## 23.7.4 Cavitation Models

This section provides information about the cavitation model used in FLUENT. You can use FLUENT's current cavitation model to include cavitation effects in two-phase flows when the mixture model is used.

Overview of the Cavitation Model

A liquid at constant temperature can be subjected to a decreasing pressure, which may fall below the saturated vapor pressure. The process of rupturing the liquid by a decrease of pressure at constant temperature is called cavitation. The liquid also contains the micro-bubbles of noncondensable (dissolved or ingested) gases, or nuclei, which under decreasing pressure may grow and form cavities. In such processes, very large and steep density variations happen in the low-pressure/cavitating regions.

The cavitation model implemented here is based on the so-called "full cavitation model", developed by Singhal et al. [ 335]. It accounts for all first-order effects (i.e., phase change, bubble dynamics, turbulent pressure fluctuations, and noncondensable gases). However, unlike the original approach [ 335] assuming single-phase, isothermal, variable fluid density flows, the cavitation model in FLUENT is under the framework of multiphase flows. It has the capability to account for multiphase (N-phase) flows or flows with multiphase species transport, the effects of slip velocities between the liquid and gaseous phases, and the thermal effects and compressibility of both liquid and gas phases. The cavitation model can be used with the mixture multiphase model (with or without slip velocities).

The complete cavitation model capability in FLUENT can be presented in two parts:

• the basic cavitation model

This includes a description of the fundamental modeling approach and the standard two-phase cavitation model.

• the extended cavitation model capability

This includes a description of the extension of the cavitation model for multiphase (N-phase) flows, or flows with multiphase species transport applications.

Basic Cavitation Model

In the standard two-phase cavitation model, the following assumptions are made:

• The system under investigation involves only two phases (a liquid and its vapor), and a certain fraction of separately modeled noncondensable gases.

• Both bubble formation (evaporation) and collapse (condensation) are taken into account in the model.

• The mass fraction of noncondensable gases is known in advance.

The cavitation model offers the following capabilities:

• The cavitation model accounts for the mass transfer between a single liquid and its vapor.

• It is compatible with all the available turbulence models in FLUENT.

• It can be solved with the mixture energy equation.

• It is fully compatible with dynamic mesh and nonconformal interfaces.

• Both liquid and vapor phases can be incompressible or compressible. The noncondensible gases are assumed to always be compressible. For compressible liquids, the density can be described using a user-defined function. See the separate UDF Manual for more information on user-defined density functions.

• The parameters used in the mass transfer model for cavitation (vaporization pressure, liquid surface tension coefficient) can be either a constant or a function of temperature.

The following limitations apply to the cavitation model in FLUENT:

• The cavitation model cannot be used with the VOF model, because the surface tracking schemes for the VOF model are incompatible with the interpenetrating continua assumption of the cavitation model.

• The cavitation model can be used only for multiphase simulations that use the mixture model. It is always preferable to solve for cavitation using the mixture model without slip velocity; slip velocities can be turned on if the problem suggests that there is significant slip between phases.

• The cavitation model can only used for cavitating flow occurring in a single liquid fluid;

• With the cavitation model, the primary phase must be liquid, the secondary phase must be vapor.

Vapor Mass Fraction and Vapor Transport

The working fluid is assumed to be a mixture of liquid, vapor and noncondensable gases. Standard governing equations in the mixture model and the mixture turbulence model describe the flow and account for the effects of turbulence. A vapor transport equation governs the vapor mass fraction, , given by:

 (23.7-12)

where is the mixture density, is the velocity vector of the vapor phase, is the effective exchange coefficient, and and are the vapor generation and condensation rate terms (or phase change rates). The rate expressions are derived from the Rayleigh-Plesset equations, and limiting bubble size considerations (interface surface area per unit volume of vapor) [ 335]. These rates are functions of the instantaneous, local static pressure and are given by:

when

 (23.7-13)

when

 (23.7-14)

where the suffixes and denote the liquid and vapor phases, is a characteristic velocity, which is approximated by the local turbulence intensity, (i.e. ), is the surface tension coefficient of the liquid, is the liquid saturation vapor pressure at the given temperature, and and are empirical constants. The default values are and .

Turbulence-Induced Pressure Fluctuations

Significant effect of turbulence on cavitating flows has been reported [ 312]. FLUENT's cavitation model accounts for the turbulence-induced pressure fluctuations by simply raising the phase-change threshold pressure from to

 (23.7-15)

where

 (23.7-16)

where is the local turbulence kinetic energy.

Effects of Noncondensable Gases

The operating liquid usually contains small finite amounts of noncondensable gases (e.g., dissolved gases, aeration). Even a very small amount (e.g., 10 ppm) of noncondensable gases can have significant effects on the cavitating flow field due to expansion at low pressures (following the ideal gas law). In the present approach, the working fluid is assumed to be a mixture of the liquid phase and the gaseous phase, with the gaseous phase comprising of the liquid vapor and the noncondensable gases. The density of the mixture, , is calculated as

 (23.7-17)

where , , and are the densities of the liquid, the vapor, and the noncondensable gases, respectively, and , , and are the respective volume fractions. The relationship between the mass fraction ( ) in Equations  23.7-12- 23.7-14 and the volume fraction ( ) in Equation  23.7-17 is

 (23.7-18)

The combined volume fraction of vapor and gas (i.e., ) is commonly referred to as the void fraction ( ).

It may be noted that the noncondensable gas is not defined as a phase or a material. When using the ideal gas law to compute the noncondensable gas density, the molecular weight and temperature are required. By default, the gas is assumed to be air and the molecular weight is set to 29.0. However, if the noncondensable gas is not air, then the molecular weight can be changed by using a text command. For more information, contact your FLUENT support engineer.

As for the temperature, the default value is set to 300 K when the energy equation is not activated. If the temperature is different, but still a constant (i.e., isothermal flow), you can change the temperature in FLUENT in the following way:

• Activate the energy equation.

Define Models Energy...

• Open the Solution Initialization panel.

Solve Initialize Initialize...

• In the Solution Initialization, set the initial value as a desired temperature.

• Open the Solution Controls panel.

Solve Controls Solution...

• In the Solution Controls panel, under Equations, turn off the energy equation by deselecting Energy in the list.. By so doing, FLUENT uses the initial values for the temperature.

Phase Change Rates

After accounting for the effects of turbulence-induced pressure fluctuations and noncondensable gases, the final phase rate expressions are written as:

when

 (23.7-19)

when

 (23.7-20)

Additional Guidelines for the Cavitation Model

In practical applications of the cavitation model, several factors greatly influence numerical stability. For instance, high pressure difference between the inlet and exit, large ratio of liquid to vapor density, and near zero saturation pressure all cause unfavorable effects on solution convergence. In addition, poor initial conditions very often lead to an unrealistic pressure field and unexpected cavitating zones, which, once present, are then usually very difficult for the model to correct. The following is a list of factors that must be considered when using the cavitation model, along with tips to help address potential numerical problems:

• relaxation factors

In general, small relaxation factors are advised for momentum equations, usually, between 0.05 - 0.4; The relaxation factor for the pressure-correction equation should usually be larger than those for momentum equations, say in the range 0.2 - 0.7. The density and the vaporization mass (source term in the vapor equation) can also be relaxed to improve convergence, Typically, the relaxation factor for density is set between the values of 0.3 and 1.0, while for the vaporization mass values between 0.1 and 1.0 may be appropriate. For some extreme cases, even smaller relaxation factors may be required for all the equations.

• initial conditions

Though no special initial condition settings are required, it is suggested that the vapor fraction is always set to inlet values. The pressure is set close to the highest pressure among the inlets and outlets to avoid unexpected low pressure and cavitating spots. Also, in complicated cases, it may be beneficial to obtain a realistic pressure field before substantial cavities are formed. This can be achieved by performing the following steps:

1.   Set near zero relaxation factors for the vaporization mass and for density, and increase them to reasonable values after a sufficient number of iterations.

2.   Obtain a converged / near-converged solution for a single phase liquid flow, and then switch on the cavitation model.

• noncondensable gases

Noncondensable gases are usually present in liquids. Even a small amount (e.g., 15 ppm) of noncondensable gases can have significant effects on both the physical realism and the convergence characteristics of the solution. A value of zero for the mass fraction of noncondensable gases should generally be avoided. In some cases, if the liquid is purified of noncondensable gases, a much smaller value (e.g., 10 ) may be used to replace the default value of 1.5 . In fact, higher mass fractions of the noncondensable gases may, in many cases, enhance numerical stability and lead to more realistic results. In particular, when the saturation pressure of a liquid at a certain temperature is zero or very small, noncondensable gases will play a crucial role both numerically and physically.

• limits for dependent variables

In many cases, setting the pressure upper limit to a reasonable value can help convergence greatly at the early stage of the solution. It is advised to always limit the maximum pressure when it is possible. By default, FLUENT sets the maximum pressure limit to 5.0 Pascal.

• the relaxation factor for the pressure correction equation

For cavitating flows, a special relaxation factor is introduced for the pressure correction equation. By default, this factor is set to 0.7, which should work well for most of the cases. For some very complicated cases, however, you may experience the divergence of the AMG solver. Under those circumstances, this value may be reduced to no less than 0.4. You can set the value of this relaxation factor by typing a text command. For more information, contact your FLUENT support engineer.

• pressure discretization schemes

As for many multiphase flows, it is more desirable to use the following pressure discretization schemes in cavitation applications:

• body force weighted

• second order

• PRESTO!

The standard and linear schemes generally are not very effective in complex cavitating flows.

Extended Cavitation Model Capability

In many practical applications, when cavitation occurs, there exist other gaseous species in the systems investigated. For instance, in a ventilated supercavitating vehicle, air is injected into a liquid to stabilize or increase the cavitation along the vehicle surfaces. Also in some cases, the incoming flow is a mixture of a liquid and some gaseous species. In order to be able to predict those type of cavitating flows, the basic two-phase cavitation model needs to be extended to a multiphase (N-phase) flows, or a multiphase species transport cavitation model.

Multiphase Cavitation Model

The multiphase cavitation model is an extension of the basic two-phase cavitation model to multiphase flows. In addition to the primary liquid and secondary vapor phase, more secondary gaseous phases can be included into the computational system under the following assumptions/limitations:

• Mass transfer (cavitation) only occurs between the first and the second phases.

• The basic cavitation model is still used to model the phase changes between the liquid and vapor.

• Only one secondary phase can be defined as compressible gas phase, while a user-defined density may be applied to all the phases.

• The predescribed noncondensable gases can still be included in the system. To exclude noncondensable gases from the system, the mass fraction needs to be set to 0, and the noncondensable gas needs to be modeled by a separate compressible gas phase.

• For an noncavitating phase , the general transport equation governing the mass fraction given by:

 (23.7-21)

where is a (user-defined) source term. By default, .

Multiphase Species Transport Cavitation Model

In some cases, there are several gas phase components in a system. It is desirable to consider them all compressible. Since only one compressible gas phase is allowed in the general multiphase approach, the multiphase species transport approach offers an option to handle these type of applications by assuming that there is one compressible gas phase with multiple species.

The detailed description of the multiphase species transport approach can be found in Section  23.8. The multiphase species transport cavitation model can be summarized as follows:

• All the assumptions/limitations for the multiphase cavitation model also apply here.

• The primary phase can only be a single liquid.

• All the secondary phases allow more than one species.

• The vapor, either as a phase or a species, must be in the second phase.

• The mass transfer between a liquid and a vapor phase/species is modeled by the basic cavitation model.

• The mass transfer between other phases or species are modeled with the standard mass transfer approach. In the standard model, the zero constant rate should be chosen.

• For the phases with multiple species, the phase shares the same pressure as the other phases, but each species has its own pressure (i.e., partial pressure). As a result, the vapor density and the pressure used in Equation  23.7-19 are the partial density and pressure of the vapor.

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Up: 23.7 Modeling Mass Transfer
Next: 23.8 Modeling Species Transport