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.
This section provides information about the following cavitation models used in Ansys Fluent.
In addition, Ansys Fluent provides an expert cavitation model by Singhal et al. [600], which has been available since release 6.1. However, due to convergence challenges, this model is not currently recommended. See Expert Cavitation Model for the description of this model.
The following assumptions are made in the standard two-phase cavitation models:
The system under investigation must consist of a liquid and a vapor phase.
Mass transfer between the liquid and vapor phase is assumed to take place. Both bubble formation (evaporation) and collapse (condensation) are taken into account in the cavitation models.
For the cavitation model, Ansys Fluent defines positive mass transfer as being from the liquid to the vapor.
The cavitation models are based on the Rayleigh-Plesset equation, describing the growth of a single vapor bubble in a liquid.
The input material properties used in the cavitation models can be constants, functions of temperature or user-defined.
The following limitations apply to the cavitation models in Ansys Fluent:
The Zwart-Gerber-Belamri and Schnerr and Sauer models do not take the effect of noncondensable gases into account by default.
Explicit VOF with cavitation is not recommended for the following reasons:
Explicit VOF is solved once per time step, therefore the mass transfer rate (which is a function of solution variables such as pressure, temperature, and volume fraction) is only evaluated once per time step. This may not give results that are consistent with those achieved using the implicit formulation. It is recommended that the volume fraction be solved every iteration, which can be enabled through the expert options in the Multiphase Model dialog box.
Explicit VOF may not be stable with sharpening schemes such as Geo-Reconstruct and CICSAM. You should use either a Sharp/Dispersed or Dispersed interface modeling option, which provide diffusive interface-capturing discretization schemes for volume fraction, such as QUICK, HRIC, and Compressive.
Explicit VOF does not model any cavitation-specific numerical treatments for added stability.
When using the implicit VOF formulation, numerical diffusion caused by turbulent effects is added by default. This added diffusion increases the solution stability, but has adverse effects on the interfacial accuracy. If interfacial sharpness is an area of concern, you can disable the diffusion as described in Cavitation Mechanism in the Fluent User's Guide.
With the multiphase cavitation modeling approach, a basic two-phase
cavitation model consists of using the standard viscous flow equations
governing the transport of mixture (Mixture model) or phases (Eulerian
multiphase), and a conventional turbulence model (k- 
model). In cavitation,
the liquid-vapor mass transfer (evaporation and condensation) is governed
by the vapor transport equation:
 | (14–580) |
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In Equation 14–580, the terms 
and 
account for
the mass transfer between the liquid and vapor phases in cavitation.
In Ansys Fluent, they are modeled based on the Rayleigh-Plesset equation
describing the growth of a single vapor bubble in a liquid.
In most engineering situations we assume that there are plenty of nuclei for the inception of cavitation. Therefore, our primary focus is on proper accounting of bubble growth and collapse. In a flowing liquid with zero velocity slip between the fluid and bubbles, the bubble dynamics equation can be derived from the generalized Rayleigh-Plesset equation as [78]
 | (14–581) |
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 = liquid kinematic viscosity | |
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Neglecting the second-order terms and the surface tension force, Equation 14–581 is simplified to
 | (14–582) |
This equation provides a physical approach to introduce the effects of bubble dynamics into the cavitation model. It can also be considered to be an equation for void propagation and, hence, mixture density.
To derive an expression of the net phase change rate, Schnerr and Sauer [585] use the following two-phase continuity equations:
Liquid phase:
 | (14–583) |
Vapor phase:
 | (14–584) |
Mixture:
 | (14–585) |
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Mixture density 
is defined as
 | (14–586) |
Combining Equation 14–583, Equation 14–584, and Equation 14–585 yields a relationship between the mixture density and
vapor volume fraction (
):
 | (14–587) |
The vapor volume fraction (
) can be related to the bubble number density (
) and the radius of bubble (
) as
 | (14–588) |
Using the Equation 14–582, and combining Equation 14–583, Equation 14–584, Equation 14–587, and Equation 14–588, the expression for the net phase change rate
(
) is finally obtained as
 | (14–589) |
 | (14–590) |
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Comparing Equation 14–589 with Equation 14–602 and
Equation 14–595, it is obvious that unlike the two previous
models, the mass transfer rate in the Schnerr and Sauer model is proportional to
. Moreover, the function 
has the interesting property that it approaches zero when
and 
, and reaches the maximum in between. Also in this model, the only
parameter that must be determined is the number of spherical bubbles per volume of
liquid. If you assume that no bubbles are created or destroyed, the bubble number
density would be constant. The initial conditions for the nucleation site volume
fraction and the equilibrium bubble radius would therefore be sufficient to specify
the bubble number density (
) from Equation 14–588 and then the phase transition
by Equation 14–589.
The final form of the model is as follows:
When 
,
 | (14–591) |
When 
,
 | (14–592) |
where 
and 
are the empirical calibration coefficients of evaporation and
condensation with the default value of 1 and 0.2, respectively. You can change these
coefficients using the expert text command option as described in Mass Transfer Mechanisms in the Fluent User's Guide.
Assuming that all the bubbles in a system have the same size, Zwart-Gerber-Belamri [741] proposed that the total interphase mass transfer rate per unit volume (
) is calculated using the bubble number densities (
), and the mass change rate of a single bubble:
 | (14–593) |
The vapor volume fraction 
is related to the bubble number density and the radius of bubble
as:
 | (14–594) |
Substituting the value of 
in Equation 14–593 into Equation 14–594, we have the expression of the net mass transfer:
 | (14–595) |
Comparing Equation 14–595 and Equation 14–602, you will notice that the difference is only
in the density terms in the mass transfer rate. In Equation 14–595, the unit volume mass transfer rate is only
related to the vapor phase density (
). Unlike Equation 14–602, 
has no relation with the
liquid phase and mixture densities in this model.
As in Equation 14–602, Equation 14–595 is derived assuming bubble growth (evaporation). To apply it to the bubble collapse process (condensation), the following generalized formulation is used:
 | (14–596) |
where 
is an empirical calibration coefficient. Though
it is originally derived from evaporation, Equation 14–596 only works well for condensation. It is physically incorrect and
numerically unstable if applied to evaporation. The fundamental reason
is that one of the key assumptions is that the cavitation bubble does
not interact with each other. This is plausible only during the earliest
stage of cavitation when the cavitation bubble grows from the nucleation
site. As the vapor volume fraction increases, the nucleation site
density must decrease accordingly. To model this process, Zwart-Gerber-Belamri
proposed to replace 
with 
in Equation 14–596. Then the final
form of this cavitation model is as follows:
If 
 | (14–597) |
If 
 | (14–598) |
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The Singhal et al. model is an expert model intended for specific applications. It can be used to account for the effect of noncondensable gases. The mass fraction of the noncondensable gases is assumed to be a known constant.
The theoretical background for this model and its limitations are provided below.
The following limitations exist with the Singhal et al. model:
The Singhal et al. model can be used only for a single cavitation process. That is, only a single liquid fluid can be subjected to cavitation when mass transfer is defined through the Singhal et al. model. This liquid can be a phase or a species in a mixture phase.
The Singhal et al. model requires the primary phase to be a liquid and the secondary phase to be a vapor. This model is only compatible with the multiphase mixture model.
The Singhal et al. model cannot be used with the Eulerian multiphase model.
The Singhal et al. model is not compatible with the LES turbulence model.
This cavitation model is based on the "full cavitation model", developed by Singhal et al. [600]. It accounts for all first-order effects (that is, phase change, bubble dynamics, turbulent pressure fluctuations, and noncondensable gases). 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. However, 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 vapor volume fraction 
is related to the bubble number density and the radius of
bubble as:
 | (14–599) |
The relationship between the mixture density and vapor volume fraction is given by:
 | (14–600) |
The expression for the net phase change rate (
) is calculated by:
 | (14–601) |
Here 
represents the vapor generation or evaporation rate, that is,
the source term 
in Equation 14–580. All terms, except
, are either known constants or dependent variables. In the
absence of a general model for estimation of the bubble number density, the
phase change rate expression is rewritten in terms of bubble radius
(
), as follows:
 | (14–602) |
Equation 14–602 indicates that the unit volume mass transfer rate
is not only related to the vapor density (
), but the function of the liquid density (
), and the mixture density (
) as well. Since Equation 14–602 is derived
directly from phase volume fraction equations, it is exact and should accurately
represent the mass transfer from liquid to vapor phase in cavitation (bubble
growth or evaporation). Though the calculation of mass transfer for bubble
collapse or condensation is expected to be different from that of bubble growth,
Equation 14–602 is often used as a first approximation to
model the bubble collapse. In this case, the right side of the equation is
modified by using the absolute value of the pressure difference and is treated
as a sink term.
It may be noted that in practical cavitation models, the local far-field
pressure 
is usually taken to be the same as the cell center pressure.
The bubble pressure 
is equal to the saturation vapor pressure in the absence of
dissolved gases, mass transport and viscous damping, that is, 
.
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Based on Equation 14–602, Singhal et al. [740] proposed a model where the vapor mass fraction is the dependent variable in the transport equation. This model accommodates also a single phase formulation where the governing equations is given by:
 | (14–603) |
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The rates of mass exchange are given by the following equations:
If 
 | (14–604) |
If 
 | (14–605) |
The saturation pressure is corrected by an estimation of the local values of the turbulent pressure fluctuations:
 | (14–606) |
The constants have the values 
and 
. In this model, the liquid-vapor mixture is assumed to be
compressible. Also, the effects of turbulence and the noncondensable gases have
been taken into account.
The influence of turbulence on the threshold pressure can optionally be included. In all the cavitation models, the threshold pressure is calculated from, Yang et al. [127]:
 | (14–607) |
where 
and 
are the liquid phase density and turbulence kinetic energy, respectively.
The recommended value for 
is 0.39 and this value
is used by default.
In practical applications of a cavitation model, several factors greatly influence numerical stability. For instance, the high pressure difference between the inlet and exit, large ratio of liquid to vapor density, and large phase change rates between the liquid and vapor all have 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 usually difficult to correct. You may consider the following factors/tips when choosing a cavitation model and addressing potential numerical problems:
Choice of the cavitation models
In Ansys Fluent, there are three available cavitation models. The Zwart-Gerber-Belamri and the Schnerr and Sauer models have been implemented following an entirely different numerical procedure from the Singhal et al. model. Numerically, these two models are robust and converge quickly. It is therefore highly recommended that you should use the Schnerr and Sauer or the Zwart-Gerber-Bleamri model. The Singhal et al. model, though physically similar to the other two, is numerically less stable and more difficult to use.
Choice of the solvers
In Ansys Fluent, both the segregated (SIMPLE, SIMPLEC, and PISO) and coupled pressure-based solvers can be used in cavitation. As usual, the coupled solver is generally more robust and converges faster, particularly for cavitating flows in rotating machinery (liquid pumps, inducers, impellers, etc). For fuel injector applications, however, the segregated solver also performs very well with the Schnerr and Sauer and the Zwart-Gerber-Belamri models.
As for the Singhal et al. model, since the coupled solver does not show any significant advantages, it is suggested that the segregated solver is used.
Initial conditions
Though no special initial condition settings are required, we suggest 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. In general, the Schnerr and Sauer and Zwart-Gerber-Belamri models are robust enough so that there is no need for specific initial conditions. But in some very complicated cases, it may be beneficial to obtain a realistic pressure field before substantial cavities are formed. This can be achieved by obtaining a converged/near-converged solution for a single phase liquid flow, and then enabling the cavitation model. Again, the Singhal et al. model is much more sensitive to initial conditions. The above mentioned treatments are generally required.
Pressure discretization schemes
As for general multiphase flows, it is more desirable to use the following pressure discretization schemes in cavitation applications in this order:
PRESTO!
body force weighted
second order
The standard and linear schemes generally are not very effective in complex cavitating flows and you should avoid using them.
Relaxation factors
Schnerr and Sauer and Zwart-Gerber-Belamri models
The default settings generally work well. To achieve numerical efficiency, however, the following values may be recommended:
The relaxation factor for vapor is 0.5 or higher unless the solution diverges or all the residuals oscillate excessively.
The density and the vaporization mass can be relaxed, but in general set them to 1.
For the segregated solver, the relaxation factor for pressure should be no less than the value for the momentum equations.
For the coupled solver, the default value for the Courant number (200) may need to be reduced to 20-50 in some complex 3D cases.
Singhal et al. model
In general, small relaxation factors are recommended for momentum equations, usually between 0.05 – 0.4. The relaxation factor for the pressure-correction equation should generally be larger than those for momentum equations, say in the range of 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.
Special tips for the Singhal et al. model
Noncondensable gases
Noncondensable gases are usually present in liquids. Even a small amount (for example, 15 ppm) of noncondensable gases can have significant effects on both the physical results 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 (for example,
) 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 advisable to always limit the maximum pressure when possible. Ansys Fluent’s default value for the maximum vapor pressure limit is five times the local vapor pressure with consideration of the turbulence and thermal effects. You can change the vapor pressure ratio (which is the ratio of the maximum vapor pressure limit to the local pressure) using the expert text command option as described in Mass Transfer Mechanisms in the Fluent User's Guide.
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 Ansys Fluent support engineer.
When cavitation occurs, in many practical applications other gaseous species exist in the systems. For instance, in a ventilated supercavitating vehicle, air is injected into a liquid to stabilize or increase the cavitation along the vehicle surfaces. In some cases, the incoming flow is a mixture of a liquid and some gaseous species. To predict those types of cavitating flows, the basic two-phase cavitation model must be extended to a multiphase (N-phase) flow, or a multiphase species transport cavitation model.
The multiphase cavitation models are the extensions of the three basic two-phase cavitation models to multiphase flows. When modeling a N-phase flow, multiple liquid and gas phases can be added in the computational system under the following assumptions:
Mass transfer (cavitation) only occurs between the liquid and vapor phase.
The basic two-phase cavitation models are still used to model the phase changes between the liquid and vapor.
In the Singhal et al. model, the predescribed noncondensable gases can be included in the system. To exclude noncondensable gases from the system, the mass fraction must be set to 0, and the noncondensable gas must be modeled by a separate compressible gas phase.
For non-cavitating phase (
), the general transport equation governing the vapor
phase is the volume fraction equation in the Zwart-Gerber-Belamri and
Schnerr and Sauer models, while in the Singhal et al. model, a mass
transfer equation is solved and the vapor must be the second
phase.
The detailed description of the multiphase species transport approach can be found in Modeling Species Transport in Multiphase Flows. The multiphase species transport cavitation model can be summarized as follows:
All the assumptions/limitations for the multiphase cavitation model apply here.
The mass transfer between a liquid and a vapor phase/species is modeled by the basic cavitation models.
For the phases with multiple species, the phase shares the same pressure as the other phases, but each species has its own pressure (that is, partial pressure). As a result, the vapor density and the pressure used in Equation 14–605 are the partial density and pressure of the vapor.