CMPS Theory Manual › VOF Interface Capturing and Coupled Discretization

VOF Interface Capturing and Coupled Discretization

Implementation alignment. The current CMPS VOF path provides Upwind, HRIC and CICSAM interface schemes. High-resolution reconstruction is applied as a bounded explicit correction while the robust differentiated Upwind phase-mass flux remains in the implicit Jacobian.

Conservative phase-mass flux

The carrier flux supplies directed positive and negative volume carriers, denoted here by \(Q_f^+\ge0\) and \(Q_f^-\le0\). For physical phase \(k\), the base flux is

\[\dot m_{k,f}=A_f\left[Q_f^+(\alpha_k\rho_k)_L+Q_f^-(\alpha_k\rho_k)_R\right].\]

The phase masses close exactly to mixture density on each side, so the sum of all physical-phase mass fluxes reproduces the carrier mass flux. Only the \(N-1\) independent phase rows are inserted into the global coupled system; the reference-phase flux follows from closure.

First-order Upwind

Upwind uses the donor cell composition selected by the sign of the complete carrier transport. It is the base implicit VOF stencil and the fallback for degenerate high-resolution reconstruction. Constituent densities and carrier terms remain coupled to velocity, pressure, temperature and independent volume fractions.

HRIC reconstruction

HRIC uses a donor \(D\), acceptor \(A\), and a virtual upstream value reconstructed from the donor gradient,

\[\alpha_U=\alpha_A-2\nabla\alpha_D\cdot(\mathbf x_A-\mathbf x_D).\]

The normalized donor variable is

\[\tilde\alpha_D=\frac{\alpha_D-\alpha_U}{\alpha_A-\alpha_U}.\]

The compressive branch used by CMPS is

\[\tilde\alpha_C= \begin{cases} 2\tilde\alpha_D,&0\le\tilde\alpha_D<0.5,\\ 1,&0.5\le\tilde\alpha_D<1,\\ \tilde\alpha_D,&\text{otherwise}. \end{cases}\]

Let \(\chi=|\hat{\mathbf n}_\alpha\cdot\hat{\mathbf n}_f|\). The angle blend is

\[\tilde\alpha_f=\tilde\alpha_D+\sqrt{\chi}\left(\tilde\alpha_C-\tilde\alpha_D\right).\]

For physical transient HRIC, the donor-cell Courant number is

\[Co_D=\frac{|u_{n,f}|A_f\Delta t}{V_D}.\]

High-resolution strength is retained for \(Co_D\le0.3\), linearly reduced between 0.3 and 0.7, and reduced to donor Upwind for \(Co_D\ge0.7\). In steady pseudo-time operation, CMPS does not substitute pseudo time into this physical Courant correction.

CICSAM reconstruction

CICSAM is a physical-time scheme in CMPS. With \(C=\min(1,\max(Co_D,\epsilon))\), the Hyper-C candidate is

\[\tilde\alpha_{HC}=\min\left(\frac{\tilde\alpha_D}{C},1\right).\]

The ULTIMATE-QUICKEST candidate used by the implementation is

\[\tilde\alpha_{UQ}=\frac{8C\tilde\alpha_D+(1-C)(6\tilde\alpha_D+3)}{8},\]

which is bounded to the CBC interval between the donor and Hyper-C candidates. With flow-direction alignment

\[\chi=\left|\frac{\nabla\alpha_D\cdot\hat{\mathbf s}_{DA}}{|\nabla\alpha_D|}\right|,\]

CMPS blends using \(\chi^2\),

\[\tilde\alpha_f=\chi^2\tilde\alpha_{HC}+(1-\chi^2)\tilde\alpha_{UQ},\]

followed by bounding to the admissible normalized-variable interval.

Multiphase simplex projection

HRIC and CICSAM reconstruct every physical phase and then project the complete phase vector onto

\[\mathcal S=\{\boldsymbol\alpha:\alpha_k\ge0,\ \sum_k\alpha_k=1\}.\]

CMPS uses the Euclidean projection onto the unit simplex rather than independent clipping. This is essential for more than two phases because component-wise clipping does not preserve the phase-sum constraint.

Deferred high-resolution correction

The high-resolution face composition produces a conservative correction to the phase-mass flux. The implicit face Jacobian remains the differentiated Upwind flux, while the HRIC/CICSAM correction is added explicitly to the residual. This provides a robust matrix stencil without discarding bounded high-resolution interface transport.

Simplex-preserving coupled update

Let \(\Delta\alpha_i\) be the solved primitive corrections. CMPS first tests the full coupled step. If the candidate leaves the phase simplex, a common scalar \(\lambda\in[0,1]\) is applied to the complete cell correction block, not just to the phase variables:

\[\mathbf V^{n+1}=\mathbf V^n+\lambda\Delta\mathbf V.\]

For every independent fraction with \(\Delta\alpha_i<0\), the lower-bound limit is \(\alpha_i/(-\Delta\alpha_i)\). If \(\sum_i\Delta\alpha_i>0\), the reference phase supplies the additional limit

\[\lambda_r=\frac{\alpha_r}{\sum_i\Delta\alpha_i}.\]

The largest admissible bound is multiplied by an interior fraction-to-boundary factor. Applying one scale to the whole correction preserves the direction produced by the coupled linear solve.

Physical and pseudo-time assembly

The VOF physical-time Jacobian uses the exact \(\partial\mathbf W/\partial\mathbf V\) block, including the flow–composition cross derivatives. For BDF time integration, the known histories remain fixed during pseudo-time subiterations. In the CMPS right-hand-side convention,

\[\mathbf B_t=\frac{V}{\Delta t}\left[-E_0\mathbf W^{(m)}+E_1\mathbf W^n-E_2\mathbf W^{n-1}\right].\]

For artificial-compressibility flow, pressure is not a physical conserved quantity, so the physical-time pressure row is omitted. The pseudo-time pressure row instead receives the artificial acoustic coefficient \(1/c_{ac}^2\).

Gradient construction and limiting

HRIC and CICSAM activate volume-fraction gradients. The VOF gradients follow the selected CMPS gradient framework, including Green–Gauss and least-square paths, and are limited consistently with partition and boundary neighbor data before high-resolution face reconstruction.

VOF boundary composition

Inlet-like boundaries prescribe a full physical-phase composition. Pressure outlets use the interior composition for outflow and a stored backflow composition for reverse flow. Walls, symmetry, axes and internal interfaces do not prescribe a new composition. The current audited VOF boundary set includes:

wall, interior wall and fluid–solid wall,
pressure outlet and symmetry,
mass-flow inlet and axis,
velocity inlet for artificial-compressibility flow,
stagnation inlet, far field and solid-propellant surface for compressible flow.

Distributed consistency and restart

Partition interfaces exchange VOF state, properties and gradients together with identity/consistency information. The paired ranks verify phase layout, equation placement, opposite face orientation and matching transferred states before face assembly. Restart data preserve the physical phase layout and transient histories required by the active formulation; incompatible phase identities or material closures are rejected rather than silently reinterpreted.