Free Surface Flow
Theory and Physics
Overview
Professor, what is free surface flow? Is it different from the VOF method?
Free surface flow refers to the general category of flows where an interface between liquid and gas (the free surface) exists. This includes dam breaks, sloshing, ship wave-making, filling processes, and surface waves. The VOF method is a representative technique for solving it, but there are also other solution methods such as the Level Set method, SPH (Smoothed Particle Hydrodynamics) method, and Phase-Field method.
What physics is unique to free surface flow?
Surface tension and contact angle become important at small scales. At large scales, the Froude number $Fr$, which is the ratio of gravity to inertia, is the governing parameter.
CSF Model for Surface Tension
How is surface tension modeled?
The CSF (Continuum Surface Force) model by Brackbill et al. (1992) is the most widely used. It calculates it as a volume force from the interface curvature $\kappa$.
I've heard that curvature calculation accuracy is problematic.
The biggest challenge of CSF is parasitic currents (spurious currents). Unphysical velocity fields arise from discretization errors at the interface. This is particularly severe in flows with low capillary number $Ca = \mu U / \sigma$ (surface tension dominated).
Comparison of Interface Tracking Methods
| Method | Interface Representation | Mass Conservation | Interface Sharpness | Computational Cost |
|---|---|---|---|---|
| VOF Method | Volume fraction $\alpha$ | Exact | Scheme dependent | Low ~ Medium |
| Level Set Method | Signed distance function $\phi$ | Non-conservative (requires reinitialization) | Sharp | Medium |
| CLSVOF | VOF + Level Set | Good | Sharp | Medium ~ High |
| Phase-Field | Order parameter | Conservative | Diffusive | High |
| SPH | Particles | Conservative | Particle resolution dependent | High |
Which method should I choose?
In industry, the VOF method is the most widely used due to its balance of cost and accuracy. For cases where interface curvature accuracy is critical (e.g., microfluidics), CLSVOF or Phase-Field methods are advantageous. For flows involving large deformation and splashing, the SPH method is also an option.
The Mechanics of Free Surfaces—Where Bernoulli Meets Capillarity
The free surface (gas-liquid interface) is a place where the richest physical phenomena in fluid mechanics are condensed. Gravity, surface tension, viscosity, and inertia all compete, and the physics governing them is determined by the combination of dimensionless numbers (Bond number, Weber number, Capillary number, Froude number) that represent their ratios. When the Bond number Bo = ρgL^2/σ exceeds 1, gravity overwhelms surface tension; conversely, when it is small, capillary phenomena dominate. In microfluidic devices, Bo << 1, and the free surface converges to a spherical shape, while for large ship waves, Bo >> 1, and surface tension can be ignored. This scale diversity spanning over 20 orders of magnitude in Bo number makes free surface CFD a technology required across all industries.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow due to heartbeats, and the flow fluctuations each time an engine valve opens and closes are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—in other words, setting this term to zero. This significantly reduces computational cost, so trying a steady-state solution first is a basic CFD strategy.
- Convection term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. The warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while they naturally mix. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is large, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In flows with low Reynolds number (slow, viscous), diffusion is dominant. Conversely, in flows with high Re number, convection overwhelms and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: The "pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, it might be due to confusion between absolute/gauge pressure.
- Source term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by a gas stove flame, Lorentz force applied to molten metal by an electromagnetic pump in a factory... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (mean free path of molecules ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (viscosity model required for non-Newtonian fluids)
- Incompressibility assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 and above, consider compressibility effects
- Boussinesq approximation (natural convection): Consider density variation only in the buoyancy term, use constant density in other terms
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C |
| Viscosity coefficient $\mu$ | Pa·s | Note confusion with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
Details of Numerical Methods
Please tell me the key numerical points for free surface flow.
Interface Courant number management and mesh design are most important. In the VOF method, the interface Courant number $Co_\alpha < 0.25$ (max 0.5) must be maintained.
Open Channel Flow
For river and channel flows, Fluent and CFX have dedicated Open Channel Flow settings. Water level can be directly specified at inlets/outlets, making water surface initialization easy.
What are the key points for setting up Open Channel Flow?
- Inlet: Specify water level (Free Surface Level) and flow rate
- Outlet: Specify water level (Pressure outlet + Open Channel)
- Initialization: Set still water surface with Flat Initialization
- Numerical Scheme: Explicit VOF + Geo-Reconstruct (high accuracy)
Wave Generation and Absorption
For ocean engineering wave simulations, wave-making and wave-absorbing boundary conditions are necessary.
| Method | Overview | Tool Support |
|---|---|---|
| Stokes wave theory | Prescribe wave velocity/water level at inlet | Fluent, STAR-CCM+ |
| 5th-order Stokes wave | Higher-order nonlinear wave | STAR-CCM+ |
| Wave damping zone | Absorb waves with damping | All tools (UDF/Field Function) |
| Numerical wave tank | Wave-maker + damping zone | OpenFOAM (waves2Foam) |
OpenFOAM's waves2Foam is famous, right?
It's a library developed by Jacobsen et al. (DTU, 2012), implementing boundary conditions and relaxation zones for wave generation/absorption. It is widely used for wave force analysis on marine structures.
Parasitic Current Countermeasures
How can parasitic currents be suppressed?
| Countermeasure | Effectiveness | Notes |
|---|---|---|
| Mesh refinement | High | Make cells near the interface sufficiently small |
| Height Function method | Very High | For structured grids, improves curvature accuracy |
| Sharp Surface Force | High | Available in STAR-CCM+ |
| CLSVOF | High | Improves curvature calculation with Level Set |
| Adjusting cAlpha | Medium | Interface compression parameter in OpenFOAM |
The Fusion of Level Set and VOF Methods—The Advent of CLSVOF
The Level Set method can accurately calculate geometric interface information (normal, curvature) but has the weakness of incomplete mass conservation. VOF excels at mass conservation but has lower accuracy in describing interface shape. The Coupled Level-Set and VOF (CLSVOF) method proposed by Sussman et al. in the 2000s combines the strengths of both, a hybrid method that calculates curvature using the Level Set function while conserving mass with VOF. In its application to sloshing analysis of aircraft fuel tanks, benchmark results showed that CLSVOF achieved equivalent accuracy with 25% fewer cells than simple VOF.
Upwind Differencing (Upwind)
1st-order upwind: Large numerical diffusion but stable. 2nd-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
2nd-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per timestep.
Residual Monitoring
Convergence is judged when residuals for the continuity equation, momentum, and energy decrease by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2~0.3, Velocity: 0.5~0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each timestep until a steady solution converges. Number of internal iterations: 5~20
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