Solidification and Casting Simulation
Theory and Physics
Overview
Professor, what does CFD for solidification calculate?
It predicts the process where molten metal cools and solidifies. It deals with flow and heat transfer problems involving solid-liquid phase changes, such as mold filling and solidification analysis in casting, shell growth in continuous casting, bead formation in welding, and molten pool behavior in metal 3D printing (PBF/DED).
You handle liquid turning into solid using fluid dynamics?
The Enthalpy Method (Enthalpy-Porosity method) is the standard approach. It does not explicitly track the solid-liquid interface; instead, it represents the progress of solidification using the liquid fraction $f_L$ (a continuous variable from 0 to 1) in each cell.
Governing Equations
Please tell me the equations for the Enthalpy Method.
The energy equation is described in terms of enthalpy $H$.
Enthalpy is the sum of sensible heat and latent heat.
$h = \int_{T_{ref}}^{T} c_p dT$ is the sensible heat, $f_L$ is the liquid fraction, and $L$ is the latent heat of solidification. The liquid fraction is determined from the liquidus temperature $T_L$ and solidus temperature $T_S$.
What happens to the flow in the solidified part?
In the Enthalpy-Porosity method, the solidified region is treated as a porous medium with high resistance. A Darcy drag source term is added to the momentum equation.
$C$ is the Mushy zone constant (typical values $10^5$ to $10^8$), $\epsilon$ is a small constant to avoid division by zero, and $\mathbf{u}_{pull}$ is the pulling speed (in continuous casting). As $f_L \to 0$ (complete solidification), the source term approaches infinity, causing the velocity to decay to zero.
Marangoni Convection
What are the driving forces for flow in a molten pool?
Marangoni convection, caused by the temperature dependence of surface tension ($d\sigma/dT$), dominates the flow pattern in the molten pool.
$$ \tau_{Ma} = \frac{d\sigma}{dT} \nabla_s T $$
For many metals, $d\sigma/dT < 0$ (surface tension is lower at higher temperatures), causing an outward flow from the center of the molten pool. However, the sign can reverse depending on oxygen or sulfur content, dramatically changing the flow pattern.
Coffee Break Trivia
The Microscopic World of Solidification—Dendrite Growth and the Stefan Problem
The physics of metal solidification is formulated as a Stefan problem: "How does the solid-liquid interface move?" For pure metals, the interface moves along the melting point isotherm, and its speed is determined by the balance between latent heat and heat flux. However, in actual alloys, constitutional supercooling occurs locally, leading to the spontaneous formation of complex dendritic solid-liquid interfaces. The Phase-Field method, which directly calculates this dendrite growth in CFD, has rapidly developed since the 1990s, and by the 2020s, dendrite calculations with 10^6 meshes became possible on GPUs at speeds close to real-time.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine 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 "state of change" is described by the temporal term. The pulsation of blood flow with each heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are 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" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's 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, it naturally mixes after a while. 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 dominates. 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 piston 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 densely 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/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings and 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 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 heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (viscosity model needed for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach number ≥ 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 $\nu = \mu/\rho$ [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.
What are the driving forces for flow in a molten pool?
Marangoni convection, caused by the temperature dependence of surface tension ($d\sigma/dT$), dominates the flow pattern in the molten pool.
For many metals, $d\sigma/dT < 0$ (surface tension is lower at higher temperatures), causing an outward flow from the center of the molten pool. However, the sign can reverse depending on oxygen or sulfur content, dramatically changing the flow pattern.
The Microscopic World of Solidification—Dendrite Growth and the Stefan Problem
The physics of metal solidification is formulated as a Stefan problem: "How does the solid-liquid interface move?" For pure metals, the interface moves along the melting point isotherm, and its speed is determined by the balance between latent heat and heat flux. However, in actual alloys, constitutional supercooling occurs locally, leading to the spontaneous formation of complex dendritic solid-liquid interfaces. The Phase-Field method, which directly calculates this dendrite growth in CFD, has rapidly developed since the 1990s, and by the 2020s, dendrite calculations with 10^6 meshes became possible on GPUs at speeds close to real-time.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Imagine 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 "state of change" is described by the temporal term. The pulsation of blood flow with each heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are 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" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There's 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, it naturally mixes after a while. 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 dominates. 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 piston 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 densely 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/gauge pressure.
- Source Term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings and 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 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 heated room in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (viscosity model needed for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach number ≥ 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 $\nu = \mu/\rho$ [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 tell me the numerical points for solidification simulation.
The biggest challenge is handling the latent heat of solidification. The release of latent heat creates nonlinearity in enthalpy and causes abrupt changes in the temperature field. Time step management and convergence of iterative calculations are key.
Influence of Mushy Zone Constant
How do you set the Mushy zone constant $C$?
The value of $C$ has a significant impact on the results.
| Value of $C$ | Effect | Application |
|---|---|---|
| $10^4$ to $10^5$ | Gentle damping | Slow solidification, continuous casting |
| $10^5$ to $10^6$ | Standard | General casting |
| $10^7$ to $10^8$ | Rapid damping | Rapid solidification, welding |
Ideally, it should be determined from experimental data on Darcy permeability, but in practice, it's common to start with $10^5$ to $10^6$ and perform a sensitivity analysis.
Coupling with VOF Method
How is the mold filling process in casting calculated?
The filling process tracks the free surface of the molten metal using the VOF method, and solidification is calculated after filling is complete (or simultaneously with filling). In Fluent or Flow-3D, simultaneous calculation with VOF + Solidification/Melting models is possible.
Flow patterns during filling, oxide film entrapment, and gas entrainment directly lead to casting defects, making filling analysis a crucial element of casting simulation.
Implementation by Tool
| Tool | Solidification Model | VOF Coupling | Stress Analysis Linkage |
|---|---|---|---|
| Ansys Fluent | Solidification/Melting | Supported | Mechanical Integration |
| Flow-3D | TruVOF + Solidification | Native | Limited |
| STAR-CCM+ | Solidification | Supported | Structural Coupling |
| ProCAST (ESI) | FEM Solidification + Filling | Dedicated Filling | Integrated Stress/Deformation Analysis |
| MAGMASOFT | Proprietary FVM | Proprietary Model | Residual Stress Prediction |
ProCAST and MAGMASOFT are dedicated casting software, right?
That's correct. They are specialized tools dedicated to casting processes, capable of integrated calculation of filling, solidification, shrinkage prediction, residual stress, and deformation. They are more equipped with casting-specific physics models (e.g., feeding, Niyama criterion) than general-purpose CFD.
Enthalpy-Porosity Method—The Standard Approach for Practical Solidification CFD
The most widely used method in industrial solidification CFD (continuous casting, mold filling) is the Enthalpy-Porosity method. It treats the solid-liquid coexistence region (mushy zone) as a "porous medium with low permeability" and reproduces the progress of solidification by suppressing local velocity using the Kozeny-Carman equation. This approach was proposed by Voller & Prakkash in 1987 and is implemented in ANSYS Fluent's solidificationMeltingSource and OpenFOAM. The model characterization constant (Amush) often uses a default value of 1.6x10^5, but if this value is inappropriate, the solidification speed can deviate by 30-50%.
Upwind Scheme
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
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 shock waves 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 time step.
Residual Monitoring
Convergence is judged when residuals for continuity, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
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 time step until a steady solution converges. Internal iteration count: 5-20 times is a guideline. If residuals fluctuate between time steps, review the time step size.
SIMPLE
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