**6. Numerical model**

**5. k-Epsilon model**

near-wall mesh is very fine.

∂ρ *Ui* ∂t þ *∂ ∂x <sup>j</sup>*

concept, so that:

where *Cμ* is a constant.

*<sup>∂</sup>*ð Þ *<sup>ρ</sup> <sup>k</sup>* ∂t þ *∂ ∂x <sup>j</sup>*

*<sup>∂</sup>*ð Þ *ρ ε* ∂t þ *∂ ∂x <sup>j</sup>*

**28**

With *<sup>C</sup><sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>09</sup> *<sup>θ</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>1</sup>*=*<sup>2</sup> *<sup>l</sup>* <sup>¼</sup> *<sup>k</sup>*3*=*<sup>2</sup>

*<sup>ρ</sup> <sup>U</sup> <sup>j</sup> <sup>ε</sup>* <sup>¼</sup> *<sup>∂</sup>*

where *Cε1*, *Cε2*, *σ<sup>k</sup>* and *σε* are constants.

tions in velocity. It has dimensions of (L2 T�<sup>2</sup>

and the momentum equation becomes:

*ρ UiU <sup>j</sup>* ¼ � *<sup>∂</sup>p*<sup>0</sup>

has dimensions of *k* per unit time (L2 T�<sup>3</sup>

continuity equation is following forms:

One of the most prominent turbulence models, the (k-epsilon) model, has been implemented in most CFD codes [20]. It has proven to be stable and numerically robust and has a well-established regime of predictive capability; the model offers a good compromise in terms of accuracy and robustness. This turbulence model uses the scalable wall-function approach to improve robustness and accuracy when the

*k* is the turbulence kinetic energy and is defined as the variance of the fluctua-

The k-ε model introduces two new variables into the system of equations. The

*ρ U <sup>j</sup>*

*μ*eff

where *SM* is the sum of body forces, *μeff* is the effective viscosity accounting for

2 <sup>3</sup> *<sup>μ</sup>*eff *∂Ui ∂x <sup>j</sup>* þ *∂U <sup>j</sup> ∂xi*

*∂Uk ∂xk*

*μ*eff ¼ *μ* þ *μ<sup>t</sup>* (25)

*<sup>ε</sup>* (26)

þ *Pk* � *ρ ε* þ *P*kb (27)

ð Þ *C<sup>ε</sup>*1*Pk* � *C<sup>ε</sup>*<sup>2</sup> *ρ ε* þ *C<sup>ε</sup>*1*P*kb (28)

lence eddy dissipation (the rate at which the velocity fluctuations dissipate), and

∂ρ ∂t þ *∂ ∂x <sup>j</sup>*

*Tribology in Materials and Manufacturing - Wear, Friction and Lubrication*

*∂xi* þ *∂ ∂x <sup>j</sup>*

> 2 <sup>3</sup> *<sup>ρ</sup> <sup>k</sup>* <sup>þ</sup>

The *k-ε* model, like the zero equation model, is based on the eddy viscosity

where *μ<sup>t</sup>* is the turbulence viscosity. The *k-ε* model assumes that the turbulence

The values of *k* and *ε* come directly from the differential transport equations for

*<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup> σk* ∂k

*∂x <sup>j</sup>*

*∂x <sup>j</sup>*

þ *ε k*

*k*2

*μ<sup>t</sup>* ¼ *Cμρ*

turbulence, and *p'* is the modified pressure as defined in Eq. (22).

*p*<sup>0</sup> ¼ *p* þ

viscosity is linked to the turbulence kinetic energy and dissipation:

*ε :*

*∂x <sup>j</sup>*

*<sup>μ</sup>* <sup>þ</sup> *<sup>μ</sup><sup>t</sup> σε* ∂ε

the turbulence kinetic energy and turbulence dissipation rate:

*<sup>ρ</sup> <sup>U</sup> <sup>j</sup> <sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>*

*∂x <sup>j</sup>*

); for example, m<sup>2</sup>

); for example, m<sup>2</sup>

/s2

/s3 .

<sup>¼</sup> <sup>0</sup> (22)

. ε is the turbu-

þ *SM* (23)

(24)

The purpose of this study is to highlight the behavior of the turbulent fluid flow fluid on the operating characteristics as well as the hydrodynamic behavior of a plain bearing This study is simulated by the CFD calculation code, which provides accuracy, reliability, speed and flexibility in potentially complex flow areas. Integrating the Reynolds equation on each control volume to derive an equation connecting the discrete variables of the elements that surround it, all of these equations eventually form a matrix system.

### **6.1 3D structure of the numerical model**

**Figure 3** illustrates the 3-D structure of the plain bearing with fluid and solid regions are shown. The supply holes are presented in a simplified manner without affecting the accuracy of the model. A tetrahedron element is adopted in the oil supply holes of the fluid region, and a hexahedral element is adopted in domain fluid. A hexahedral element is also applied to the solid region such as the bearing and the shaft (**Figure 4**).

The geometrical and operating parameters of the plain journal bearing is presented in the **Table 1**. As well as, parameters of the lubricant are showed in **Table 2**.

#### **6.2 Boundary conditions of the numerical model**

Boundary conditions of the numerical model of the plain bearing are shown in **Figure 5**, definite as follows: 1: the rotating speed is applied to the outer wall surface

**Figure 3.** *3D structure of the non-textured plain bearing. (a) Non-textured bearing. (b) textured bearing.*

#### **Figure 4.**

*Mesh of the plain bearing. (a) Non-textured bearing. (b) textured bearing.*


5: the two ends of the plain bearing domain, and the pressure is set to one bar; and is

The setting is done by a graphical interface. The mesh used is a mixed mesh which understood elements of tetrahedral type with 6 nodes and hexahedral elements with 8 nodes. It's necessary to choose an appropriate mesh, consequently, a mesh independence study is carried out, and calculation results are shown in **Figure 6**. When the nodes number is greater than 4815, the evolution of the pressure stabilizes in the angular coordinate 205° of the plain bearing. Therefore, the number of nodes chosen for this numerical analysis corresponds to a number of nodes equal to 4815. The nodes number for textured bearing is 65,172. Convergence criterion of the numerical results is calculated for a maximum number of iterations

**6.3 Validation of the mesh independence of the numerical model**

*Turbulent Flow Fluid in the Hydrodynamic Plain Bearing to a Non-Textured…*

*DOI: http://dx.doi.org/10.5772/intechopen.94235*

*Evolution max pressure according to the nodes number of the shaft mesh.*

considered as symmetry.

**Figure 5.**

**Figure 6.**

**31**

*Boundary conditions.*

#### **Table 1.**

*Geometrical and operating parameters of the plain bearing.*


#### **Table 2.**

*Parameters of the lubricant.*

of the shaft; 2: the inner wall surface of the bushing is stationary; 3: the domain is simulated by the fluid region. The slip of the interface is ignored; 4: the oil supply pressure is 0.08 MPa and supply temperature is 40°C, are set in oil supply holes;

*Turbulent Flow Fluid in the Hydrodynamic Plain Bearing to a Non-Textured… DOI: http://dx.doi.org/10.5772/intechopen.94235*

**Figure 5.** *Boundary conditions.*

5: the two ends of the plain bearing domain, and the pressure is set to one bar; and is considered as symmetry.
