**2.3 Cavitation model**

According to [17, 28], the differential equation describing the transport of the vapor fraction is given by

$$\frac{\partial}{\partial t} \left( \rho\_m \boldsymbol{\xi}\_v \right) + \nabla \cdot \left( \rho\_m \boldsymbol{\xi}\_v \vec{\boldsymbol{U}}\_v \right) = \mathbf{R}\_\mathbf{e} - \mathbf{R}\_\mathbf{e} \tag{8}$$

Here *Re* and *Rc* denote the evaporation and condensation rates, respectively. The rates *Re* and *Rc* depend on the static pressure and the velocity as well as on the fluid properties. They depend also on the pressure fluctuations generated by the turbulence as well as on the turbulent kinetic energy *k*.

The following relations are usually adopted to express the rates *Re* and *Rc* [29].

$$R\_{\epsilon} = C\_{\epsilon} \frac{\sqrt{k}}{\sigma} \rho\_{\epsilon} \rho\_{v} \left(\frac{2}{3} \frac{p\_{v} - p}{\rho\_{l}}\right)^{1/2} \left(1 - \xi\_{v} - \xi\_{\mathfrak{g}}\right) \qquad \qquad p \le p\_{v} \tag{9}$$

$$R\_c = \mathbf{C}\_c \frac{\sqrt{k}}{\sigma} \rho\_\ell \rho\_v \left(\frac{2}{3} \frac{p - p\_v}{\rho\_l}\right)^{1/2} f\_v \tag{10}$$

where *Ce* = 0.02 and *Cc* = 0.01 are calibration constants, *fv* and *fg* are vapor mass fraction and non-condensable gas mass fraction. The phase-change threshold pressure *pv* is estimated from the following equation [17]:

$$p\_v = p\_{sat} + \frac{p\_t}{2} \tag{11}$$

where *psat* present the vapor saturation pressure and *pt = 0.39ρmk* is the turbulence pressure [17]. The above relations show that the evaporation and condensation happen at the vapor pressure *pv* and not at the saturation pressure *psat* as in laminar flows. The diffusion phenomenon between phases is neglected.

#### **2.4 Entropy production**

Taking into account the following assumptions:


The entropy production rate for the mixture is then written [30]:

$$\bar{\dot{P}}\_{sm} = \frac{1}{T} \left( \Phi\_{\mu m} + \Phi\_{tm} + \Phi\_{Dm} \right) \tag{12}$$

where Φ*<sup>μ</sup><sup>m</sup>* is the average entropy production for the mixture, Φ*tm* is the entropy generation due to turbulent shear stress and Φ*Dm* is the entropy production due to the turbulent dissipation term.

The mean of the entropy production and the entropy production due to the turbulent shear stress can be written as follows [30]:

$$\boldsymbol{\Phi}\_{\rm effm} = \boldsymbol{\Phi}\_{\mu m} + \boldsymbol{\Phi}\_{tm} = \mu\_{\rm effm} \left[ \left( \nabla . \left\langle \boldsymbol{\bar{U}}\_{m} \right\rangle + \left( \nabla . \left\langle \boldsymbol{\bar{U}}\_{m} \right\rangle \right)^{\rm T} \right) : \nabla . \left\langle \boldsymbol{\bar{U}}\_{m} \right\rangle \right] \tag{13}$$

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

where *μeffm* ¼ *μ<sup>m</sup>* þ *μtm* � � is the effective viscosity of the mixture.

In a two-dimensional flow in an injection orifice, the entropy production in cylindrical coordinates (2D) is then written [30]:

$$\bar{P}\_{sm} = \frac{\mu\_{\text{eff}m}}{T} \left[ 2 \left( \left( \frac{\partial u\_m}{\partial r} \right)^2 + \left( \frac{u\_m}{r} \right)^2 + \left( \frac{\partial v\_m}{\partial z} \right)^2 \right) + \left( \frac{\partial u\_m}{\partial z} + \frac{\partial v\_m}{\partial r} \right)^2 \right] + \frac{\rho\_m \bar{c}\_m}{T} \tag{14}$$

## **3. Numerical method and nozzle geometry**

To simulate the cavitating flow, the numerical code Fluent was used. This code is based on implicit finite volume scheme. The SIMPLE algorithm [31] is used for the pressure–velocity coupling. Grid generation process for performing finite volume simulations were carried out using GAMBIT (v2.3.16) program available with the commercial code Fluent. A first order implicit temporal discretization and a first order upwind differentiating scheme have been used. All under-relaxation factors range between to 0.2–0.4.

#### **3.1 Nozzle geometry and boundary conditions.**

**Figure 1** illustrates the nozzle geometry of diesel injector in 2D axisymmetric. The geometric parameters of the nozzle are R1 = 0.3 mm, R2 = 0.1 mm and L1 = 0.5 mm. The transition radius between inlet pipe and orifice is rc.

In this study, a stationary single phase fluid is assumed as initial conditions. Uniform inlet and outlet static pressure were adopted as boundary conditions. A value of *k0* and ε<sup>0</sup> are imposed at the inlet.

#### **3.2 Effect of the grid resolution**

the mesh presents an essential parameter in fluid mechanics problems for the convergence of the solution. The existence of an important gradient of the physical (pressure, velocity and vapor fraction) requires a concrete study of the sensitivity of the mesh on the solution. The mesh sensitivity on the solution was studied under the following injection conditions: inlet pressure *pin* = 5 bar and outlet pressure *pout* = 1 bar.

The mesh grid is adapted several times (six times: see **Table 1**). In **Table 1**, n and m represents the number of meshes according to z and r axis in the orifice.

**Figure 1.** *2D-axisymmetric configuration of the diesel injector with boundary conditions.*

#### *Applications of Computational Fluid Dynamics Simulation and Modeling*


**Table 1.**

*Meshes tested in the simulation.*

**Figure 2.**

*Pressure profile near the wall using different mesh and b-mass flow rate and discharge coefficient for several grid resolutions.*

The mesh effect on the local field (e.g the pressure field) and on the global coefficient (e.g discharge coefficient) along the wall of the nozzle is presented in **Figure 2(a,b)**. **Figure 2(a)**, indicates the pressure field distribution along the near wall at the orifice for several grids resolutions. It is clear that the local minimum pressure is strongly affected by the mesh size. The local pressure near the wall decrease with mesh size until it reaches a minimum value lower than the vapor saturation pressure.

In this region, the cavitation phenomenon appears. According to Raleigh-Plesset Equation [32], cavitation is governed by the local pressure. The effect of mesh on the masse flow rate and discharge coefficient is presented in **Figure 2b** (**i** & **ii**). We notice a variation in the discharge coefficient *Cd* with the mesh. For example, *Cd* undergoes a variation of 1.6% going from 1 to 4 and then remains constant. Mesh N° 3 can be used for simulation.

### **4. Results and discussions**

#### **4.1 Steady flow**

In this part, we study the effect of geometric parameters such as L2/d2, rc/d2 ratios and Reynolds number Re on the cavitation phenomenon in steady state.

#### *4.1.1 Influence of the Nozzle length*

The pressure drop in the injector is described by the discharge coefficient *Cd* that is the ratio of the effective mass flow rate to the theoretical maximum flow rate:

$$\mathbf{C}\_d = \frac{\dot{m}\_{\text{eff}}}{\dot{m}\_{ideal}} \tag{15}$$

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

where *<sup>m</sup>*\_ *eff* <sup>¼</sup> ÐÐ *<sup>ρ</sup><sup>m</sup> <sup>v</sup>* ! *<sup>m</sup>: dA*�! is the effective mass flow rate and *<sup>m</sup>*\_ *ideal* <sup>¼</sup> *πR*<sup>2</sup> 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*ρ<sup>l</sup> pin* � *pout* <sup>q</sup> � � is the ideal flow rate.

**Figure 3** shows the distributions of vapor volume fraction for several nozzle lengths (*pin* = 10 bar and *pout* = 1 bar). The latter (nozzle lengths) has a significant effect on the cavitation area. The cavitation region decreases when the length of nozzle increases.

We perceive a transition on the cavitation nature from fully developed cavitation (for L2 = d2 or 2d2) to incipient cavitation (for L2 > 2d2).

For a length L2 ≤ 0.6 mm, cavitation still takes place and occupies a large part of the orifice. Cavitation is highly developed in this situation. However for L2 ≥ 0.6 mm, the cavitation remains very confined. It is useful to also examine the effect of length at high injection pressures and taking into account the presence of the combustion chamber.

**Figure 4**, shows the discharge coefficient *Cd* as a function of L2/d2 ratio for a nozzle with rc/d2 = 0. it is clear that the discharge coefficient varies linearly with the L2/d2 ratio. When the L2/d2 ratio increases from 2 to 8, we notice a reduction of 13%

**Figure 3.** *Contour of vapor volume fraction for several nozzle lengths.*

**Figure 4.** *Discharge coefficient Cd as a function of L2/d2 ratio.*

in *Cd*. The discharge coefficient is weakly dependent on the length of the orifice which can be explained by the dominance singular pressure losses generated at the entrance over regular pressure drops provoked by wall friction. These results have been shown experimentally [33] and numerically [34].

For the low values of Re, we must take into account the linear and singular pressure losses.
