Liquid Film Model
Theory and Physics
Overview
Professor, what is a liquid film model?
It's a model that calculates the flow, evaporation, and splashing of a thin liquid film formed on a wall surface. It predicts the behavior of liquid films flowing on walls, such as rainwater on an automobile windshield, icing on aircraft wings, fuel films on engine interior walls, and paint coatings.
Is it different from solving for a wall film using the VOF method?
Since the film thickness is extremely thin, ranging from tens of micrometers to a few millimeters, directly resolving it with the VOF method would require an impractically fine mesh. The liquid film model describes the film using 2D shell equations on the wall surface, allowing for efficient computation independent of the 3D mesh.
Governing Equations
Please explain the equations for the liquid film.
The equation describing the mass conservation of the liquid film (change in film thickness) is as follows.
$h$ is the liquid film thickness, $\bar{\mathbf{u}}_f$ is the film-thickness-averaged liquid film velocity, and $\nabla_s$ is the gradient operator along the wall surface. The source terms on the right-hand side represent mass changes due to droplet impingement, evaporation, and splashing, respectively.
How is the liquid film velocity determined?
We use the thin-film approximation (lubrication theory). The velocity profile inside the film becomes a parabolic distribution due to the no-slip condition at the wall and the balance of shear forces (shear stress $\tau_g$ from the airflow) at the film surface. Averaging over the film thickness gives:
The first term is the driving force due to the pressure gradient and the tangential component of gravity, and the second term is the driving force due to airflow shear. The energy equation for the liquid film is also solved similarly using the thin-film approximation to calculate the evaporation rate.
Droplet-Wall Interaction
How is the behavior of a droplet impacting a wall surface modeled?
The impact regime is determined by the Weber number and wall temperature.
| Regime | Condition | Behavior |
|---|---|---|
| Stick | $We < We_{cr,low}$ | Adheres to the wall |
| Rebound | High-temperature wall | Reflects elastically |
| Spread | Moderate $We$ | Spreads and forms a liquid film |
| Splash | $We > We_{cr,high}$ | Splashes and generates secondary droplets |
The Stanton-Rutland model and the Bai-Gosman model are representative and are implemented in Fluent and STAR-CCM+.
The Complexity Born from Thinness—Governing Equations at the µm Scale
A wall film is an extremely thin liquid layer with a thickness of 1–1000 µm, appearing in diverse locations from aircraft icing and engine wall cooling to the gastric mucus layer. The Thin Film Approximation assumes a parabolic velocity profile in the thickness direction, reducing the three-dimensional Navier-Stokes equations to two-dimensional thin-film equations. Marangoni convection (flow driven by surface tension differences due to temperature or concentration gradients) occurring on the film surface directly affects coating uniformity in painting processes and film non-uniformity in heat exchangers, making it a phenomenon of high practical importance.
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 the heartbeat, 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"—meaning this term is set to zero. Since computational cost is significantly reduced, starting with a steady-state solution 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. 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 things" → 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, it naturally mixes, right? 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 high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, 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 plunger side is high pressure, and 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: "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 and 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" and are expressed by source terms. What happens if you forget a source term? In natural convection analysis, forgetting to include buoyancy means 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 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (viscosity model required for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. For Mach numbers above 0.3, compressibility effects must be considered.
- Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms.
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc., required)
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 | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion 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 explain the numerical solution method for the liquid film model.
The liquid film is solved on the wall surface mesh. Independently of the 3D CFD mesh, the transport equations for film thickness, velocity, and temperature are solved using the 2D connectivity information of the wall boundary faces.
The coupling between the gas-phase CFD ↔ liquid film model is performed via the following information exchange.
| Gas Phase → Liquid Film | Liquid Film → Gas Phase |
|---|---|
| Wall shear stress $\tau_g$ | Mass source due to evaporation |
| Temperature & concentration near the wall | Heat source due to evaporation |
| DPM droplet wall impingement | Splashed droplets from the film |
| Wall pressure distribution | Roughness effect of the film surface |
So DPM droplets impact the wall and become a film, and then they can break up and become droplets again.
Implementation by Tool
| Tool | Liquid Film Model Name | Main Features |
|---|---|---|
| Ansys Fluent | Eulerian Wall Film | Liquid film flow, evaporation, splash, DPM coupling |
| STAR-CCM+ | Thin Film Model | Liquid film flow, heat transfer, evaporation, splashing |
| OpenFOAM | regionFaModel | Finite Area Method, basic liquid film flow |
| Ansys CFX | Wall Film (limited) | Basic liquid film tracking |
So Fluent and STAR-CCM+ are well-developed in this area.
Since the demand for liquid film models is high in the automotive and aerospace industries, these two tools have the most mature implementations. OpenFOAM's regionFaModel is based on the Finite Area Method and is suitable for research-oriented customization.
Thin-Film Numerical Solution—Unified Treatment of Wall Curvature and Gravity
In CFD implementation of wall films, the shell element approach is effective for liquid film flow on complex-shaped walls. The integral method, integrating in the wall-normal direction, derives transport equations for film thickness h and average velocity. ANSYS Fluent's wall film model treats gravity, pressure gradient, shear stress, evaporation, and condensation uniformly as source terms and is widely used for predicting engine wall oil film behavior. However, for thick films (h > ~1 mm) or turbulent films, the thin-film approximation may break down, necessitating a switch to 3D VOF.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing (Central Differencing)
Second-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 methods: CFL ≤ 1 is the stability condition. Implicit methods: 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 typically judged when the residuals for the continuity equation, momentum, and energy drop by 3–4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Typical initial values: Pressure: 0.2–0.3, Velocity: 0.5–0.7. Reduce the factor if divergence occurs. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each timestep until a steady solution converges. Internal iteration count: 5–20 iterations is a guideline. If residuals fluctuate between timesteps, review the timestep size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and then velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—this discretization method reflects the physics that upstream information determines downstream conditions. Although it's first-order accurate, it is highly stable because it correctly captures the flow direction.
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