DEM-CFD Coupling
DEM-CFD Coupling: Theoretical Foundations
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.
Cundall's RevolutionβIn 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 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.
Computational Methods for DEM-CFD Coupling
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 particleβfluid 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.