22. Cavitation Population Source Models

The number balance in Homogeneous Cavitation Modeling 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. (15.6.1.2).

22.1. 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

(22.1.1)\[\Delta G(R)=4\pi\sigma R^2-\frac{4\pi}{3}\Delta p R^3 \qquad R_* =\frac{2\sigma}{\Delta p} \qquad G_* =\frac{16\pi\sigma^3}{3k_BT\Delta p^2}\]

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

(22.1.2)\[J_h=n_l\sqrt{\frac{2\sigma}{\pi m_a}}\exp(-G_*)\]

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

(22.1.3)\[\begin{split}\begin{aligned} m_\theta&=-\cos\theta_b \qquad F_\theta=\frac14(2+m_\theta)(1-m_\theta)^2\\ J_w&=\frac{1-m_\theta}{2}n_l^{2/3} \sqrt{\frac{2\sigma}{\pi m_a}} \frac{\exp(-F_\theta G_*)}{\sqrt{F_\theta+\epsilon_\theta}} \end{aligned}\end{split}\]

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. (15.6.1.1). This rate may be capped by a prescribed birth-rate limit. Births vanish for nonpositive driving pressure; the exponential barrier also suppresses them as \(\Delta p\to0^+\).

22.2. 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 [PB90] is

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

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 [LS96]:

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

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.

22.3. References

[LS96]

Hean Luo and Hallvard F. Svendsen. Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE Journal, 42(5):1225–1233, 1996. doi:10.1002/aic.690420505.

[PB90]

M. J. Prince and H. W. Blanch. Bubble coalescence and break-up in air-sparged bubble columns. AIChE Journal, 36(10):1485–1499, 1990. doi:10.1002/aic.690361004.