Homogeneous Volume-of-Fluid (VOF) Model

Model scope. The NeuralFlow homogeneous VOF model uses one shared velocity, pressure and temperature for all VOF constituents and solves independent phase fractions. It is a homogeneous one-fluid formulation and is distinct from the dilute dispersed-phase model.

Model scope and shared fields

NeuralFlow treats the VOF constituents as a homogeneous one-fluid mixture. For N physical phases, all constituents share

The phase topology is represented through volume fractions . Only fractions are solved independently. A selected reference phase is reconstructed from the simplex condition

This reference-phase representation removes one redundant transport equation and makes the phase-sum constraint exact at the state-definition level.

Primitive and conserved variables

The compact VOF primitive state is

The corresponding compact conservative mapping used by the physical-time operator is

Thus the transported VOF quantities are constituent partial densities , not bare volume fractions. The volume fractions remain the primitive composition variables corrected by the coupled nonlinear update.

Mixture thermodynamic closure

Each constituent is evaluated at the common . The homogeneous mixture density is the volume-weighted sum

NeuralFlow uses volume-weighted dynamic viscosity and thermal conductivity,

The volumetric sensible enthalpy and internal energy are

and the specific mixture quantities follow by division by . The total energy density is

Likewise, the volumetric heat capacities are and .

Exact composition derivatives

Because the reference fraction changes when an independent fraction changes, the fixed- mixture derivatives have a difference form. If physical phase corresponds to independent fraction ,

The same reference-phase subtraction is used for , , , and . These derivatives feed the complete flow–composition Jacobian, including momentum–VOF and energy–VOF cross-couplings.

Supported constituent equations of state

The active homogeneous VOF closure accepts ideal-gas, stiffened-gas, and constant-density incompressible constituents. Every constituent is evaluated independently at the common mixture pressure and temperature.

Ideal-gas constituent

Stiffened-gas constituent

For constant and , define

Then

Constant-density constituent

The active carrier formulation is derived from the complete material set. If every constituent has , NeuralFlow selects artificial compressibility. Otherwise the homogeneous mixture is treated as compressible and pressure-dependent and constant-density phases may coexist.

For detailed caloric relations, internal-energy derivatives and shifted total-pressure formulas, see Equations of State and Caloric Models.

Frozen-composition acoustics

For acoustic/preconditioning calculations, the composition is frozen using phase mass fractions

The common-pressure EOS derivatives at fixed phase mass fractions are evaluated as

With , the frozen-composition acoustic compressibility is

No arithmetic or volume-weighted average of constituent sound speeds is used. An exactly constant-density mixture has zero physical acoustic compressibility and is handled by the artificial-compressibility formulation.

Low-speed preconditioning

For compressible low-speed VOF, NeuralFlow extends the Smith preconditioning structure to the complete coupled flow–VOF state. With reference velocity ,

The pseudo-time matrix retains the exact conservative Jacobian and modifies only the pressure column through a rank-one correction. In compact form,

where contains the conservative pressure-correction direction, including phase mass fractions. This preserves the solved flow–VOF cross derivatives while modifying the acoustic pseudo-time scaling.

Current model-compatibility limits

• Compressible VOF uses the AUSM carrier-flux path.
• The current VOF production path is not combined with agglomeration multigrid.
• Split volumetric reaction integration is not activated with VOF.
• Ordinary, interior and fluid–solid VOF walls are currently restricted to adiabatic thermal treatment.
• CICSAM is available only for physical transient calculations because its construction uses the physical Courant number.
• VOF uses the final equation block after the carrier, turbulence, dispersed-phase and species slots.