Equations of State and Caloric Models

Formulation convention. NeuralFlow evaluates thermodynamic material states primarily from pressure and temperature. The equations below reproduce the active ideal-gas, stiffened-gas and constant-density closures used by the current carrier and homogeneous-VOF material-state paths.

Thermodynamic state and notation

For a fluid material, the primitive thermodynamic state is described by static pressure and temperature . Density and caloric quantities are derived from the selected equation of state (EOS):

The total specific energy stored by the compressible finite-volume formulation is

For pressure-based derivatives used in the coupled Jacobian, NeuralFlow requires the thermodynamic sensitivities

Ideal-gas EOS

For a single ideal-gas material with molecular weight ,

The density derivatives are therefore

The caloric relation is evaluated through the configured heat capacity. In the ideal-gas path,

For a calorically perfect ideal gas with constant heat capacities and a zero sensible-energy reference, this reduces to

The acoustic speed used by the material closure is

Stiffened-gas EOS

The stiffened-gas model introduces a pressure offset . Define the thermodynamic pressure

The currently validated NeuralFlow stiffened-gas material state requires constant and constant . Its effective gas constant and constant-volume heat capacity are

The pressure-temperature form of the EOS is

NeuralFlow also exposes the equivalent density/internal-energy form

With the current constant- enthalpy path,

This identity is important for liquids modeled as weakly compressible stiffened gases: the pressure offset contributes to internal energy even when and are constant.

The exact density derivatives used by the material-state Jacobian are

Because NeuralFlow defines , the corresponding derivatives are

The stiffened-gas sound speed follows from the shifted thermodynamic pressure,

The admissible state must satisfy

Constant-density incompressible closure

For the constant-density material model, density is a material constant rather than a thermodynamic unknown:

The material caloric path sets

No finite physical acoustic speed is manufactured from this closure. NeuralFlow therefore uses the artificial-compressibility formulation for a carrier or homogeneous mixture that is entirely constant density.

Heat-capacity and enthalpy models

For a constant heat capacity, the active material evaluator uses

For a polynomial range ,

The corresponding integrated enthalpy evaluator is

Outside the complete tabulated temperature interval, the current property path clamps evaluation to the nearest endpoint rather than extrapolating the polynomial indefinitely.

Total-state relations

For ideal and stiffened gases the total temperature relation used by the material model is

For an ideal gas, the isentropic total-pressure relation is

For a stiffened gas the same relation is applied to the shifted pressure:

The entropy-like isentropic invariant used by the material class has the common form

for a stiffened gas, with recovering the ideal-gas expression.

Primitive-to-conservative thermodynamic Jacobian

For a compact compressible carrier state and , the EOS derivatives generate the pressure and temperature columns. For momentum,

For the total-energy density,

These relations explain why a thermodynamic model in NeuralFlow is more than a density formula: consistent derivatives propagate directly into the fully coupled implicit block.

Embedding the EOS in homogeneous VOF

Every VOF constituent is evaluated at the common using one of the supported material closures above. Mixture density is

At fixed phase mass fractions, the mixture density sensitivities are

With and , the frozen-composition acoustic closure is

This uses the constituent EOS derivatives directly; NeuralFlow does not obtain by averaging constituent sound speeds.

Model selection and admissibility

The compressible material-state path accepts ideal-gas and stiffened-gas closures; constant-density materials use the artificial-compressibility branch when the complete active carrier/VOF composition is incompressible. The dilute Eulerian particle phase has a separate pressureless/regularized formulation and should not be interpreted as another carrier thermodynamic EOS.

Before a material state is accepted, density, heat capacities and thermodynamic derivatives are required to be finite and physically admissible. This validation is especially important for stiffened-gas states because a numerically positive static pressure alone does not guarantee .