**5. k-Epsilon model**

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 near-wall mesh is very fine.

*k* is the turbulence kinetic energy and is defined as the variance of the fluctuations in velocity. It has dimensions of (L2 T�<sup>2</sup> ); for example, m<sup>2</sup> /s2 . ε is the turbulence eddy dissipation (the rate at which the velocity fluctuations dissipate), and has dimensions of *k* per unit time (L2 T�<sup>3</sup> ); for example, m<sup>2</sup> /s3 .

The k-ε model introduces two new variables into the system of equations. The continuity equation is following forms:

$$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \text{ (}\rho \text{ U}\_j\text{)} = \mathbf{0} \tag{22}$$

*Pkb* and *Pε<sup>b</sup>* represent the influence of the buoyancy forces, which are described below. *Pk* is the turbulence production due to viscous forces, which is modeled using:

> *∂Uk ∂xk*

3*μ<sup>t</sup>*

*∂Uk ∂xk*

þ *ρ k* (29)

*∂x <sup>j</sup>* � 2 3

The term 3 *μ<sup>t</sup>* in Eq. (37) is based on the "frozen stress" assumption. This

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

**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

The geometrical and operating parameters of the plain journal bearing is presented

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

in the **Table 1**. As well as, parameters of the lubricant are showed in **Table 2**.

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

prevents the values of *k* and *ε* becoming too large through shocks.

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

*Pk* ¼ *μ<sup>t</sup>*

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

equations eventually form a matrix system.

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

**6.2 Boundary conditions of the numerical model**

**6. Numerical model**

and the shaft (**Figure 4**).

**Figure 3.**

**29**

*∂Ui ∂x <sup>j</sup>* þ *∂U <sup>j</sup> ∂xi ∂Ui*

and the momentum equation becomes:

$$\frac{\partial \rho \ U\_i}{\partial \mathbf{t}} + \frac{\partial}{\partial \mathbf{x}\_j} \left( \rho \ U\_i U\_j \right) = -\frac{\partial p'}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \mu\_{\text{eff}} \left( \frac{\partial U\_i}{\partial \mathbf{x}\_j} + \frac{\partial U\_j}{\partial \mathbf{x}\_i} \right) \right] + \mathbf{S}\_{\mathcal{M}} \tag{23}$$

where *SM* is the sum of body forces, *μeff* is the effective viscosity accounting for turbulence, and *p'* is the modified pressure as defined in Eq. (22).

$$p' = p + \frac{2}{3} \,\rho \, k + \frac{2}{3} \,\mu\_{\rm eff} \, \frac{\partial U\_k}{\partial \mathbf{x}\_k} \tag{24}$$

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

$$
\mu\_{\rm eff} = \mu + \mu\_t \tag{25}
$$

where *μ<sup>t</sup>* is the turbulence viscosity. The *k-ε* model assumes that the turbulence viscosity is linked to the turbulence kinetic energy and dissipation:

$$
\mu\_t = \mathcal{C}\_{\mu}\rho \, \frac{k^2}{\varepsilon} \tag{26}
$$

where *Cμ* is a constant.

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> *ε :*

The values of *k* and *ε* come directly from the differential transport equations for the turbulence kinetic energy and turbulence dissipation rate:

$$\frac{\partial(\rho \,\mathrm{k})}{\partial \mathbf{t}} + \frac{\partial}{\partial \mathbf{x}\_j} \left(\rho \,\, U\_j \,\, \mathrm{k}\right) = \frac{\partial}{\partial \mathbf{x}\_j} \left[\left(\mu + \frac{\mu\_t}{\sigma\_k}\right) \,\, \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j}\right] + P\_k - \rho \,\, \mathrm{e} + P\_{\mathrm{kb}} \tag{27}$$

$$\frac{\partial(\rho \,\, \varepsilon)}{\partial \mathbf{t}} + \frac{\partial}{\partial \mathbf{x}\_j} \left(\rho \,\, U\_j \,\, \varepsilon\right) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left(\mu + \frac{\mu\_t}{\sigma\_\varepsilon}\right) \,\, \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \frac{\varepsilon}{k} (\mathbf{C}\_{\varepsilon 1} \mathbf{P}\_k - \mathbf{C}\_{\varepsilon 2} \,\, \rho \,\, \varepsilon + \mathbf{C}\_{\varepsilon 1} \mathbf{P}\_{\mathrm{kb}}) \tag{28}$$

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

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

*Pkb* and *Pε<sup>b</sup>* represent the influence of the buoyancy forces, which are described below. *Pk* is the turbulence production due to viscous forces, which is modeled using:

$$P\_k = \mu\_t \left(\frac{\partial U\_i}{\partial \mathbf{x}\_j} + \frac{\partial U\_j}{\partial \mathbf{x}\_i}\right) \frac{\partial U\_i}{\partial \mathbf{x}\_j} - \frac{2}{3} \frac{\partial U\_k}{\partial \mathbf{x}\_k} \left(3\mu\_t \frac{\partial U\_k}{\partial \mathbf{x}\_k} + \rho \not\models \mathbf{k}\right) \tag{29}$$

The term 3 *μ<sup>t</sup>* in Eq. (37) is based on the "frozen stress" assumption. This prevents the values of *k* and *ε* becoming too large through shocks.
