CMPSTheory & Implementation Manual
Homogeneous Volume-of-Fluid (VOF) Model
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Homogeneous Volume-of-Fluid (VOF) Model

Source alignment. This chapter documents the homogeneous, fully coupled VOF formulation present in the 13 August 2026 CMPS source. The VOF constituents share one velocity, one pressure and one temperature. The formulation is distinct from the dilute dispersed-phase model described elsewhere in this manual.

Model scope and shared fields

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

\[\mathbf{u}_1=\cdots=\mathbf{u}_N=\mathbf{u},\qquad p_1=\cdots=p_N=p,\qquad T_1=\cdots=T_N=T.\]

The phase topology is represented through volume fractions \(\alpha_k\). Only \(N-1\) fractions are solved independently. A selected reference phase is reconstructed from the simplex condition

\[\alpha_r=1-\sum_{k\ne r}\alpha_k,\qquad 0\le \alpha_k\le1,\qquad \sum_{k=1}^{N}\alpha_k=1.\]

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

\[\mathbf{V}_{\mathrm{VOF}}=[u_1,\ldots,u_d,p,T,\alpha_1,\ldots,\alpha_{N-1}]^T.\]

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

\[\mathbf{W}_{\mathrm{VOF}}=[\rho u_1,\ldots,\rho u_d,\rho,\rho E,\alpha_1\rho_1,\ldots,\alpha_{N-1}\rho_{N-1}]^T.\]

Thus the transported VOF quantities are constituent partial densities \(\alpha_k\rho_k\), 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 \((p,T)\). The homogeneous mixture density is the volume-weighted sum

\[\rho=\sum_k \alpha_k\rho_k(p,T).\]

CMPS uses volume-weighted dynamic viscosity and thermal conductivity,

\[\mu=\sum_k\alpha_k\mu_k,\qquad k=\sum_k\alpha_k k_k.\]

The volumetric sensible enthalpy and internal energy are

\[\rho h=\sum_k\alpha_k\rho_k h_k,\qquad \rho e=\sum_k\alpha_k\rho_k e_k,\]

and the specific mixture quantities follow by division by \(\rho\). The total energy density is

\[\rho E=\sum_k\alpha_k\rho_k e_k+\frac12\rho\lvert\mathbf{u}\rvert^2.\]

Likewise, the volumetric heat capacities are \(\rho C_p=\sum_k\alpha_k\rho_k C_{p,k}\) and \(\rho C_v=\sum_k\alpha_k\rho_k C_{v,k}\).

Exact composition derivatives

Because the reference fraction changes when an independent fraction changes, the fixed-\(p,T\) mixture derivatives have a difference form. If physical phase \(i\) corresponds to independent fraction \(\alpha_i\),

\[\frac{\partial\rho}{\partial\alpha_i}=\rho_i-\rho_r,\qquad \frac{\partial\mu}{\partial\alpha_i}=\mu_i-\mu_r,\qquad \frac{\partial k}{\partial\alpha_i}=k_i-k_r.\]

The same reference-phase subtraction is used for \(\rho h\), \(\rho e\), \(\rho C_p\), and \(\rho C_v\). 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

\[\rho_k=\frac{p}{R_kT},\qquad \rho_{k,p}=\frac1{R_kT},\qquad \rho_{k,T}=-\frac{\rho_k}{T}.\]
\[e_k=h_k-R_kT,\qquad a_k^2=\gamma_k\frac{p}{\rho_k}.\]

Stiffened-gas constituent

For constant \(C_{p,k}\) and \(\gamma_k\), define

\[R_{s,k}=C_{p,k}\frac{\gamma_k-1}{\gamma_k}.\]

Then

\[\boxed{\rho_k=\frac{p+p_{\infty,k}}{R_{s,k}T}},\qquad \rho_{k,p}=\frac1{R_{s,k}T},\qquad \rho_{k,T}=-\frac{\rho_k}{T}.\]
\[p=(\gamma_k-1)\rho_ke_k-\gamma_kp_{\infty,k},\qquad a_k^2=\gamma_k\frac{p+p_{\infty,k}}{\rho_k}.\]

Constant-density constituent

\[\rho_k=\rho_{0,k},\qquad \rho_{k,p}=0,\qquad \rho_{k,T}=0.\]

The active carrier formulation is derived from the complete material set. If every constituent has \(\rho_{k,p}=\rho_{k,T}=0\), CMPS 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

\[Y_k=\frac{\alpha_k\rho_k}{\rho}.\]

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

\[\rho_p^Y=\rho\sum_k\alpha_k\frac{\rho_{k,p}}{\rho_k},\qquad \rho_T^Y=\rho\sum_k\alpha_k\frac{\rho_{k,T}}{\rho_k}.\]

With \(C_p^Y=\sum_kY_k C_{p,k}\), the implemented frozen-composition acoustic compressibility is

\[\frac{1}{a_m^2}=\rho_p^Y+\frac{\rho_T^Y}{\rho C_p^Y}.\]

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, CMPS extends the Smith preconditioning structure to the complete coupled flow–VOF state. With reference velocity \(u_r\),

\[\Theta=\frac{1}{u_r^2}-\frac{\rho_T^Y}{\rho C_p^Y},\qquad \beta_s=\frac12\left(1-\frac{u_r^2}{a_m^2}\right).\]

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

\[\Gamma=\frac{\partial\mathbf W}{\partial\mathbf V}+ (\Theta-\rho_p^Y)\,\mathbf z\,\mathbf e_p^T,\]

where \(\mathbf z\) 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.

Isentropic-Mach acceleration. The general low-Mach preconditioning path can raise the VOF reference-velocity floor using the isentropic-Mach estimate described in Section 5.2.1. In the current homogeneous-VOF branch, the local pressure ratio for that limiter does not apply a stiffened-gas pressure shift; the mixture acoustic speed is still used for the velocity scaling.

Current model-compatibility limits