(app_cavitation_population)=
# Cavitation Population Source Models

The number balance in {ref}`chap_cavitation` uses a birth rate $J$, a
binary-breakup frequency $b$, and a coalescence kernel $K$. This appendix
gives the optional closures for these terms. All bubbles share the radius
$R_b$ obtained from the vapor-plus-gas volume and number density $N_b$ in
Eq. {eq}`cav_transported_geometry`.

## Classical Nucleation

An optional birth model uses the work of forming a spherical vapor nucleus.
For $\Delta p=p_s-p>0$, the surface and pressure contributions are

```{math}
:label: cav_nucleation_barrier
\Delta G(R)=4\pi\sigma R^2-\frac{4\pi}{3}\Delta p R^3
\qquad R_* =\frac{2\sigma}{\Delta p}
\qquad G_* =\frac{16\pi\sigma^3}{3k_BT\Delta p^2}
```

Here $R_*$ is the critical radius from $d\Delta G/dR=0$, and $G_*$
is the barrier divided by thermal energy $k_BT$, where $k_B$ is
Boltzmann's constant. Let $m_a$ be molecular mass and
$n_l=\rho_l^{\mathrm{sat}}/m_a$ the molecular number density of liquid.
The homogeneous birth rate per liquid volume is

```{math}
:label: cav_homogeneous_birth
J_h=n_l\sqrt{\frac{2\sigma}{\pi m_a}}\exp(-G_*)
```

For a wall bubble contact angle $\theta_b$, measured so that $\theta_b=\pi$
is the fully wetted limit, define

```{math}
:label: cav_wall_birth
\begin{aligned}
m_\theta&=-\cos\theta_b
\qquad F_\theta=\frac14(2+m_\theta)(1-m_\theta)^2\\
J_w&=\frac{1-m_\theta}{2}n_l^{2/3}
\sqrt{\frac{2\sigma}{\pi m_a}}
\frac{\exp(-F_\theta G_*)}{\sqrt{F_\theta+\epsilon_\theta}}
\end{aligned}
```

The small dimensionless $\epsilon_\theta$ regularizes the vanishing
geometric factor. $J_w$ is number per wall area per time. In a cell,
volume births contribute $\alpha_lJ_h\Omega_P$ and adjacent wall faces
contribute $\alpha_lJ_wA_w$, where $A_w$ is wall-face area. Dividing their
sum by cell volume $\Omega_P$ gives $J$ in Eq. {eq}`cav_number_balance`.
This rate may be capped by a prescribed birth-rate limit. Births vanish
for nonpositive driving pressure; the exponential barrier also suppresses
them as $\Delta p\to0^+$.

## Breakup and Coalescence

The simplest population closure prescribes nonnegative constants $b,K$.
An optional SST RANS closure evaluates them from turbulent dissipation
$\varepsilon=\beta^*k\omega$, where $\beta^*$ is the SST coefficient.
Let $d=2R_b$ and $\nu_l=\mu_l/\rho_l$.

The equal-size turbulent collision closure based on Prince and Blanch
{cite}`cavpop-prince1990bubble` is

```{math}
:label: cav_coalescence_kernel
\begin{aligned}
E_c&=\exp\left[-\sqrt{\frac{R_b^3\rho_l}{16\sigma}}
 \ln\!\left(\frac{h_0}{h_f}\right)
 \frac{\varepsilon^{1/3}}{R_b^{2/3}}\right]\\
K&=C_K\,0.356\pi(2d)^2\varepsilon^{1/3}
 \sqrt{2d^{2/3}}\,E_c
\end{aligned}
```

The ratio $h_0/h_f>1$ represents film drainage from initial to rupture
thickness. $C_K$ is a nonnegative multiplier. This diameter-based
collision convention includes turbulent collisions, with no slip-driven
or mean-shear collision term.

The reduced breakup closure evaluates the Luo–Svendsen frequency for
equal-volume daughters {cite}`cavpop-luo1996theoretical`:

```{math}
:label: cav_breakup_kernel
\begin{aligned}
\xi_{\min}&=\frac{11.4}{d}
 \left(\frac{\nu_l^3}{\varepsilon}\right)^{1/4}\\
B_\sigma&=\frac{12(2^{1/3}-1)\sigma}
 {2.05\rho_l\varepsilon^{2/3}d^{5/3}}\\
b&=C_b\,0.923\alpha_l
 \left(\frac{\varepsilon}{d^2}\right)^{1/3}
 \int_{\xi_{\min}}^1
 \frac{(1+\xi)^2}{\xi^{11/3}}
 \exp\left(-\frac{B_\sigma}{\xi^{11/3}}\right)\,d\xi
\end{aligned}
```

Here $\xi$ is eddy size divided by bubble diameter and $C_b\geq0$ is
a multiplier. Breakup is zero if $\xi_{\min}\geq1$. The turbulent
population rates also vanish when dissipation, bubble size, liquid volume,
or void volume vanishes. These are reduced single-size kernels; daughter
sizes and the resulting size distribution are not separately transported.

## References

```{bibliography}
:filter: docname in docnames
:keyprefix: cavpop-
```