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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:



Basic Cavitation Model


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

The cavitation model offers the following capabilities:

The following limitations apply to the cavitation model in FLUENT:

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, $f$, given by:


 \frac{\partial}{\partial t} (\rho f) + \nabla (\rho \vec{v_v} f) = \nabla (\gamma \nabla f) + R_e - R_c (23.7-12)

where $\rho$ is the mixture density, $\vec{v_v}$ is the velocity vector of the vapor phase, $\gamma$ is the effective exchange coefficient, and $R_e$ and $R_c$ 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 $p < p_{sat}$


 R_e = C_e \frac{V_{ch}}{\sigma} \rho_l \rho_v \sqrt{\frac{2(p_{\rm sat}-p)}{3\rho_l}}(1-f) (23.7-13)

when $p > p_{\rm sat}$


 R_c = C_c \frac{V_{\rm ch}}{\sigma} \rho_l \rho_v \sqrt{\frac{2(p-p_{\rm sat})}{3\rho_l}}f (23.7-14)

where the suffixes $l$ and $v$ denote the liquid and vapor phases, $V_{\rm ch}$ is a characteristic velocity, which is approximated by the local turbulence intensity, (i.e. $V_{\rm ch} = \sqrt{k}$), $\sigma$ is the surface tension coefficient of the liquid, $p_{\rm sat}$ is the liquid saturation vapor pressure at the given temperature, and $C_e$ and $C_c$ are empirical constants. The default values are $C_e = 0.02$ and $C_c = 0.01$.

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 $p_{\rm sat}$ to


 p_v = \frac{1}{2} (p_{\rm sat} + p_{\rm turb}) (23.7-15)

where


 p_{\rm turb} = 0.39 \rho k (23.7-16)

where $k$ 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, $\rho$, is calculated as


 \rho = \alpha_v \rho_v + \alpha_g \rho_g + (1 - \alpha_v - \alpha_g) \rho_l (23.7-17)

where $\rho_l$, $\rho_v$, and $\rho_g$ are the densities of the liquid, the vapor, and the noncondensable gases, respectively, and $\alpha_l$, $\alpha_v$, and $\alpha_g$ are the respective volume fractions. The relationship between the mass fraction ( $f_i$) in Equations  23.7-12- 23.7-14 and the volume fraction ( $\alpha_i$) in Equation  23.7-17 is


 \alpha_i = f_i \frac{\rho}{\rho_i} (23.7-18)

The combined volume fraction of vapor and gas (i.e., $\alpha_v + \alpha_g$) is commonly referred to as the void fraction ( $\alpha$).

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:

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 $p < p_v$


 R_e = C_e \frac{\sqrt{k}}{\sigma} \rho_l \rho_v \sqrt{\frac{2(p_v-p)}{3\rho_l}}(1-f_v-f_g) (23.7-19)

when $p > p_v$


 R_c = C_c \frac{\sqrt{k}}{\sigma} \rho_l \rho_l \sqrt{\frac{2(p-p_v)}{3\rho_l}}f_v (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:



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:

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:


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