*4.1.1 Equation of continuity*

The continuity equation can be expressed by the relationship (1) [20].

$$\nabla \left( \rho \stackrel{\rightarrow}{U} \right) = \stackrel{\rightarrow}{\mathbf{0}} \tag{1}$$

*ρ u ∂v ∂x* þ *v ∂v ∂y* þ *w ∂v ∂z*

*ρ u* ∂w <sup>∂</sup><sup>x</sup> <sup>þ</sup> *<sup>v</sup>*

*ρ* is fluid density.

expressed as follows:

� �

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

∂w ∂y þ *w* ∂w ∂z

� �

¼ � *<sup>∂</sup><sup>p</sup> ∂y* þ *μ*

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

¼ � <sup>∂</sup><sup>p</sup> ∂z þ *μ*

*Bx* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

*By* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

governing differential equation into this control volume.

*4.1.3 Discretization of governance equations*

Cartesian coordinate system (x, y, z):

*∂ ∂X <sup>j</sup>*

ð *s*

**Figure 2**.

**25**

*ρ U jUi* � � ¼ � <sup>∂</sup><sup>P</sup>

*<sup>ρ</sup> <sup>U</sup> jUidn <sup>j</sup>* ¼ �<sup>ð</sup>

Considering the Z axis as the axis of rotation, the components of B can be

*<sup>Z</sup> rx* <sup>þ</sup> <sup>2</sup>*ωzv* � �

*<sup>Z</sup> ry* <sup>þ</sup> <sup>2</sup>*ωzu* � �

*Bz* ¼ 0

The main step of the finite volume method is the integration of governing equations for each control volume [20]. The algebraic equations deduced from this integration make the resolution of the transport equations simpler. Each node is surrounded by a set of surfaces that has a volume element. All the variables of the problem and the properties of the fluid are stored at the nodes of this element. The equations governing the flow are presented in their averaged forms in a

*ρ U <sup>j</sup>*

Eqs. (6) and (7) can be integrated into a control volume, using the Gaussian divergence theorem to convert volume integrals to surface integrals as follows:

*μ*

*∂Ui ∂X <sup>j</sup>* þ *∂U <sup>j</sup> ∂Xi* � �*dn <sup>j</sup>* <sup>þ</sup>

*∂Ui ∂Xi* þ *∂U <sup>j</sup> ∂Xi*

� � � �

*∂ ∂X <sup>j</sup>*

*∂Xi* þ *∂ ∂X <sup>j</sup>*

ð *s*

> ð *s μ*

The next step is to discretize the known m's of the problem as well as the differential operators of this equation. All these mathematical operations will lead to obtaining, on each volume of control, a discretized equation that will link the variables of a cell to those of neighboring cells. All of these discretized equations will eventually form a matrix system. Considering an element of an isolated mesh,

*s P dn <sup>j</sup>* þ

The finite volume method used to solve the continuity and Navier-Stokes equations consists in subdividing the physical domain of the flow into elements of more or less regular volumes; it converts the general differential equation into a system of Algebraic equations by relating the values of the variable under consideration to the adjacent nodal points of a typical control volume. This is achieved by integrating the

*∂*2 *v ∂x*<sup>2</sup> þ

*∂*2 *w ∂x*<sup>2</sup> þ

*∂*2 *v ∂y*<sup>2</sup> þ

� �

*∂*2 *w ∂y*<sup>2</sup> þ

� �

*∂*2 *v ∂z*<sup>2</sup>

> *∂*2 *w ∂z*<sup>2</sup>

� � <sup>¼</sup> <sup>0</sup> (6)

*ρ U jdn <sup>j</sup>* ¼ 0 (8)

*<sup>S</sup>*uidv �

ð *V*

þ *Bx* (7)

(9)

þ *By*

þ *Bz*

(5)

where *U* ! = *U* ! *(u,* v, *w*) is the velocity vector. Eq. (1) can also be written as follows:

$$\frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}}{\partial \mathbf{y}} + \frac{\partial \mathbf{w}}{\partial \mathbf{z}} = \mathbf{0} \tag{2}$$

### *4.1.2 Navier-Stokes equations*

The Navier-Stokes equation can be defined in the following form (2003):

$$\partial \nabla \cdot \left( \vec{U} \otimes \vec{U} \right) = -\nabla p + \mu \nabla \cdot \left( \nabla \vec{U} + \left( \nabla \vec{U} \right)^{T} \right) + B \tag{3}$$

With *P* static pressure (thermodynamic); *U* velocity; *μ* dynamic viscosity.

For fluids in a rotating frame with constant angular velocity ω source term B can be written as follows:

$$B = -\rho \left( \mathbf{2} \,\overrightarrow{\boldsymbol{\phi}} \times \overrightarrow{\mathbf{U}} + \overrightarrow{\boldsymbol{\phi}} \times \left( \overrightarrow{\boldsymbol{\phi}} \times \overrightarrow{\boldsymbol{r}} \right) \right) \tag{4}$$

Eq. (1) can also be expressed in the form:

$$\rho \left( u \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + v \frac{\partial \mathbf{u}}{\partial \mathbf{y}} + w \frac{\partial \mathbf{u}}{\partial \mathbf{z}} \right) = -\frac{\partial \mathbf{p}}{\partial \mathbf{x}} + \mu \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial \mathbf{z}^2} \right) + B\_{\mathbf{x}}$$

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

$$\begin{aligned} \rho \left( u \frac{\partial v}{\partial \mathbf{x}} + v \frac{\partial v}{\partial \mathbf{y}} + w \frac{\partial v}{\partial \mathbf{z}} \right) &= -\frac{\partial p}{\partial \mathbf{y}} + \mu \left( \frac{\partial^2 v}{\partial \mathbf{x}^2} + \frac{\partial^2 v}{\partial \mathbf{y}^2} + \frac{\partial^2 v}{\partial \mathbf{z}^2} \right) + B\_\mathbf{y} \\ \rho \left( u \frac{\partial \mathbf{w}}{\partial \mathbf{x}} + v \frac{\partial \mathbf{w}}{\partial \mathbf{y}} + w \frac{\partial \mathbf{w}}{\partial \mathbf{z}} \right) &= -\frac{\partial \mathbf{p}}{\partial \mathbf{z}} + \mu \left( \frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial \mathbf{y}^2} + \frac{\partial^2 w}{\partial \mathbf{z}^2} \right) + B\_\mathbf{z} \end{aligned} \tag{5}$$

*ρ* is fluid density.

**4.1 Equation fluid flow**

*4.1.1 Equation of continuity*

*4.1.2 Navier-Stokes equations*

be written as follows:

*ρ u* ∂u <sup>∂</sup><sup>x</sup> <sup>þ</sup> *<sup>v</sup>*

**Figure 1.**

**24**

where *U* ! = *U* !

The continuity equation can be expressed by the relationship (1) [20].

∇ *ρ U* !

> ∂v ∂y þ ∂w

The Navier-Stokes equation can be defined in the following form (2003):

¼ �∇*p* þ *μ*∇*:* ∇*U*

With *P* static pressure (thermodynamic); *U* velocity; *μ* dynamic viscosity. For fluids in a rotating frame with constant angular velocity ω source term B can

> ! � *U* ! þ *ω* ! � *ω*

¼ � <sup>∂</sup><sup>p</sup> <sup>∂</sup><sup>x</sup> <sup>þ</sup> *<sup>μ</sup>*

*Schematization of plain bearing. (a) Non-textured plain bearing. (b) Textured plain bearing.*

!

!

þ ∇*U* ! *<sup>T</sup>*

! � *r*

*∂*2 *u ∂y*<sup>2</sup> þ

*∂*2 *u ∂z*<sup>2</sup>

þ *Bx*

*∂*2 *u ∂x*<sup>2</sup> þ

*(u,* v, *w*) is the velocity vector.

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

*B* ¼ �*ρ* 2 *ω*

Eq. (1) can also be expressed in the form:

∂u ∂y þ *w* ∂u ∂z

∂u ∂x þ

Eq. (1) can also be written as follows:

*∂*∇*: U* ! ⊗ *U* ! ¼ 0 !

<sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>0</sup> (2)

þ *B* (3)

(1)

(4)

Considering the Z axis as the axis of rotation, the components of B can be expressed as follows:

$$B\_{\mathbf{x}} = \left(a\_Z^2 \, r\_{\mathbf{x}} + 2a\_{\mathbf{z}}v\right)$$

$$B\_{\mathbf{y}} = \left(a\_Z^2 \, r\_{\mathbf{y}} + 2a\_{\mathbf{z}}u\right)$$

$$B\_{\mathbf{z}} = \mathbf{0}$$

The finite volume method used to solve the continuity and Navier-Stokes equations consists in subdividing the physical domain of the flow into elements of more or less regular volumes; it converts the general differential equation into a system of Algebraic equations by relating the values of the variable under consideration to the adjacent nodal points of a typical control volume. This is achieved by integrating the governing differential equation into this control volume.

#### *4.1.3 Discretization of governance equations*

The main step of the finite volume method is the integration of governing equations for each control volume [20]. The algebraic equations deduced from this integration make the resolution of the transport equations simpler. Each node is surrounded by a set of surfaces that has a volume element. All the variables of the problem and the properties of the fluid are stored at the nodes of this element.

The equations governing the flow are presented in their averaged forms in a Cartesian coordinate system (x, y, z):

$$\frac{\partial}{\partial X\_j} \left( \rho \, U\_j \right) = 0 \tag{6}$$

$$\frac{\partial}{\partial \mathbf{X}\_j} \left( \rho \, U\_j U\_i \right) = -\frac{\partial \mathbf{P}}{\partial \mathbf{X}\_i} + \frac{\partial}{\partial \mathbf{X}\_j} \left( \mu \left( \frac{\partial U\_i}{\partial \mathbf{X}\_i} + \frac{\partial U\_j}{\partial \mathbf{X}\_i} \right) \right) + B\_{\mathbf{x}} \tag{7}$$

Eqs. (6) and (7) can be integrated into a control volume, using the Gaussian divergence theorem to convert volume integrals to surface integrals as follows:

$$\int\_{s} \rho \, U\_{j} d\boldsymbol{n}\_{j} = \mathbf{0} \tag{8}$$

$$\int\_{\mathfrak{s}} \rho \, U\_j \mathbf{U}\_i dn\_j = -\int\_{\mathfrak{s}} P \, dn\_j + \int\_{\mathfrak{s}} \left( \mu \left( \frac{\partial U\_i}{\partial \mathbf{X}\_j} + \frac{\partial U\_j}{\partial \mathbf{X}\_i} \right) dn\_j + \int\_{V} \mathbf{S}\_{\text{ul}} \mathbf{dv} \right) \tag{9}$$

The next step is to discretize the known m's of the problem as well as the differential operators of this equation. All these mathematical operations will lead to obtaining, on each volume of control, a discretized equation that will link the variables of a cell to those of neighboring cells. All of these discretized equations will eventually form a matrix system. Considering an element of an isolated mesh, **Figure 2**.

**Figure 2.** *Integration point in an element of a control volume control.*

After the discretization and rearrangement of Eqs. (8) and (9) the following forms will be obtained:

$$\sum\_{\text{ip}} \left( \rho \ U\_j \,\Delta n\_j \right)\_{\text{ip}} = \mathbf{0} \tag{10}$$

*ϕ* ¼

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

*x* ¼

*y* ¼

*z* ¼

*∂ϕ ∂x* � � � � *i* <sup>¼</sup> <sup>X</sup> *n*

Stokes equations involves the evaluation of the following expression:

*<sup>P</sup>*ip <sup>¼</sup> <sup>X</sup> *n*

of the constraint equations Reynolds RMS).

(Detached Eddy Simulation, DES).

which case they can be written in the following way:

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

different flows is as follows:

*4.1.6 Pressure gradients*

where:

**27**

Node X *i*¼1

These functions are also us5ed for the calculation of various geometric quantities, such as positions, coordinates of the integration point (ip), surfaces and different vectors. Form equations are also applicable for Cartesian coordinates, in

> Node X *i*¼1

> Node X *i*¼1

> Node X *i*¼1

The shape functions are also used to evaluate the partial derivatives of the flow terms on the control surfaces and for each direction, the general formula of the

> *∂Nn ∂x* � � � � *ip*

The integration of the pressure gradient (P) on the control volume in the Navier-

For the improved treatment of fluctuations induced by turbulence in the motion

of a particle of fluid, there are three methods of approach to address the notion turbulence. The first method is to decompose the field of velocity and temperature in a mean component and a turbulent fluctuation, to make a variety of models are now available, ranging from the simple model equation to zero to complex (model

The second is a method in which all the structures of turbulence (macro and micro-structures) are solved directly and models the effect of small structures by models more or less simple, so-called sub-grid models. This method is known as the large eddy simulation (Large Eddy Simulation, LES). The third method is a hybrid approach combines the advantages qm large eddy simulation (LES), with good results in highly separated zones, and model Reynolds-Averaged Navier-Stokes (RANS), which are most effective in areas close to the walls. The method is called

*P Δnip* � �

*Ni* ¼ 1 (15)

*Ni xi* (16)

*Ni yi* (17)

*Ni zi* (18)

*ϕ<sup>n</sup>* (19)

*ip* (20)

*Nn S*ip, *t*ip, *u*ip � �*Pn* (21)

$$\sum\_{\text{ip}} m\_{\text{ip}} \left( U\_i \right)\_{\text{ip}} = \sum\_{\text{ip}} \left( P \, \Delta n\_j \right)\_{\text{ip}} + \sum\_{\text{ip}} \left( \mu \left( \frac{\partial U\_i}{\partial \mathbf{X}\_j} + \frac{\partial U\_j}{\partial \mathbf{X}\_i} \right) \Delta n\_j \right) + \overline{\mathbf{S}\_{u\_i} V} \qquad \text{(11)}$$

$$N\_j = \begin{Bmatrix} \mathbf{1} & i = j \\ \mathbf{0} & i \neq j \end{Bmatrix}$$

#### *4.1.4 Coupling pressure-velocity*

The method of pressure interpolation in pressure-velocity coupling is similar to that used by Rhie and Chow (1982). This method is among the methods that best save memory space and computation time. If the pressure is known, the discretized equations are easily solved [20]:

$$
\left(\frac{\partial \mathbf{U}}{\partial \mathbf{x}}\right)\_i + \frac{\Delta \mathbf{x}^3 A}{4m} \left(\frac{\partial^4 P}{\partial \mathbf{x}^4}\right) \mathbf{0} \tag{12}
$$

where:

$$
\rho \mathfrak{m} = \rho \ U\_i \,\Delta n\_j \tag{13}
$$

#### *4.1.5 Form functions*

The physical quantity *ϕ* (p, u, v, w and p) of the flow in a volume element is a function of those in the nodes of the element is given by the following relation:

$$\phi = \sum\_{i=1}^{\text{Node}} N\_i \,\phi\_i \tag{14}$$

where *Ni* is the form function for node *i* and *ϕi* the value of the variable in the same node. A particularity of the form factors makes sure that:

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

$$\phi = \sum\_{i=1}^{\text{Node}} N\_i = \mathbf{1} \tag{15}$$

These functions are also us5ed for the calculation of various geometric quantities, such as positions, coordinates of the integration point (ip), surfaces and different vectors. Form equations are also applicable for Cartesian coordinates, in which case they can be written in the following way:

$$\mathbf{x} = \sum\_{i=1}^{\text{Node}} N\_i \,\mathbf{x}\_i \tag{16}$$

$$\mathcal{y} = \sum\_{i=1}^{\text{Node}} N\_i \mathcal{y}\_i \tag{17}$$

$$z = \sum\_{i=1}^{\text{Node}} N\_i \, z\_i \tag{18}$$

The shape functions are also used to evaluate the partial derivatives of the flow terms on the control surfaces and for each direction, the general formula of the different flows is as follows:

$$\left. \frac{\partial \phi}{\partial \mathbf{x}} \right|\_i = \sum\_n \frac{\partial N\_n}{\partial \mathbf{x}} \bigg|\_{ip} \phi\_n \tag{19}$$

#### *4.1.6 Pressure gradients*

The integration of the pressure gradient (P) on the control volume in the Navier-Stokes equations involves the evaluation of the following expression:

$$\left(P\,\Delta n\_{ip}\right)\_{ip} \tag{20}$$

where:

After the discretization and rearrangement of Eqs. (8) and (9) the following

ip *<sup>ρ</sup> <sup>U</sup> <sup>j</sup> <sup>Δ</sup><sup>n</sup> <sup>j</sup>* � �

*<sup>N</sup> <sup>j</sup>* <sup>¼</sup> <sup>1</sup> *<sup>i</sup>* <sup>¼</sup> *<sup>j</sup>*

ip *<sup>μ</sup>*

0 *i* 6¼ *j* � �

The method of pressure interpolation in pressure-velocity coupling is similar to that used by Rhie and Chow (1982). This method is among the methods that best save memory space and computation time. If the pressure is known, the discretized

> *Δx*<sup>3</sup>*A* 4*m*

The physical quantity *ϕ* (p, u, v, w and p) of the flow in a volume element is a function of those in the nodes of the element is given by the following relation:

where *Ni* is the form function for node *i* and *ϕi* the value of the variable in the

Node X *i*¼1

*∂*<sup>4</sup>*P ∂x*<sup>4</sup> � �

*∂Ui ∂X <sup>j</sup>* þ *∂U <sup>j</sup> ∂Xi*

� �

� �

ip <sup>þ</sup><sup>X</sup>

ip ¼ 0 (10)

þ *Sui*

0 (12)

*m* ¼ *ρ Ui Δn <sup>j</sup>* (13)

*Ni ϕ<sup>i</sup>* (14)

*V* (11)

*Δn <sup>j</sup>*

X

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

∂U ∂x � �

*i* þ

*ϕ* ¼

same node. A particularity of the form factors makes sure that:

ip *P Δn <sup>j</sup>* � �

*Integration point in an element of a control volume control.*

forms will be obtained:

ip*m*ip ð Þ *Ui* ip <sup>¼</sup> <sup>X</sup>

*4.1.4 Coupling pressure-velocity*

equations are easily solved [20]:

X

**Figure 2.**

where:

**26**

*4.1.5 Form functions*

$$P\_{\rm ip} = \sum\_{n} \mathcal{N}\_{n} \left( \mathcal{S}\_{\rm ip}, t\_{\rm ip}, u\_{\rm ip} \right) P\_{n} \tag{21}$$

For the improved treatment of fluctuations induced by turbulence in the motion of a particle of fluid, there are three methods of approach to address the notion turbulence. The first method is to decompose the field of velocity and temperature in a mean component and a turbulent fluctuation, to make a variety of models are now available, ranging from the simple model equation to zero to complex (model of the constraint equations Reynolds RMS).

The second is a method in which all the structures of turbulence (macro and micro-structures) are solved directly and models the effect of small structures by models more or less simple, so-called sub-grid models. This method is known as the large eddy simulation (Large Eddy Simulation, LES). The third method is a hybrid approach combines the advantages qm large eddy simulation (LES), with good results in highly separated zones, and model Reynolds-Averaged Navier-Stokes (RANS), which are most effective in areas close to the walls. The method is called (Detached Eddy Simulation, DES).
