Resolved CFD-DEM#

In resolved CFD-DEM, the incompressible Navier-Stokes equations are solved on a mesh that is significantly finer than the particle size. The particle-fluid coupling is then obtained directly by imposing a no-slip boundary condition on the surface of the moving particles [1].

Sharp-interface immersed boundary method#

Lethe uses a Sharp-Interface Immersed Boundary Method (SIBM) to impose the no-slip boundary condition on moving solids while coupling the solid dynamics and fluid flow implicitly. This implicit formulation keeps the resolved CFD-DEM solver stable even for large fluid-to-solid density ratios.

The SIBM retains the order of convergence of the underlying finite element discretization (up to fourth order) for both stationary and moving boundaries [2] [3] and its full algorithmic details are presented in [4].

Hydrodynamic forces and lubrication correction#

The hydrodynamic force \(\mathbf{F}^\mathrm{pf}\) and torque \(\mathbf{M}^\mathrm{pf}\) acting on each particle are evaluated by integrating the fluid stress tensor over the particle surface (see the particle momentum balance in Granular Flows - Discrete Element Method (DEM)). When two particles approach, the finite grid resolution can under-resolve the lubrication force, so Lethe adds the correction proposed by [5]:

\[\begin{split}\mathbf{F}^\mathrm{l_c}_{ij} &= \mathbf{F}^\mathrm{l}_{ij} - \mathbf{F}^\mathrm{l}_{ij} \big|_{l=\varepsilon_0} \\ \mathbf{F}^\mathrm{l}_{ij} &= \frac{3}{2} \pi \mu_f \left (\frac{d_{p_i} d_{p_j}}{d_{p_i} + d_{p_j}} \right )^2 \frac{1}{l} \left (\mathbf{v}_{ij} \cdot \mathbf{e}_{ij} \right ) \mathbf{e}_{ij}\end{split}\]

where:

  • \(\mu_f\) is the fluid dynamic viscosity;

  • \(d_{p}\) is the particle diameter;

  • \(l\) is the gap between the particles;

  • \(\varepsilon_0\) is a minimum gap used to regularize the lubrication force;

  • \(\mathbf{v}_{ij}\) is the relative velocity between particles \(i\) and \(j\);

  • \(\mathbf{e}_{ij}\) is the unit vector pointing from particle \(i\) to \(j\).

The model is derived for spheres; it can be enabled for non-spherical solids, but results should be interpreted with care.

Signed distance functions for complex solids#

The SIBM supports multiple solid descriptions: spheres, cylinders, boxes, CAD geometries in step/iges/stl formats, Radial Basis Function (RBF) surfaces [6], and composites defined using boolean operations. Each solid is represented through a Signed Distance Function (SDF) \(\lambda\). For every node adjacent to a cut cell where \(\lambda(\mathbf{x})=0\), the method extrapolates along the outward normal

\[\mathbf{n}(\mathbf{x}) = \nabla \lambda (\mathbf{x})\]

using Lagrange polynomials. The extrapolated values are used both to enforce the no-slip boundary condition at the moving interface and to evaluate stresses on the particle surface.