Primary variables
- \(\rho\) mixture density
- \(\mathbf{V}\) mixture velocity
- \(f_g\) non-condensable gas mass fraction
- \(\rho_g\) NCG density
- \(\alpha_g\) NCG volume fraction
- \(\alpha_l\) liquid volume fraction
This note takes the Singhal et al. (2002) homogeneous-mixture framework and explicitly turns off the vapor/cavitation part (no vapor phase, no phase change). What remains is a two-constituent EVET mixture: liquid + non-condensable gas (NCG).
The model is a homogeneous mixture (Equal-Velocity / Equal-Temperature; EVET) formulation:
- One mixture momentum equation (variable density)
- One scalar transport equation for non-condensable gas mass fraction
- Algebraic mixture closure for density (liquid + NCG)
Mixture closure (liquid + NCG only)
With vapor ignored (set \(f_v=0\) everywhere), the mixture density relation reduces to:
\[ \frac{1}{\rho}=\frac{f_g}{\rho_g}+\frac{1-f_g}{\rho_l} \]
NCG density is computed as:
\[ \rho_g=\frac{P}{R_{air}T} \]
The corresponding volume fractions follow from the same algebra used in the full model:
\[ \alpha_g=f_g\,\frac{\rho}{\rho_g} \]
\[ \alpha_l=1-\alpha_g \]
NCG mass fraction transport (no phase change)
In this vapor-off limit there are no evaporation/condensation source terms. The transported scalar is the NCG mass fraction \(f_g\):
\[ \frac{\partial}{\partial t}(\rho f_g)+\nabla\cdot(\rho\,\mathbf{V}\,f_g)=\nabla\cdot(\Gamma\,\nabla f_g) \]
Here \(\Gamma\) is an effective diffusion coefficient used for the scalar transport (typically a turbulence-model-based closure in CFD implementations).
What this is (and is not)
- This is still an EVET homogeneous mixture (one velocity, one temperature, variable density).
- It is not a cavitation model anymore in the classic fluid-mechanics sense, because cavitation refers to the formation of vapor-filled cavities when local pressure falls below the liquid vapor pressure, and the associated collapse when pressure recovers (Brennen (1995)).
In hydraulics practice it’s common to associate cavitation damage with “air” because:
- Entrained/dissolved gas provides nuclei and strongly influences how cavities grow/collapse.
- Some phenomena are more properly called aeration/ventilation (gas ingestion/entrained bubbles) rather than vapor cavitation; Brennen explicitly distinguishes air entrainment (“ventilation”) from vaporization-driven cavities in the historical context of propellers (Brennen (1995)).
So: a liquid+air (NCG) mixture model can still be useful for predicting density/compressibility effects, but by itself it does not model vaporization/condensation.
References
Brennen, Christopher Earls. 1995. Cavitation and Bubble Dynamics. Oxford University Press. http://caltechbook.library.caltech.edu/1/.
Singhal, Ashok K., Mohan M. Athavale, Huiying Li, and Yong Jiang. 2002. “Mathematical Basis and Validation of the Full Cavitation Model.” Journal of Fluids Engineering 124 (3): 617–24.