription 1 means oil and 2 means gaseous phase.

This CFD analysis can analyze the internal flow of an oil-filler port with the bearing clearance simultaneously because the inertia term is considered in the basic equation.

On the other hand, the energy equation is shown as follows:

$$
\rho c\_p \left[ \overrightarrow{u} \,\nabla(T) \right] = \lambda \nabla T + \mathcal{S}\_h + \mu \left( \overrightarrow{u} \,\nabla^2 \overrightarrow{u} \right) \tag{6}
$$

where *cp* is the specific heat and *λ* thermal conductivity. Moreover, the second term in the right side means the volume heat term and the third term in the right side means the viscous dissipation term.

#### **2.2 Surface tension and cavitation model**

To consider the effect of surface tension, the continuum surface force (CSF) model proposed by Brackbill et al. [25] was used as the surface tension model implemented in FLUENT out-of-the-box. The last term of ∇*σ* þ *σγn* ! *<sup>δ</sup>int* in Eq. (2) represents the surface tension. In Brackbill et al.'s CSF model, the effect of surface tension is included as the surface tension term in the Navier–Stokes equation.

In addition, the cavitation model proposed by Schnerr and Sauer [20] was also used. The equation for the volume fraction of fluid is as follows:

$$\frac{\partial}{\partial t}(F\rho\_2) + \nabla(F\rho\_2) = \frac{\rho\_1\rho\_2}{\rho}\frac{DF}{Dt} \tag{7}$$

The vapor volume fraction ? is related to the number of bubbles *nb* per unit volume of liquid and bubble radius *Rb* as shown in the following equation:

$$F = \frac{n\_b \frac{4}{3} \pi R\_b^{-3}}{1 + n\_b \frac{4}{3} \pi R\_b^{-3}} \tag{8}$$

six-layer grids in the direction of the bearing clearance. The total mesh number under flooded and starved lubrication conditions were 64 <sup>10</sup><sup>5</sup> and 18 <sup>10</sup><sup>6</sup>

results were obtained as a pre-test study.

**Table 1.**

**Figure 2.**

**Table 2.**

**197**

*Calculation conditions.*

*Calculation model of the journal bearing [1].*

Fluid property Density (*ρ*) [kg/m<sup>3</sup>

*Specifications of bearing.*

*DOI: http://dx.doi.org/10.5772/intechopen.92421*

respectively. The confirmation of mesh size was conducted, and enough calculation

**Multiphase model Volume of fluid Calculation procedure Implicit method**

Surface tension (*S*) [N/m] 0.04

Viscosity (*μ*) [Pas] Oil 0.019

Specific heat (*Cp*), J/kg<sup>K</sup> Oil 19.5 <sup>10</sup> <sup>2</sup>

Thermal conductivity (*f*), [W/mK] Oil 0.13

] Oil 860

Air 1.23

Air 1.75 <sup>10</sup> <sup>5</sup>

Air 0.024

Air 10.1 <sup>10</sup> <sup>2</sup>

Calculation conditions Vaporization pressure (*Pv*) [Pa] 0

Diameter (*D*) [mm] 25.0 Length (*L*) [mm] 14.5 Clearance (*Cr*) [mm] 0.125 Width-diameter ratio (*L*/*D*) 0.58 Diameter of oil-filler hole (*Dp*) [mm] 8.2

*The Multiphase Flow CFD Analysis in Journal Bearings Considering Surface Tension and Oil…*

**Table 2** shows the calculation conditions. The tension between oil and air was also considered while performing the calculations. The vapor pressure is set zero the same as the ambient pressure. Because, in this study, the side of the bearing and side

,

where *nb* is the number of bubbles which was set as 10<sup>13</sup> in this study.

While considering the vapor pressure, the volume of air that is dissolved in oil expanded caused by a negative pressure which was observed in the journal bearing. Therefore, the vapor pressure was set to zero in this study. Moreover, the flow is laminar, and the analysis was conducted in a steady-state condition.

#### **2.3 Calculation model**

**Figure 1** depicts the outline of a bearing treating in this study. We chose a full circular-type journal bearing in this study. The upper position of the bearing is provided with an oil-filler port, allowing the lubricating oil to flow into the bearing clearance by using gravity. **Table 1** lists its major dimensions. This model is one of the typical bearings for a small-size rotating machinery.

The model of bearing CFD calculation in this study is depicted in **Figure 2**. The bearing clearance with oil-filler port and oil-supply groove were set as flow calculation regions. The symmetrical configuration against the bearing center is used in reducing the calculation cost. The clearance around the minimum part contains

**Figure 1.** *Geometry of the test journal bearing [1].*

*The Multiphase Flow CFD Analysis in Journal Bearings Considering Surface Tension and Oil… DOI: http://dx.doi.org/10.5772/intechopen.92421*


#### **Table 1.**

where *cp* is the specific heat and *λ* thermal conductivity. Moreover, the second term in the right side means the volume heat term and the third term in the right

To consider the effect of surface tension, the continuum surface force (CSF) model proposed by Brackbill et al. [25] was used as the surface tension model

represents the surface tension. In Brackbill et al.'s CSF model, the effect of surface tension is included as the surface tension term in the Navier–Stokes equation.

*<sup>F</sup>ρ*<sup>2</sup> ð Þþ <sup>∇</sup> *<sup>F</sup>ρ*<sup>2</sup> ð Þ¼ *<sup>ρ</sup>*1*ρ*<sup>2</sup>

The vapor volume fraction ? is related to the number of bubbles *nb* per unit

1 þ *nb* 4 <sup>3</sup> *πRb*

4 <sup>3</sup> *πRb* 3

While considering the vapor pressure, the volume of air that is dissolved in oil expanded caused by a negative pressure which was observed in the journal bearing. Therefore, the vapor pressure was set to zero in this study. Moreover, the flow is

**Figure 1** depicts the outline of a bearing treating in this study. We chose a full circular-type journal bearing in this study. The upper position of the bearing is provided with an oil-filler port, allowing the lubricating oil to flow into the bearing clearance by using gravity. **Table 1** lists its major dimensions. This model is one of

The model of bearing CFD calculation in this study is depicted in **Figure 2**. The bearing clearance with oil-filler port and oil-supply groove were set as flow calculation regions. The symmetrical configuration against the bearing center is used in reducing the calculation cost. The clearance around the minimum part contains

volume of liquid and bubble radius *Rb* as shown in the following equation:

*<sup>F</sup>* <sup>¼</sup> *nb*

where *nb* is the number of bubbles which was set as 10<sup>13</sup> in this study.

laminar, and the analysis was conducted in a steady-state condition.

the typical bearings for a small-size rotating machinery.

**2.3 Calculation model**

**Figure 1.**

**196**

*Geometry of the test journal bearing [1].*

*ρ*

*DF*

In addition, the cavitation model proposed by Schnerr and Sauer [20] was also

!

*Dt* (7)

<sup>3</sup> (8)

*δint* in Eq. (2)

implemented in FLUENT out-of-the-box. The last term of ∇*σ* þ *σγn*

used. The equation for the volume fraction of fluid is as follows:

*∂ ∂t*

side means the viscous dissipation term.

*Computational Fluid Dynamics Simulations*

**2.2 Surface tension and cavitation model**

*Specifications of bearing.*

#### **Figure 2.**

*Calculation model of the journal bearing [1].*


#### **Table 2.** *Calculation conditions.*

six-layer grids in the direction of the bearing clearance. The total mesh number under flooded and starved lubrication conditions were 64 <sup>10</sup><sup>5</sup> and 18 <sup>10</sup><sup>6</sup> , respectively. The confirmation of mesh size was conducted, and enough calculation results were obtained as a pre-test study.

**Table 2** shows the calculation conditions. The tension between oil and air was also considered while performing the calculations. The vapor pressure is set zero the same as the ambient pressure. Because, in this study, the side of the bearing and side of the oil-supply groove are open to the outside, the outside gas is easy to flow into the bearing and easy to generate the gaseous-phase cavitation at the position of negative-pressure generation. The surface tension was set to 0.04 N/m, which was measured while using the du Noüy method (ASTM 971–50).
