Artificial Compressibility

Typically, for casting simulations, the gas phase is considered to be compressible and the ideal gas law is applied. The melt is a liquid and has a constant density. Therefore, the multiphase mixture is not compressible throughout the entire computational domain. This inhomogeneity results in regions with finite and infinite speeds of sound, which can lead to numerical instabilities and convergence problems.

To overcome such robustness issues caused by combining a constant density VOF melt phase with an ideal gas VOF gas phase, the artificial compressibility method can be applied. The artificial compressibility for the VOF melt phase ensures a compressible mixture in the entire fluid domain.

There are two options for defining the artificial compressibility of the melt:

  • Pressure Dependent Density
  • Pressure and Temperature Dependent Density

Pressure Dependent Density

If the melt density does not depend on temperature, such as for isothermal simulations, the density is defined as:

Figure 1. EQUATION_DISPLAY
ρMelt=max(ρMeltmin,ρMeltref+ddp(ρMelt)p)
(378)

where:

Symbol Description Value
ρ Melt Density of the melt
ρ Melt m i n Minimum melt density 1 k g / m 3
ρ Melt r e f Reference melt density that is linearly interpolated from the temperature-dependent density table of the Casting Material Database at a temperature that closely matches the Casting Temperature.
ddp(ρMelt) Pressure derivative of the melt density 10 - 5 P a
p Relative pressure

and

T(ρMelt)=0

Pressure and Temperature Dependent Density

If the melt density depends on both pressure and temperature, the density is given as:

Figure 2. EQUATION_DISPLAY
ρMelt=max(ρMeltmin,ρMelt(T)+p(ρMelt)p)
(379)

where:

Symbol Description Value
ρ Melt Density of the melt
ρ Melt m i n Minimum melt density 1 k g / m 3
ρMelt(T) Temperature-dependent density of the melt that is linearly interpolated from the Casting Material database.
p(ρMelt) Pressure derivative of the melt density 10 - 5 P a
p Relative pressure

and

T(ρMelt))0

is computed from the tabular melt density values of the Casting Material Database.