Distributed Lagrange Multiplier / Fictitious Domain

At the micro scale, i.e., the level of the particle, we use a Distributed Lagrange Multiplier / Fictitious Domain (DLM/FD) method to simulate the dynamics of suspension flows. The method has been proposed in the literature in the late nineties by Glowinski and co-workers and assorted variants have been suggested since the original version. Other options in the literature include Immersed Boundary method (IBM, very popular in the community), Force Coupling method, lattice Boltzmann method and Arbitrary Lagrangian-Eulerian method. The DLM/FD method belongs to the family of fixed mesh approaches. The fixed mesh feature implies that the mesh around solid bodies in non-boundary fitted and that additional body terms are added to the right hand side term of the fluid momentum equations. Though DLM/FD is based on different arguments, it can be viewed as a kind of IBM in which the addtional body terms, i.e., the hydrodynamic force and torque, and the particles velocity are implicitly determined by solving a saddle-point problem with an Uzawa algorithm. In contrast, in the popular direct forcing variant of IBM, these terms are explicity determined. Among the assorted variants of DLM/FD, a direct forcing version has also been suggested (in the same spirit as direct forcing IBM) by some authors, as well as a non-Lagrange multiplier based FD.

We list below some specific features of our DLM/FD implementation:

• a collocation point method for the discretization of the Lagrange multiplier space combined with a Uzawa algorithm for the solution of the saddle-point problem,

• 2 discretization schemes are available: Finite Element and Staggered Grid/Finite Volume, the latter being restricted to structured, cartesian and orthogonal grids,

• collisions between particles are handled with the aforementioned DEM method (PeliGRIFF is linked to our granular solver Grains3D), which enables us to consider non-spherical particles and true geometric contact in our suspension flows simulations,

• our DLM/FD method is implemented both for momentum and heat transfer, which implies that non-isothermal suspension flows can be studied,

• the whole solution algorithm is of the operator-splitting type. The incompressible Navier & Stokes sub-problem can be solved with 2nd order time accuracy but the whole accuracy is only 1st order due to the 1st order decoupling of the DLM/FD problem from the incompressible Navier & Stokes equations, 

• the incompressible Navier & Stokes equations are solved with a 2nd order time accurate classical L2 projection method in which the pressure laplacian linear system is solved by a powerful Boomer AMG preconditioned conjugate gradient algorithm,

High fidelity DNS at the micro scale is expected to supply computed solutions of high accuracy since the  mesh is much finer than the smallest particle in the particle (cf Figure 1). Provided the method is properly validated, these simulations are deemed to be reliable and serve as a reference in the multi-scale analysis. Their purposes is twofold: (i) help to understand the fundamental (both momentum and heat) transfer in suspension flows and thus extend our comprehension of the assorted intricate mechanisms involved in such flows, and (ii) provide enhanced closure laws for meso-scale simulations.

Fig 1. Particles on an Eulerian grid in micro-scale DNS

  Two-way Euler / Lagrange (DEM-CFD)

At the meso scale, particles are still individually tracked in a Lagrangian way as in purely granular flows or DNS, while the incompressible Navier & Stokes equations are solved in an average way. While at the micro scale the complete kinematics around each solid body is resolved and hence no closure law is required for the momentum transfer, meso-scale simulations imply to use such a closure law to compute the hydrodynamic force & torque. Actually, in most implementations, the hydrodynamic torque is neglected while the hydrodynamic force is often limited to the hydrodynamic drag. This numerical model is generally named two-way Euler-Lagrange or DEM-CFD in the literature.

At the discrete level, while DNS require the grid size to be at most one tenth the smallest particle diameter, DEM-CFD simulations at the meso-scale require that the grid size is at least twice the largest particle diameter (in order to compute proper averages at the cell level, cf Figure 2). Hence, much larger simulations in terms of number of particles (and thus of system size) can be performed at the meso-scale but with a lower accuracy since they strongly depend on the chosen hydrodynamic closure law. A fair number of hydrodynamic closure laws is available in the literature, the most popular being Ergun's, with variable validity ranges in terms of volume fraction and Reynolds number.

Our DEM-CFD model is implemented with the Staggered Grid/Finite Volume scheme only and is thus restricted to box-like flow domains in 3D. Up to now, it is essentially employed in our group to look at fluidized bed problems.

Fig 2. Particles on an Eulerian grid in meso-scale DEM-CFD simulations