**Nomenclature**

178 Mass Transfer - Advanced Aspects

b. Steady state cavitating behaviour of the studied inducers.

cavitating behaviour for these inducers.

a. Cavitating flow through venturi geometry.

a. Steady state performances: pressure head coefficient and efficiency versus flow

a. Steady and unsteady states performances: pressure head coefficient and efficiency

c. Vapour distributions and other numerical results, which enable to explain the

The numerical simulations were performed using the commercial code *Fluent*, which is based on a cell-centred finite-volume method. The cavitation model used for the calculations assumes a thermal equilibrium between the phases. It is based on the classical conservation equations of the vapour phase and a mixture phase, with mass transfer due to the cavitation, which appears as a source term and a sink term in the vapour mass fraction equation. The mass transfer rate is derived from a simplified Rayleigh–Plesset model for bubble dynamics. The experimental tests were carried out at the DynFluid Laboratory of Arts et Métiers

Next, the different cases studied in this work, to understand the cavitating behaviour of the

the intrinsic instabilities were detected and compared to experimental data. b. Analysis of cavitating flow in two-dimensional inducers – blades cascades.

cavitation, alternate blade cavitation and rotating blade cavitation.

regime, and the Table 1 presents the main characteristics of these inducers.

This part presents the numerical validation of the cavitating flow through a 2D simple geometry. On the one hand, the numerical results demonstrate the influence of the turbulence model to predict instabilities in non-homogeneous flows. On the other hand,

This part presents the steady and unsteady numerical study carried out on two blades cascades: first, on a two-blade aircraft inducer, and then, on a three-blade industrial inducer, see Fig. 1. This analysis confirms the influence of blades number and the solidity on the behaviour of the instabilities of cavitating flow. Various forms and behaviours of vapour have been observed in the blades cascades, such as: stable blade

c. Experimental study and numerical analysis of the cavitating flow in three-dimensional

The commercial code used for all simulations was *Fluent*. This code employs a cell–centred finite–volume method that allows the use of computational elements with arbitrary

This part presents the steady and unsteady numerical simulations, and experimental investigations in steady state of the cavitating flow through of the three inducers presented in the Fig. 1. The flow behaviour in the inducers is modified by the appearance of the cavitation on the leading edge. These cavitating behaviours change with respect to the operating conditions of the inducer: flow rates and cavitation levels. Fig. 1 shows the three inducers used for the numerical and experimental study in cavitating

b. Steady and unsteady states cavitating behaviour of the studied inducers.

d. The fluid flow instabilities generated by the presence of vapour.

1. Experimental results concerning:

2. Numerical results concerning:

versus flow rates.

rates.

ParisTech.

inducers, are described:

inducers.

**2. Numerical method** 

*D* diameter **Greek Subscript** 


	-
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	- Φflow coefficient *cav* cavitation
	- Ψ
	-

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$$\frac{1}{\rho} = \frac{\gamma\_v}{\rho\_v} + \frac{\gamma\_g}{\rho\_g} + \frac{1 - \gamma\_v - \gamma\_g}{\rho\_l} \tag{1}$$

$$\gamma\_v = \frac{a\_v \rho\_v}{\rho}, \quad \gamma\_g = \frac{a\_g \rho\_g}{\rho}, \quad \text{and} \quad \gamma\_l = \frac{a\_l \rho\_l}{\rho} = 1 - \gamma\_v - \gamma\_g \tag{2}$$

$$\frac{\partial}{\partial t}(\rho \gamma) + \nabla \cdot (\rho \nu \gamma) = \nabla \cdot (\Gamma \nabla \gamma) + R\_e - R\_c \tag{3}$$

$$R\_B \frac{D^2 R\_B}{Dt^2} + \frac{2}{3} \left(\frac{D R\_B}{Dt}\right)^2 = \frac{P\_B - P}{\rho\_l} - \frac{4\nu\_l}{R\_B} \dot{R\_B} - \frac{2\sigma\_s}{\rho\_l R\_B} \tag{4}$$

$$R = (\hbar 4\pi)^{1/3} \frac{\rho\_v \rho\_l}{\rho} \left[ \frac{2}{3} \left( \frac{P\_B - P}{\rho\_l} \right) - \frac{2}{3} R\_B \frac{D^2 R\_B}{Dt^2} \right]^{1/2} \tag{5}$$

$$\frac{\partial}{\partial t}(\rho \gamma) + \nabla \cdot (\rho \gamma \vec{v}) = (n4\pi)^{1/3} (3a)^{2/3} \frac{\rho\_v \rho\_l}{\rho} \left[ \frac{2}{3} \left( \frac{P\_B - P}{\rho\_l} \right) \right]^{1/2} \tag{6}$$

$$
\alpha\_v = n \cdot (4/3) \cdot \pi \cdot R\_B^3 \tag{7}
$$

$$P\_v = P\_{sat} + 0.195 \cdot \rho \cdot \kappa \tag{8}$$

$$R\_e = C\_e \frac{\sqrt{\kappa}}{\sigma\_s} \rho\_v \rho\_l \left[\frac{2}{3} \frac{P\_v - P}{\rho\_l}\right]^{1/2} \left\{1 - \chi\_v - \chi\_g\right\} \tag{9}$$

$$R\_c = C\_c \frac{\sqrt{\kappa}}{\sigma\_\mathcal{S}} \rho\_l \rho\_l \left[\frac{2}{3} \frac{P - P\_v}{\rho}\right]^{1/2} \chi\_v \tag{10}$$

$$\frac{\partial}{\partial t}(\rho \kappa) + \frac{\partial}{\partial \mathbf{x}\_l}(\rho \kappa u\_l) = \frac{\partial}{\partial \mathbf{x}\_l} \left[ (\mu + \frac{\mu\_\mathbf{f}}{\sigma\_\mathbf{x}}) \frac{\partial \kappa}{\partial \mathbf{x}\_l} \right] + P\_\mathbf{k} - \rho \varepsilon \tag{11}$$

$$\frac{\partial}{\partial t} \{\rho \varepsilon\} + \frac{\partial}{\partial \mathbf{x}\_l} \{\rho \varepsilon u\_l\} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + C\_{1\varepsilon}^\* \frac{\varepsilon}{\kappa} P\_\mathbf{k} - C\_{2\varepsilon} \rho \frac{\varepsilon^2}{\kappa} \tag{12}$$

$$\mathcal{C}\_{1\varepsilon}^{\*} = \mathcal{C}\_{1\varepsilon} - \frac{\mathcal{C}\_{\kappa}\eta^{3}(1 - \eta/\eta\_{0})}{1 + \beta\eta^{3}}, \quad \eta = \frac{\mathcal{S}\kappa}{\varepsilon}, \quad \text{and} \quad \mathcal{S} = \left(2\mathcal{S}\_{lj}\mathcal{S}\_{lj}\right)^{1/2}$$

$$P\_{\kappa} = -\rho \overline{u\_l'} \overline{u\_l'} \frac{\partial u\_l}{\partial u\_l} \tag{13}$$

$$
\mu\_t = \rho \mathcal{C}\_\mu \frac{\kappa^2}{\varepsilon} \tag{14}
$$

$$
\mu\_t = f(\rho) \mathcal{C}\_\mu \frac{\kappa^2}{\varepsilon} \tag{15}
$$

$$f(\rho) = \rho\_v + \left(\frac{\rho\_v - \rho}{\rho\_v - \rho\_l}\right)^n (\rho\_l - \rho\_v) \quad where \quad n > 1\tag{16}$$

Numerical and Experimental Study of Mass Transfer Through Cavitation in Turbomachinery 185

Fig. 4. Contours of vapour volume fraction (*α*≥10%). Vapour detachment cycle calculated for

three cavitation numbers. Calculations using RNG κ–ε modified model, *tref=65.8E-3 s*
