Cavitation modelling

Phase change models represent the mass transfer between the phases. In cavitation, pressure is responsible for the mass transfer between liquid and vapor phases. This is in contrast to boiling, in which the temperature causes the phase change. There are two general categories of phase change models used for cavitation: the barotropic models and equilibrium models. This section will briefly discuss the advantages and disadvantages of each type.

If the pressure is greater than vapor pressure, then the fluid is liquid, otherwise vapor. This means density of liquid water is considered as the density of fluid if the pressure is greater than vapor pressure and the density of water vapor is considered when pressure is less than vapor pressure of water at the ambient temperature.

The equilibrium model requires the solution of the energy equation. The equation for state of water is used, with the energy absorbed or released by phase change creating local temperature gradients which control the rate of phase change.

Several models for the bubble dynamics have been proposed:

The Rayleigh model is the oldest, dating from 1917. This model was derived by Lord Rayleigh^[1] It describes an empty space in the water, influenced by a constant external pressure. His assumption of an empty space led to the name cavity still used. The Rayleigh equation, derived from the Navier-Stokes equation for a spherically symmetric bubble convected with the flow with constant external pressure, reads

${\displaystyle R{\ddot {R}}+{\frac {3}{2}}{\dot {R}}^{2}={\frac {p(R)-p_{\infty }}{\rho _{L}}}}$

Building on the work of Lord Rayleigh, Plesset^[2] included the effects of viscosity, surface tension and a non-constant external pressure to the equation. This equation reads

${\displaystyle R{\ddot {R}}+{\frac {3}{2}}{\dot {R}}^{2}={\frac {p_{i}-p_{\infty }-{\frac {2\sigma }{R}}-{\frac {4\mu }{R}}{\dot {R}}}{\rho _{L}}}}$

The equation by Gilmore accounted for the compressibility of the liquid. In its derivation, the viscous term is only present as a product with the compressibility. This term is neglected. The resulting term is:

${\displaystyle (1-{\frac {{\dot {R}}(t)}{c(R)}})R(t){\ddot {R}}(t)+{\frac {3}{2}}(1-{\frac {{\dot {R}}(t)}{3c(R)}}){\dot {R}}^{2}(t)=(1+{\frac {{\dot {R}}(t)}{c(R)}})H(R)+(1-{\frac {\dot {R}}{c(R)}}){\frac {R}{c(R)}}{\dot {H}}(R)}$

In which:

${\displaystyle H={\frac {n}{n-1}}{\frac {p_{\infty }(t)+B}{\rho _{L}}}\left[({\frac {P+B}{p_{\infty }(t)+B}})^{\frac {n-1}{n}}-1\right]}$

${\displaystyle c=c_{0}\left({\frac {p_{g}(t)-2\sigma /R+B}{p_{\infty }(t)+B}}\right)^{\frac {n-1}{2n}}}$

${\displaystyle {\dot {H}}={\frac {D}{p_{\infty }(t)+B}}H-{\frac {D}{\rho }}({\frac {P+B}{p_{\infty }(t)+B}})^{\frac {n-1}{n}}+{\frac {\dot {R}}{\rho _{L}R}}\left[{\frac {p_{\infty }(t)+B}{P+B}}\right]^{\frac {1}{n}}\left[{\frac {2\sigma }{R}}-3kp_{g}(t)\right]}$

Over the years, several other models have been developed by making different assumptions in the derivation of the Navier-Stokes equations.