State of Art of SPH

SPH, similar to traditional CFD, is limited handling low compressible fluid without making a fully incompressible assumption and Boussinesq-like approximation. Currently, the buoyancy force can only be applied in the momentum equation through a Boussinesq-like approximation.

\[ \rho = \rho_0 - \alpha\rho_0\Delta T \]

with \( \Delta T\) the temperature difference between reference state (when density \( \rho = \rho_0 \)) and current temperature. However, this Boussinesq approximation implicitly assumes a linear correlation between the buoyancy force and the temperature difference. That is why traditional SPH cannot handle the situations with large temperature difference and highly transient dynamics. On the other hand, SPH has its limitation to achieve a stable result when the liquid has low viscosity (e.g., water and air).

Another approach to predict thermal properties in CFD using SPH is applying modified Equation of State (usually Tait Water EoS).

\[ P = \frac{c_0 ^2 \rho_0}{ \gamma} [(\frac{\rho}{\rho_0})^\gamma - 1] \]

with \( \gamma=7 \). And \(c_0\) the reference sound of spped, \( \rho_0 = 1000 kg m^{-3} \) being the reference density. In these EoS, compressibility of water is enlarged by 1000 times to compensate numerical instabilities.