### 5.13.5. Cavitation Model

The tendency for a flow to cavitate is characterized by the cavitation number, defined as:

| (5–259)

where is a reference pressure for the flow (for example, inlet pressure), is the vapor pressure for the liquid, and the denominator represents the dynamic pressure. Clearly, the tendency for a flow to cavitate increases as the cavitation number is decreased.

Cavitation is treated separately from thermal phase change, as the cavitation process is typically too rapid for the assumption of thermal equilibrium at the interface to be correct. In the simplest cavitation models, mass transfer is driven by purely mechanical effects, namely liquid-vapor pressure differences, rather than thermal effects. Current research is directed towards models that take both effects into account.

In CFX, the Rayleigh Plesset model is implemented in the multiphase framework as an interphase mass transfer model. User-defined models can also be implemented.

For cavitating flow, the homogeneous multiphase model is typically used.

#### 5.13.5.1. The Rayleigh Plesset Model

The Rayleigh-Plesset equation provides the basis for the rate equation controlling vapor generation and condensation. The Rayleigh-Plesset equation describing the growth of a gas bubble in a liquid is given by:

| (5–260)

where represents the bubble radius, is the pressure in the bubble (assumed to be the vapor pressure at the liquid temperature), is the pressure in the liquid surrounding the bubble, is the liquid density, and is the surface tension coefficient between the liquid and vapor. Note that this is derived from a mechanical balance, assuming no thermal barriers to bubble growth. Neglecting the second order terms (which is appropriate for low oscillation frequencies) and the surface tension, this equation reduces to:

| (5–261)

The rate of change of bubble volume follows as:

| (5–262)

and the rate of change of bubble mass is:

| (5–263)

If there are bubbles per unit volume, the volume fraction may be expressed as:

| (5–264)

and the total interphase mass transfer rate per unit volume is:

| (5–265)

This expression has been derived assuming bubble growth (vaporization). It can be generalized to include condensation as follows:

| (5–266)

where is an empirical factor that may differ for condensation and vaporization, designed to account for the fact that they may occur at different rates (condensation is usually much slower than vaporization). For modeling purposes the bubble radius will be replaced by the nucleation site radius .

Despite the fact that Equation 5–266 has been generalized for vaporization and condensation, it requires further modification in the case of vaporization.

Vaporization is initiated at nucleation sites (most commonly non-condensable gases). As the vapor volume fraction increases, the nucleation site density must decrease accordingly, because there is less liquid. For vaporization, in Equation 5–266 is replaced by to give:

| (5–267)

where is the volume fraction of the nucleation sites. Equation 5–266 is maintained in the case of condensation.

To obtain an interphase mass transfer rate, further assumptions regarding the bubble concentration and radius are required. The Rayleigh-Plesset cavitation model implemented in CFX uses the following defaults for the model parameters:

For an illustration and a validation of the Rayleigh-Plesset cavitation model see Bakir et al. [148]

#### 5.13.5.2. User Defined Cavitation Models

Additional information on creating a user-defined model is available. For details, see User Defined Cavitation Models in the CFX-Solver Modeling Guide.

When using a user-defined cavitation model, the CFX-Solver will perform generic linearizations for the volume fraction and volume continuity equations to help stability and convergence. The saturation pressure is used by the CFX-Solver in linearizing the cavitation rate against pressure.

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