DEM-CFD連成
Theory and Physics
Overview
Professor, what is DEM-CFD coupling? Do you calculate particles and fluid together?
Exactly. It tracks the motion of individual particles using DEM (Discrete Element Method) and solves the fluid field using CFD. By coupling them bidirectionally, it reproduces the particle-fluid interactions.
How is it different from Lagrangian particle tracking (DPM)?
The decisive difference is the treatment of particle-particle contacts. In DPM, collisions between particles are either simplified (stochastic collisions) or ignored, but in DEM, contact forces between particles are rigorously calculated using an elastic spring-dashpot model. Therefore, it is suitable for systems where inter-particle forces are important, such as powders and granules.
Governing Equations
Please tell me the equations on the DEM side.
It tracks the translational and rotational motion of each particle $i$.
$\mathbf{F}_{c,ij}$ is the contact force with particle $j$, and the Hertz-Mindlin model is representative. The normal contact force is expressed as follows.
Here, $E^*$ is the equivalent Young's modulus, $R^*$ is the equivalent radius, $\delta_n$ is the overlap amount, and $\gamma_n$ is the damping coefficient.
What about the CFD side?
It solves the locally averaged Navier-Stokes equations. The void fraction $\varepsilon_f$ due to the presence of particles is considered.
$\mathbf{S}_p$ is the reaction force from particles to fluid (momentum source term), which is the sum of fluid forces acting on all particles within a CFD cell divided by the volume.
Drag Model
How do you model the fluid forces acting on particles?
The most important is drag, using a drag model that depends on the local void fraction.
Model Applicable Range Features Ergun $\varepsilon_f < 0.8$ For packed beds Wen-Yu $\varepsilon_f > 0.8$ For dilute regions Gidaspow All ranges Switches between Ergun + Wen-Yu Di Felice All ranges Continuous transition Koch-Hill All ranges Lattice Boltzmann method database
Are there forces other than drag?
Pressure gradient force, virtual mass force, Saffman lift force, Magnus force, etc., can also be considered, but for density ratios $\rho_p / \rho_f \gg 1$ (e.g., powder-air systems), drag is dominant, and other forces can often be omitted.
Coffee Break Yomoyama Talk
Cundall's Revolution—In 1979, Particles Were Formulated to "Collide"
The founder of DEM (Discrete Element Method), Peter Cundall, while researching rock fracture mechanics, published a paper in 1979 proposing the idea of "modeling the contact force of each particle as a spring-dashpot system." Initially aimed at analyzing rock block collapse, 30 years later, DEM-CFD coupled with CFD rapidly spread as a design tool for fluidized beds, mixers, and tablet coating machines in the pharmaceutical and chemical industries. Cundall himself is said to have remarked in later years, "I never thought it would be used so widely," a prime example of a simple model changing an industry.
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 unstable and splashing, but after a while, it becomes a steady flow, right? This "during the change" is described by the temporal term. The pulsation of blood flow from a heartbeat, the fluctuation of flow each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled"—meaning setting this term to zero. Since computational cost drops significantly, solving first with steady-state 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 end of a room is also because the "carrier," air, transports heat by convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, when 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" → 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, after a while, it naturally mixes, right? That's molecular diffusion. Now, next question—honey and water, which flows easier? 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 low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms, and diffusion becomes 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? The plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—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. When you switch to compressible analysis and results suddenly become strange, 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's pushed up 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 acting on molten metal in a factory's electromagnetic pump... 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 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- 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 changes only in the buoyancy term, using constant density in other terms
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock wave 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, be careful with cross-sectional area units Pressure $p$ Pa Distinguish between gauge pressure 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 $Re = \rho u L / \mu$. Indicator for laminar/turbulent transition CFL number Dimensionless $CFL = u \Delta t / \Delta x$. Directly related to time step stability
How do you model the fluid forces acting on particles?
The most important is drag, using a drag model that depends on the local void fraction.
| Model | Applicable Range | Features |
|---|---|---|
| Ergun | $\varepsilon_f < 0.8$ | For packed beds |
| Wen-Yu | $\varepsilon_f > 0.8$ | For dilute regions |
| Gidaspow | All ranges | Switches between Ergun + Wen-Yu |
| Di Felice | All ranges | Continuous transition |
| Koch-Hill | All ranges | Lattice Boltzmann method database |
Are there forces other than drag?
Pressure gradient force, virtual mass force, Saffman lift force, Magnus force, etc., can also be considered, but for density ratios $\rho_p / \rho_f \gg 1$ (e.g., powder-air systems), drag is dominant, and other forces can often be omitted.
Cundall's Revolution—In 1979, Particles Were Formulated to "Collide"
The founder of DEM (Discrete Element Method), Peter Cundall, while researching rock fracture mechanics, published a paper in 1979 proposing the idea of "modeling the contact force of each particle as a spring-dashpot system." Initially aimed at analyzing rock block collapse, 30 years later, DEM-CFD coupled with CFD rapidly spread as a design tool for fluidized beds, mixers, and tablet coating machines in the pharmaceutical and chemical industries. Cundall himself is said to have remarked in later years, "I never thought it would be used so widely," a prime example of a simple model changing an industry.
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 unstable and splashing, but after a while, it becomes a steady flow, right? This "during the change" is described by the temporal term. The pulsation of blood flow from a heartbeat, the fluctuation of flow each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? Looking only at "after sufficient time has passed and the flow has settled"—meaning setting this term to zero. Since computational cost drops significantly, solving first with steady-state 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 end of a room is also because the "carrier," air, transports heat by convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, when 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" → 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, after a while, it naturally mixes, right? That's molecular diffusion. Now, next question—honey and water, which flows easier? 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 low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re number flows, convection overwhelms, and diffusion becomes 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? The plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—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. When you switch to compressible analysis and results suddenly become strange, 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's pushed up 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 acting on molten metal in a factory's electromagnetic pump... 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 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- 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 changes only in the buoyancy term, using constant density in other terms
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock wave 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, be careful with cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge pressure 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 | $Re = \rho u L / \mu$. 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
How is the coupling between DEM and CFD achieved?
Let me explain the basic coupling scheme.
1. Solve the fluid field with CFD time step $\Delta t_{CFD}$
2. Interpolate fluid velocity and pressure at each particle position
3. Calculate fluid forces (drag, etc.) and apply to each particle
4. Update particles with DEM time step $\Delta t_{DEM}$ (multiple substeps)
5. Recalculate void fraction from particle positions
6. Reflect particle→fluid reaction force into CFD source term
7. Proceed to next CFD step
Are the DEM and CFD time steps different?
The DEM time step is very small for contact force calculation, with a guideline of 20-30% of the Rayleigh time.
Typically, DEM executes 100 to 1000 substeps for one CFD step. This is the main cause of computational cost in DEM-CFD.
Void Fraction Calculation
How do you calculate the void fraction?
It is calculated from the volume of particles contained in a CFD cell. For particles straddling cell boundaries, the following approaches exist for handling.
| Method | Overview | Features |
|---|---|---|
| Cell Centre | Assigns entire volume to the cell containing the particle center | Simple but discontinuous |
| Divided Volume | Distributes particle volume among cells | Smoother |
| Diffusion-based | Smooths with a kernel function | Smoothest but high computational cost |
For unresolved DEM-CFD, the CFD cell size must be at least 3 to 5 times the particle diameter. If cells are smaller than particles, the definition of void fraction breaks down.
resolved vs. unresolved DEM-CFD
What is the difference between resolved and unresolved?
In unresolved, the particle diameter is smaller than the CFD mesh, and interactions are expressed using drag models. In resolved, the particle diameter is larger than the mesh, and the flow field around the particle surface is directly resolved. This is achieved using methods like Immersed Boundary or Overset Mesh, but the number of particles is limited to around several hundred.
Major Software
What tools are available for DEM-CFD coupling?
What is the difference between resolved and unresolved?
In unresolved, the particle diameter is smaller than the CFD mesh, and interactions are expressed using drag models. In resolved, the particle diameter is larger than the mesh, and the flow field around the particle surface is directly resolved. This is achieved using methods like Immersed Boundary or Overset Mesh, but the number of particles is limited to around several hundred.
What tools are available for DEM-CFD coupling?