Multiphase Chapter 3: Volume-of-Fluid (VOF) Method
The Volume-of-Fluid (VOF) method is a numerical technique used in Computational Fluid Dynamics (CFD) to capture and track interfaces between immiscible fluids, most often gas-liquid systems such as air-water. It is a cornerstone of free-surface modeling in ANSYS Fluent, widely used for problems involving sloshing, jet breakup, filling, or dam-break flows.
The model solves a single set of momentum and continuity equations for all fluids and tracks the volume fraction of each phase within each computational cell, which defines the interface position. This approach allows precise representation of evolving free surfaces without deforming the computational mesh
Introduction and Physical Meaning
When two immiscible fluids coexist, say air and water, the interface separating them (the “free surface”) evolves dynamically with the flow. Modeling this requires a way to determine, at any instant, which part of each cell belongs to each fluid. The VOF method does exactly that: it introduces a scalar variable called the volume fraction, representing how much of each fluid occupies a cell.
A value of 0 means the cell contains only phase A (e.g., air).
A value of 1 means the cell contains only phase B (e.g., water).
Values between 0 and 1 correspond to cells cut by the interface.
Each cell, therefore, “knows” how much of each fluid it contains, and the interface is implicitly defined as the surface where the volume fraction transitions between 0 and 1.
VOF belongs to the Eulerian family of methods: the computational mesh remains fixed (or prescribed to move in a known way), while the fluids move through it. The motion of the interface results from solving an advection equation for the volume fraction, using the velocity field obtained from the Navier–Stokes equations.
Physically, this method is best suited to separated flows with distinct interfaces (e.g., free-surface flows, sloshing, wave impact), as opposed to dispersed flows (e.g., bubbly or droplet flows) where the mixture model or Eulerian multiphase model is more appropriate.
Theoretical Basis and Model Concept
Core Idea
VOF assumes that at any given point, each fluid either fully or partially occupies the local control volume. The method enforces that the sum of all phase volume fractions equals one, ensuring full occupancy of the domain (no voids).
Unlike mixture or Eulerian models, which may allow slip velocities and interpenetration, VOF enforces no slip between phases: both share the same velocity and pressure fields.
Governing Principles
Mass conservation: Each phase’s volume fraction changes only because of the flow of that phase into or out of a cell.
Momentum conservation: A single set of Navier–Stokes equations is solved using mixture properties, obtained by volume-fraction weighting the density and viscosity of each phase.
Interface tracking: The evolution of the interface is determined by transporting the volume fraction field using the computed velocity field.
This means that once the flow velocity is known, the interface motion can be predicted, allowing one to compute how the water surface moves, tilts, or breaks apart.
Physical Limitations
Fluids are assumed immiscible (no mixing or phase change).
Each cell must contain at least one fluid - voids are not permitted.
The interface thickness is limited by the grid resolution: the finer the mesh, the sharper the interface representation.
Because all phases share one velocity field, the model cannot capture slip or drag between them (for that, the multi-fluid VOF or Eulerian model is required).
Numerical Formulation and Computational Aspects
Time Discretization: Explicit vs Implicit
VOF can be solved with two time-integration approaches:
Explicit scheme
Directly advances the volume fraction using known values from the previous step.
Conditionally stable: the time step is limited by the Courant number (typically below 0.25).
Excellent for transient free-surface dynamics, where sharp interface tracking is critical (e.g., sloshing, jet breakup).
Enables Geo-Reconstruct and CICSAM interface schemes, which give crisp interfaces.
Implicit scheme
Solves the VOF equation iteratively together with momentum and pressure at each time step.
Unconditionally stable: can use larger time steps or even steady-state mode.
Slightly diffusive; the interface becomes smoother but less accurate.
Suitable for steady or quasi-steady free-surface configurations (e.g., open-channel flow at equilibrium).
Guideline:
Use explicit + Geo-Reconstruct for accurate interface tracking;
use implicit + Compressive/HRIC for steady or large-time-step cases.
Spatial Discretization
Fluent employs the finite-volume method. Each control volume exchanges fluxes with its neighbors through cell faces. For the volume fraction equation, these fluxes determine how much of a fluid crosses from one cell to another.
Different interpolation schemes are available to approximate these fluxes:
First-order upwind: robust but very diffusive; interface smears out.
Second-order upwind: higher accuracy, but can overshoot near discontinuities.
QUICK: suitable for structured meshes; moderately accurate.
Modified HRIC (High-Resolution Interface Capturing): blends upwind and downwind schemes to maintain boundedness and moderate sharpness.
Compressive or CICSAM schemes: more accurate, maintain sharper interfaces.
Geo-Reconstruct (PLIC): reconstructs the interface geometry directly from local gradients; gives the sharpest interface - about one cell thick.
Interface Reconstruction and Capturing
At the heart of VOF lies how the interface is represented:
In geometric reconstruction (PLIC), the local interface is approximated by a plane (in 3D) whose position and orientation are derived from the volume fraction gradient. This ensures accurate advection of the interface and conservation of volume.
In donor–acceptor or algebraic capturing schemes, fluid transfer between cells is computed algebraically without reconstructing geometry, less sharp but faster.
Compressive and BGM (Bounded Gradient Maximization) schemes further enhance sharpness for specific conditions (BGM for steady cases).
For complex 3D unstructured meshes, Fluent uses the Geo-Reconstruct algorithm internally - currently, the most accurate and widely used method.
Stability and Convergence Guidelines
Multiphase calculations are numerically demanding. Stability depends on:
A reasonable initial field, consistent with the physical situation.
Small time steps during startup, especially if gravity or high velocity gradients are present.
Appropriate Courant number: default ≈ 0.25 for explicit VOF.
Under-relaxation factors: keep moderate (0.3–0.5 for volume fraction when using SIMPLE; higher for PC-SIMPLE).
Mesh quality: poor skewness can destabilize the interface; use refined mesh near the free surface.
Solver type: pressure-based, either segregated (SIMPLE/PISO) or coupled.
Fluent also allows variable time stepping, adjusting the step size automatically based on the global Courant number to accelerate convergence while maintaining stability.
Surface Tension and Wall Adhesion
Physical Role
Surface tension arises from molecular cohesion at fluid interfaces. It tends to minimize the surface area, shaping droplets and stabilizing waves. In CFD, it manifests as a force acting along the interface, typically modeled as a pressure jump across curved surfaces.
VOF includes surface tension via two main formulations:
Continuum Surface Force (CSF) - converts surface forces into equivalent volumetric forces distributed near the interface. This is the classical Brackbill model, most common in Fluent.
Continuum Surface Stress (CSS) - a conservative variant that avoids explicit curvature computation, better for under-resolved or corner regions.
Implementation Aspects
Fluent computes curvature from the gradient of the smoothed volume fraction field.
Node-based smoothing improves robustness for irregular meshes.
Surface tension effects are more reliable on hexahedral or quadrilateral meshes.
For high Weber-number flows (where inertia dominates), surface tension can often be neglected; for small-scale or microfluidic problems, it is essential.
Wall Adhesion
At solid boundaries, the fluid–solid contact angle defines how the interface meets the wall:
The contact angle specifies the wall wettability (e.g., hydrophobic > 90°, hydrophilic < 90°).
Fluent modifies the local surface normal near the wall according to this angle to model wetting behavior.
The same concept applies to porous jumps (Jump Adhesion).
Modeling Issues and Remedies
Inaccurate curvature estimation may produce spurious currents (false velocities near interfaces).
Excessive smoothing or poor grid resolution can degrade accuracy.
Density weighting of surface-tension forces may introduce unphysical behavior (especially for light bubbles in heavy fluids).
Remedies include finer mesh, node-based smoothing, or coupled level-set + VOF method for curvature accuracy.
Boundary Conditions, Solver Setup, and Strategies
Boundary Conditions
Velocity or mass-flow inlet: only one phase should enter; set its volume fraction = 1.
Pressure inlet: one phase specified as entering; patch adjacent cells with its volume fraction = 1 to avoid sharp gradients.
Pressure outlet: ensure backflow volume fraction corresponds to the lighter (usually gas) phase; enlarge the domain to reduce gradients near outlet.
Walls: usually slip or no-slip depending on case; slip walls help in large-scale free-surface flows like dams or tanks.
Operating density: set equal to the fluid dominating the pressure outlet region (e.g., air for open free surfaces).
Initial Conditions
Initialize velocity = 0; then patch regions with correct phase fractions (e.g., α = 1 for water below, 0 for air above).
Solver Recommendations
Always use pressure-based solver (VOF not available for density-based).
Explicit scheme: PISO algorithm; all under-relaxation factors ≈ 1.
Implicit scheme: HRIC/Compressive interface; URF for VOF around 0.5.
Use PRESTO or Body-Force Weighted pressure discretization when gravity is important.
Turbulence damping can be enabled to limit over-diffusion of turbulence near interfaces (especially with k-ω models).
Bounded 2nd-order implicit time stepping is available for added stability and accuracy in steady or long transient runs.
Coupled VOF Solver (solves pressure, velocity, and volume fraction together) improves robustness for steady-state free-surface problems.
Time-Step Estimation
Start small and increase gradually:
Use minimum cell volume and expected flow velocity to estimate a safe Δt.
Ideally, each time step should converge in ~10–15 iterations.
Sub-time steps (VOF CFL ≈ 0.25) are automatically managed by Fluent.
Model Assumptions, Limitations, and Variants
Assumptions
Immiscible phases with no mass transfer.
Shared pressure and velocity fields.
Incompressible (unless one phase defined as compressible gas).
Constant properties within each phase.
Limitations
Only pressure-based solver can be used.
No void regions allowed.
Only one compressible ideal gas phase permitted.
Not compatible with most combustion or particle models.
Interface accuracy limited by grid resolution.
Level-Set + VOF cannot be used on polyhedral meshes.
Variants
Multi-Fluid VOF (Eulerian): hybrid between Eulerian and VOF models; allows separate velocity fields and interfacial drag, used for sharper interfaces with slip.
Coupled Level-Set + VOF (CLSVOF): combines mass conservation of VOF with geometric accuracy of level-set, ideal for surface-tension-dominated flows.
BGM and Compressive schemes: additional high-resolution interface capturing methods.
Real-World Applications and Examples
Dam-Break Flow – the classic benchmark for VOF validation.
Demonstrates accurate tracking of a smooth, transient-free surface.
Air–water, laminar, explicit VOF + Geo-Reconstruct, pressure outlet boundary.
Slip wall BCs simplify computation while preserving correct free-surface evolution.
Falling Box – solid–fluid interaction combining VOF with 6DOF motion.
The box falls into water, generating strong interface deformation and air entrainment.
Highlights the influence of solver schemes:
Explicit + Geo-Reconstruct: sharp interface but expensive.
Implicit + First-Order Upwind: diffused interface.
Explicit + Compressive: good balance of sharpness and cost.
Turbulence modeled with k-ε; mesh dynamically deformed.
Automotive Fuel-Tank Sloshing – evaluates liquid stability under acceleration; internal baffles reduce oscillations and prevent air ingestion.
Wave–Structure Interaction – uses Moving/Deforming Mesh (MDM), 6DOF, and Open-Channel Wave BCs for offshore and marine problems.
Microfluidic Applications – surface-tension-dominated, requiring fine meshes and accurate curvature computation (contact angle effects, droplet motion, emulsions).
Summary
The Volume-of-Fluid method provides an efficient and robust framework for simulating immiscible fluid interfaces without deforming the mesh. It captures transient free-surface motion by tracking the local fluid volume fraction, ensuring mass conservation and compatibility with complex geometries.
Its accuracy hinges on three main factors:
Interface reconstruction scheme (Geo-Reconstruct or Compressive),
Time integration approach (Explicit for sharp transients, Implicit for steady flows), and
Mesh and Courant control (refinement near interface, CFL ≈ 0.25).
Through examples like the dam-break, sloshing tanks, and falling box, the VOF model demonstrates its power to reproduce real-world multiphase behavior ranging from gentle waves to violent splashing, provided the numerical settings and physical assumptions are chosen carefully.
Cheat-Sheet Summary: VOF Workflow
Model Selection
→ Pressure-based Solver → Multiphase → VOF (Explicit or Implicit)
Phases Setup
→ Define primary (e.g., air) and secondary (e.g., water) phases
Interface Schemes
→ Explicit + Geo-Reconstruct (for sharp, transient)
or Implicit + Compressive/HRIC (for steady)
Boundary & Initial Conditions
→ Patch α = 1 in liquid zone, specify correct outlet density
Solver Controls
→ PISO / PC-SIMPLE / Coupled, Courant ≤ 0.25 (explicit)
→ Under-relaxation ≈ 0.5 (implicit), variable Δt enabled
Post-Processing
→ Plot α contours to visualize interface;
→ Check mass conservation and residuals < 1e-4.