*4.3.1 Case of single phase and laminar flow*

#### *4.3.1.1 Mesh sensitivity*

In this part, we study the effect of the mesh on the axial component of the velocity field. The studied geometry is discretized in n m meshes of rectangular shape. The calculation is carried out using Ansys software based on the finite volume method.

In order to assess the accuracy of the numerical method, we performed several studies based on systematic mesh refinement until there were negligible changes in the variation of the axial velocity. **Table 2** present the different mesh used in this simulation for single phase laminar flow inside the diesel injector and for two Reynold values Re = 381 and 1000, respectively.

**Figure 9(a, b)** shows the axial velocity profile for different mesh and for two Reynolds number Re = 381 and 1000, respectively. From this figure, the axial velocity exhibits oscillations for the first and the second mesh. This oscillation shows that the solution is not stable and that convergence has not yet been achieved. The last mesh (1453 meshes) seems the best since it shows that convergence has indeed been achieved.


**Table 2.** *Mesh used in this simulation.*

*Analysis of Geometric Parameters of the Nozzle Orifice on Cavitating Flow and Entropy… DOI: http://dx.doi.org/10.5772/intechopen.99404*

**Figure 9.** *Axial velocity profile for several mesh and for two values of Re = 381 (left) and Re = 1000 (right).*

#### *4.3.1.2 Entropy production*

**Figure 10** shows that the entropy generation increases from zero at the center of the channel to a maximum value on the wall. Thistrend indicates that the maximum entropy produced on the wall is mainly due to the irreversibility of fluid friction contributed by the wall velocity gradient near top while towards the center of the channel with zero velocity gradient, the generation of entropy due to friction of the fluid.

The total entropy is obtained by taking the integral of the local entropy over the total volume, given by the following expression:

$$\mathbf{S}\_{\text{tot}} = \mathbf{g} \mathbf{f} \mathbf{S} \mathbf{d} \mathbf{V} \tag{16}$$

**Figure 11** illustrates the total entropy as a function of the inlet injection velocity v0. Its clear that the total entropy production varies quadratically with the inlet velocity.

**Figure 10.** *Local entropy profile inside the orifice for different values of average inlet velocity v0 = 0.1, 0.325, 0.55 et 1 mm/s.*

**Figure 11.** *Evolution of total entropy production as a function of inlet velocity.*

The interpolation in Lagrange polynomial of degree 2 of Stot as a function of inlet velocity *v*0:

$$\mathcal{S}\_{tot}(nW/K) = \mathcal{C}\_0 + \mathcal{C}\_1 \times \mathcal{v}\_0 + \mathcal{C}\_2 \times \mathcal{v}\_0^2 \tag{17}$$

where C0 = 0.0078 nW/K, C1 = -0.08 nW.s/K.m and C2 = 10.4 nW.s<sup>2</sup> /K.m<sup>2</sup> .

#### *4.3.2 Case of single phase and turbulent flow*

The same geometry already seen in the previous part will be used in this part. However, the flow will be considered turbulent. The main objective of this study is to simulate the entropy generation within the injector for high injection pressure taking into account the turbulent behavior of the flow. The mathematical model used consists of the Navier–Stokes equations coupled with the k-ε turbulence model taking into account the following assumptions:


Water with a density ρ = 1000 kg/m<sup>3</sup> and a dynamic viscosity μ = 10�<sup>3</sup> Pa.s is used as a fluid.

For the velocity field and the pressure, the same boundary conditions that will be used previously at the inlet and outlet of the injector. On the other hand, at the level of the walls we use a wall law.

For the k-ε turbulence model, the boundary conditions used are as follows:

• On the axis of symmetry (r = 0)

$$\frac{\partial k}{\partial r} = \mathbf{0} \,\text{et} \,\frac{\partial \varepsilon}{\partial r} = \mathbf{0} \tag{18}$$

*Analysis of Geometric Parameters of the Nozzle Orifice on Cavitating Flow and Entropy… DOI: http://dx.doi.org/10.5772/intechopen.99404*

• At the inlet

$$l = C\_{\mu} \frac{k^{3/2}}{\varepsilon} \approx 3.89 \% D\_h \tag{19}$$

$$I = \mathbf{0}.\mathbf{1}\mathbf{6}(Re)^{-1/8} \tag{20}$$

• At the outlet

$$
\overrightarrow{n} \, \overrightarrow{grad} \, \vec{k} = 0,\\
\overrightarrow{n} \, \overrightarrow{grad} \, \varepsilon = \mathbf{0} \tag{21}
$$

In order to assess the accuracy of the numerical method, we performed an automatic adaptive mesh refinement. The mesh adaptation is done automatically in areas with a significant gradient (pressure, velocity, turbulent kinetic energy, kinetic energy dissipation rate).

**Figures 12**–**15** illustrate the radial profiles of local entropy within the orifice for different axial positions. **Figures 12(a)**–**15(a)** show the radial entropy profiles which arise from dissipation due to the mean flow movement for various inlet velocity u0 ranging from 10 m/s to 60 m/s. The logarithmic representation of the average viscous dissipation is shown in **Figures 12**–**15**. For such a single-phase viscous flow, which is both laminar and turbulent, the viscous dissipation is a function of the velocity gradient. Near the wall vicinity, the velocity gradient has a maximum value. Axially, the local entropy is important at the entrance of the orifice where the velocity gradient is important. We also notice that laminar entropy increases considerably with the inlet volocity.

**Figures 12(b)**–**15(b)** show the irreversibility due to the Reynolds shear stress resulting from the velocity fluctuation. This irreversibility can also be interpreted as

#### **Figure 12.**

*Radial entropy profile at different position along the z-axis for inlet velocity v0 = 10 m/s. (a) Laminar entropie, (b) turbulent entropie et (c) total entropy.*

**Figure 13.**

*Radial entropy profile at different position along the z-axis for inlet velocity v0 = 30 m/s. (a) Laminar entropie, (b) turbulent entropie et (c) total entropy.*

the work done by force in the direction of the flow. From **Figures 12(a,b)-15(a,b)**, we can conclude that the entropy due to shear stress is dominant.

**Figures 12(c)**–**15(c)** illustrate the radial profiles of the total local entropy (laminar entropy and turbulent entropy) for different axial positions. As it is clear, the total entropy is important at the level of the constriction zone having a large velocity fluctuation.

**Figure 16** show the local entropy distribution inside the orifice for various inlet velocity 10, 30, 50 and 60 m/s, respectively.

It is clear that the entropy is maximum in the vicinity of the wall at the contraction zone having a large velocity gradient. The greater the injection velocity, the more the irreversibility zone spreads out.

**Figure 17** illustrates the total entropy as a function of the inlet injection velocity v0. Its clear that the total entropy production varies quadratically with the inlet velocity.

The interpolation in Lagrange polynomial of degree 2 of Stot as a function of inlet velocity *v*0:

*Analysis of Geometric Parameters of the Nozzle Orifice on Cavitating Flow and Entropy… DOI: http://dx.doi.org/10.5772/intechopen.99404*

**Figure 14.** *Radial entropy profile at different position along the z-axis for inlet velocity v0 = 50 m/s. (a) Laminar entropie, (b) turbulent entropie et (c) total entropy.*

$$\mathcal{S}\_{tot}(\mathcal{M}\mathcal{W}/\mathcal{K}) = \mathcal{C}\_0 + \mathcal{C}\_1 \times \mathcal{v}\_0 + \mathcal{C}\_2 \times \mathcal{v}\_0^2 \tag{22}$$

where C0 = 0.29 MW/K, C1 = -0.037 MW.s/K.m and C2 = 0.0012 MW.s<sup>2</sup> /K.m2 .

#### *4.3.3 Case of two-phase turbulent cavitating flow*

Our study is based on the same configuration seen in the previous sections to study the entropy generation. In fact, in this part we take into account the effect of cavitation within the orifice of the diesel injector. This will allow us to make an exhaustive study of the topology of the flow associated with the cavitation that has appeared in the injector. For this, we take into account the homogeneous mixture approach for the modeling of two-phase flows. The fluid used in this study is therefore water, the properties of which are illustrated in the following **Table 3**.

To analyze entropy generation within the injector, we have studied the effect of the injection pressure. In this case, simulations were carried out for inlet pressure varying between 1.9 bar and 1000 bar and for a fixed downstream pressure of 0.95 bar. **Figure 18** illustrates the vapor fraction and local entropy distributions inside the injector for various values of cavitation number K.

We notice from **Figure 18** (left) that the state of the fluid changes as it enters the orifice and as the injection pressure is increased. The change in the fluid state is due to the cavitation phenomenon that is created for a significant local depression. The 2D results for the vapor fraction show that cavitation is triggered when K ≈ 1.45 confirms the previous results.

**Figure 15.** *Radial entropy profile at different position along the z-axis for inlet velocity v0 = 60 m/s. (a) Laminar entropie, (b) turbulent entropie et (c) total entropy.*

In the center of the orifice, the fluid is formed by a dense core of fluid (liquid) and a dispersed phase (vapor), which is analogous to the experimental results of Yan and Thorpe [38]. For high injection pressures, the cavitation zone extends to the outlet leading to the formation of the hydraulic flip.

According to the results of **Figure 19** (right) show the local entropy distribution for different K values. It is clear that the entropy generation takes place near the orifice edge with a very high intensity. These results can be explained by the effect that near the wall, the radial component is important. Thus, the low pressure zone essentially gives rise to the formation of bubbles and the recirculation zone is very limited. The velocity gradient in the recirculation zone is very important causing viscous effects. This trend indicates that the maximum entropy produced near the edge of the orifice is mainly due to the irreversibility of fluid friction contributed by the velocity gradient due to the abrupt change of injector section.

For z ≥ D1, the flow is strongly developed. The turbulent velocity components promote the transfer of momentum between adjacent layers of the fluid and tend to reduce the average velocity gradient and subsequently a decrease in the degree of irreversibility.

**Figure 19** shows the evolution of total entropy as a function of the number of cavitation K. These results prove that the degree of irreversibility is proportional to the injection pressure. For high injection pressures, entropy production is important.

*Analysis of Geometric Parameters of the Nozzle Orifice on Cavitating Flow and Entropy… DOI: http://dx.doi.org/10.5772/intechopen.99404*

#### **Figure 16.**

*Total entropy distribution for various inlet velocities vo = 10, 30, 50 and 60 m/s. the results are presented in logarithmic decimal scale.*

#### **Figure 17.**

*Evolution of total entropy production as a function of inlet velocity.*


**Table 3.** *Water properties.*
