Theory of Cavitation Models
In the vapor transport equation 5.172, 
and 
are the mass transfer source terms connected to the growth and collapse of the vapor bubbles in cavitating flows. These terms account for the mass exchange between the vapor and liquid phases during a cavitation process. As in all the mechanistic cavitation models,
and 
are, in Simerics-MP, modelled based on the Rayleigh-Plesset equation describing the growth of a single vapor bubble in a liquid.
Liquid-Vapor Mass Transfer
To derive an expression of the net phase change rate in cavitation, we consider an incompressible liquid-vapor two phase flow with zero-slip velocity (non-condensable gas is not considered). Introducing 
to represent the net rate of the mass transfer from the liquid to vapor, we have the liquid and vapor volume fraction equations and the total mass continuity equation (see Multiphase module) written as follows:
Liquid Phase:
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Vapor Phase:
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Mixture (Total Mass Continuity):
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With the liquid-vapor two-phase system, the mixture density, 
can be expressed in terms of the vapor volume fraction and phase densities:
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Since both the liquid and vapor density are assumed to be constant (incompressible), a relation between the velocity gradient and vapor volume fraction can be derived from equation 5.184 and equation 5.185:
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Combining equation 5.182 and equation 5.186 we get the expression for net mass source term as:
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Substituting equation 5.187 into equation 5.183, the equation for the vapor volume fraction can be rewritten in the general form:
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Applying the relation between the mass fraction and volume fraction of the vapor equation 5.176, we have equation 5.188 in terms of the vapor mass fraction:
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From equation 5.188 and equation 5.189, it is clear that under the condition of zero velocity slip between the liquid and vapor phase, cavitation can be modelled as a single-phase flow with an additional vapor mass fraction transport equation, or a Eulerian multiphase mixture flow with liquid-vapor mass transfer. In fact, without considering the effect of diffusion and phase velocity difference, the two approaches are identical mathematically. As mentioned above, at present, Simerics-MP adopts the single-phase approach to model cavitating flows.
Bubble Dynamics Consideration
It is commonly recognized that for the most of natural occurrences and engineering systems, there are adequate number of nuclei (bubbles, non-condensable gases, etc.) in the liquid for cavitation inception. To model the cavitation process, consequently, the focus is primarily on properly accounting for the bubble growth and collapse. Assuming that in a flowing liquid there is zero velocity slip between the liquid and vapor bubbles, the bubble dynamics equation can be derived from the generalized Rayleigh-Plesset equation describing the growth of a gas bubble in a liquid1Brennen, C.E., “Cavitation and Bubble Dynamics”, Oxford: Oxford University Press, 1995. 2Kubota, A., Kato, H. and Yamaguchi, H., “A New modelling of Cavitating Flows: A Numerical Study of Unsteady Cavitation on a Hydrofoil Section”, J. of Fluid Mech., Vol.240, pp.59-96, 1992. :
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where 
represents the bubble radius; 
is the pressure in the bubble (assumed to be the vapor pressure at the liquid temperature in the absence of other gases); 
is the pressure in the liquid surrounding the bubble; and 
is the surface tension coefficient between the liquid and vapor. Note that this equation is derived from the mechanical balance (no thermal barriers to bubble growth).
Neglecting the second order time directive (which is appropriate for low oscillation frequencies), the viscous damping term and surface tension force, we have a reduced expression of equation 5.190 that is valid for the asymptotic state:
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This reduced Rayleigh-Plesset equation provides a physical approach to introduce the effects of bubble dynamics into cavitation models. Note that the bubble radius can increase or decrease depending on the signs of (
): the bubble would grow if 
, while it would collapse when 
. Therefore, equation 5.191 may be rewritten as:
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If 
is the vapor bubble number density in a liquid (the number of bubbles present per unit volume), and all the vapor bubbles are perfect spheres with the same radius 
, we then have the vapor phase volume fraction as:
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Further, it is assumed that the vapor bubbles can be neither created nor destroyed in a liquid, but the bubbles can grow (evaporation) and collapse (condensation) during a cavitation process. In other words, in equation 5.193, the vapor bubble number density (
) remains a constant, but the bubble radius (
) may increase or decrease. Then the time derivative of vapor volume fraction can be calculated as:
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Substituting equation 5.194 into equation 5.189, we have transport equation governing the vapor phase mass fraction:
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And applying equation 5.192, the net rate of mass transfer per unit volume between the liquid and vapor has the form:
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equation 5.196 indicates that in cavitation, the unit volume mass transfer rate (
) is not only the function of (proportional to) both vapor and liquid phase density, buts also inversely proportional to the mixture density. Since equation 5.195 is derived directly from the phase and mixture mass continuities, it is exact and should accurately represent the mass transfer between the liquid and vapor phase in cavitation. Furthermore, with the introduction of bubble dynamics, equation 5.196 adopts the similar approach to model the two opposite (and physically different) mass transfer processes: from liquid to vapor (bubble growth or evaporation), and from vapor to liquid (bubble collapse or condensation). For the vapor mass fraction transport equation 5.195, the bubble growth is a source term, while the bubble collapse 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 (
, a material property) in the absence of dissolved gases, mass transport and viscous damping, 
.
Comparing equation 5.195 and equation 5.196 with the general vapor mass fraction equation 5.172, we can have the source terms 
and 
as follows:
If the local flow pressure is below the saturation vapor pressure, 
, only evaporation occurs so that:
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If the local flow pressure is above the saturation vapor pressure 
, there is only condensation:
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equation 5.197 and equation 5.198 form the basis of almost all the available mechanistic two-phase cavitation models3Zwart, P. J., Gerber, A.G., and Belamri, T., “A Two-Phase Flow Model for Predicting Cavitation Dynamics”, ICMF2004, Yokohama, Japan, 2004.4Li, H., Kelecy, F. J., Egelja-Maruszewski, A., and Vasquez, S.A., “Advanced Computational modelling of Steady and Unsteady Cavitating Flows”, Paper No. IMECE2008-67450, Boston, Massachusetts, USA, November, 2008. 5Singhal, A.K., Athavale, M. M., Li, H., and Jiang, Y., “Mathematical Basis and Validation of the Full Cavitation Model”, ASME Fluids Engineering Division Summer Meeting, New Orleans, USA, 2001. . In Simerics-MP, the Singhal et al modelling approach has been adopted.
Gas Absorption/Dissolution and Release
As mentioned above, non-condensable gases are almost always present in a working fluid and could have significant impact on cavitation. In many situations, a non-condensable gas not only freely transports with the flow, but it can also be dissolved into or released from a liquid, naturally aiming towards a dynamic equilibrium of the mass concentrations between the liquid and gas phases. The gas absorption/dissolution and release in a liquid is also a liquid-gas mass transfer phenomenon, which is driven by the mass concentration differences / gradients. To model cavitating flows, therefore, it is necessary to also account for the effect of the non-condensable gas and the possible liquid-gas mass transfer in the mixture flow.
Assuming that in a liquid-gas two phase flow, a non-condensable gas (e.g., air, oxygen, etc.) exists in both liquid phase (dissolved gas) and gas phase (free gas), we write the gas mass fraction transport equations in each phase as follows:
Free Gas (gas phase):
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Dissolved Gas (liquid phase):
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where, 
and 
are the mass fractions of the free gas and dissolved gas, respectively. 
and 
are the external / user-defined sources. 
and 
are the diffusivity of the free gas and dissolved gas. If the mass fraction of a non-condensable gas is pre-described as 
, then we have:
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or its transport in space and time can be obtained by solving the following equation:
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Note that for equation 5.199, equation 5.200 and equation 5.202only two of them need to be solved directly.
In equation 5.199 and equation 5.200, the source 
indicates the release rate of the dissolved gas, while the term 
represents the absorption / dissolution rate of the free gas. When the two phases are in contact, there is a tendency for the free gas and dissolved gas “
" and "
" transporting from one phase to the other to achieve a dynamic equilibrium between the two phases. The equilibrium models assume that the volumetric rates of the mass transfers depend on the mass-concentration gradients / differences6Subramanian, R. S, “Convective Mass Transfer”, Dept. of Chemical and Biomolecular Engineering, Clarkson University.:
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where 
is the liquid-gas interfacial area, defined in Multiphase module; 
(=
) is the bulk volumetric mass transfer coefficient; 
(=
) and 
(= 
) represent local mass concentrations of the dissolved and free gas; And 
and 
are the equilibrium mass concentrations of the dissolved and free gas on their hosting phases, respectively. Note that the term (
) has the unit of inversed time, 1/s, an indicator of the mass transfer efficiency. Hence, equation 5.203 and equation 5.204 have also been written in the following forms:
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Typically, 
and 
are not the same (discontinuity), but there is a well-defined equilibrium curve between the two concentrations, which depends on temperature, pressure, and mixture compositions7Rhim, J-A, “Equilibrium concentration and overall Henry’s Law constant of the dissolved ozone”, Environ. Eng., Res Vo. 9, No. 2, pp. 88-95, 2004.. The curve is usually monotonic and nonlinear, and it is often expressed as a quasi-linear relation with coefficient 
:
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where 
is usually decided using physical laws/empirical correlations. One common approach is to follow the Henry’s law that provides a generalized equilibrium relationship: for a liquid mixture in contact with the gas phase, the partial pressure of the free gas 
is equal to the product of the equilibrium mole fraction of the dissolved gas in the liquid phase, 
and the Henry constant, 
:
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If the free gas phase follows the idea gas law, then the Dalton’s law of partial pressure gives:
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Using equation 5.207 - equation 5.209, one can obtain the equilibrium ratio as:
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Clearly, the Henry’s constant has a unit of pressure. For this reason, one may consider it as a reference pressure. For example, for an ideal liquid mixture in contact with a gas, the Henry’s constant is the saturation vapor pressure 
. Therefore, equation 5.210 can also be written as:
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where 
is defined as the reference pressure for the dissolved equilibrium mass fraction. Then equation 5.207 may be rewritten as:
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In equation 5.205 and equation 5.206, the terms 
and 
are the mass concentration differences/gradients, the driving forces of the absorption/dissolution and release of the non-condensable gases. It indicates that the non-condensable gas transport across two phases requires a departure of mass concentration from its equilibrium state. The direction of the mass transfer tends to move the system towards equilibrium depending on local and equilibrium mass concentrations on both phases. Specifically, from the equilibrium model, we have the following:
Gas Absorption/Dissolution into a Liquid:
For an absorption/dissolution process, the mass transfer is from the gas phase (free gas) to the liquid phase (dissolved gas). The equilibrium model assumes that the free gas at the gas phase is at the equilibrium state: 
. From equation 5.207 and equation 5.212, we then have:
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And from equation 5.205 and equation 5.206, the mass transfer source terms are:
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Gas Release from a Liquid:
For the gas release, the mass transfer is from the liquid phase (dissolved gas) to the gas phase (free gas). In this process, the equilibrium model assumes that the dissolved gas in the liquid phase is always at the equilibrium state: 
. From equation 5.205-equation 5.207 and equation 5.212, we then have the rates of mass transfer as:
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Finally, it may be noted that when a part of the non-condensable gas is dissolved into the liquid, the freely expandable gas is only the portion remaining in the gas phase, 
. Hence the mixture density is computed as:
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And the volume fraction of the free gas is as:
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