December 20, 2025
This post is a code-oriented restatement of the Full Cavitation Model from Singhal et al. (2002), with the core equations collected in one place.
The model is a homogeneous mixture (Equal-Velocity / Equal-Temperature; EVET) formulation:
When including a non-condensable gas (NCG):
Mixture density is related to vapor mass fraction \(f\) by:
\[ \frac{1}{\rho}=\frac{f}{\rho_v}+\frac{1-f}{\rho_l} \]
Vapor volume fraction is then:
\[ \alpha \equiv f\,\frac{\rho}{\rho_v} \]
The vapor mass fraction is governed by:
\[ \frac{\partial}{\partial t}(\rho f)+\nabla\cdot(\rho\,\mathbf{V}\,f)=\nabla\cdot(\Gamma\,\nabla f)+R_e-R_c \]
Here \(\Gamma\) is an effective diffusion coefficient used for the scalar transport (typically a turbulence-model-based closure in CFD implementations).
Evaporation (vapor generation) rate:
\[ R_e=C_e\,\frac{V_{ch}}{\sigma}\,\rho_l\,\rho_v\left[\frac{2}{3}\,\frac{(P_v-P)}{\rho_l}\right]^{1/2}(1-f) \]
Condensation rate:
\[ R_c=C_c\,\frac{V_{ch}}{\sigma}\,\rho_l\,\rho_l\left[\frac{2}{3}\,\frac{(P-P_v)}{\rho_l}\right]^{1/2}f \]
Where:
The authors recommend estimating \(V_{ch}\) from turbulent kinetic energy:
\[ V_{ch}=\sqrt{k} \]
The phase-change threshold pressure is raised using turbulent pressure fluctuations:
\[ P'_{turb}=0.39\,\rho\,k \]
\[ P_v=\left(P_{sat}+\frac{P'_{turb}}{2}\right) \]
With NCG, the mixture density relation becomes:
\[ \frac{1}{\rho}=\frac{f_v}{\rho_v}+\frac{f_g}{\rho_g}+\frac{1-f_v-f_g}{\rho_l} \]
NCG density is computed as:
\[ \rho_g=\frac{P}{R_{air}T} \]
Volume fractions are updated as:
\[ \alpha_g=f_g\,\frac{\rho}{\rho_g} \]
\[ \alpha_l=1-\alpha_v-\alpha_g \]
The paper then rewrites the phase-change rates (replacing \(V_{ch}\) with \(\sqrt{k}\)) as:
\[ R_e=C_e\,\frac{\sqrt{k}}{\sigma}\,\rho_l\,\rho_v\left[\frac{2}{3}\,\frac{(P_v-P)}{\rho_l}\right]^{1/2}\left(1-f_v-f_g\right) \]
and
\[ R_c= C_c\,\frac{\sqrt{k}}{\sigma}\,\rho_l\,\rho_l\left[\frac{2}{3}\,\frac{(P-P_v)}{\rho_l}\right]^{1/2}f_v \]
There are a couple of error in Eq. (24) of the original paper. Eq (24) is supposed to be a restatement of Eq. (16) with \(V_{ch}\) is replaced with \(\sqrt{k}\). However, there are two mistakes.
Eq. (16)
\[ R_c=C_c\,\frac{V_{ch}}{\sigma}\,\rho_l\,\rho_l\left[\frac{2}{3}\,\frac{(P-P_v)}{\rho_l}\right]^{1/2}f_v \]
Eq. (24) as printed \[ R_c= {\color{red}{C_e}}\,\frac{\sqrt{k}}{\sigma}\,\rho_l\,\rho_l\left[\frac{2}{3}\,\frac{(P-P_v)}{\color{red}{\rho}}\right]^{1/2}f_v \]
Eq. (24) corrected: \[ R_c= C_c\,\frac{\sqrt{k}}{\sigma}\,\rho_l\,\rho_l\left[\frac{2}{3}\,\frac{(P-P_v)}{\rho_l}\right]^{1/2}f_v \]
\[ C_e=0.02\qquad C_c=0.01 \]