Kai Höfler

Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations

The extension of Kynchs kinematic theory of sedimentation of monodisperse suspensions to polydisperse mixtures leads to a nonlinear system of conservation laws for the volume fractions of each species. In this paper, we show that a second-order central (Riemann-solver-free) scheme for the solution of systems of conservation laws can be employed as an efficient tool for the simulation of the settling and the separation of polydisperse suspensions. This is demonstrated by comparison with a published experimental study of the settling of a bidisperse suspension. In addition, we compare the prediction of the one-dimensional kinematic sedimentation model with a three-dimensional particle-scale simulation.

Computation of particle settling speed and orientation distribution in suspensions of prolate spheroids

A numerical technique for the dynamical simulation of three-dimensional rigid particles in a Newtonian fluid is presented. The key idea is to satisfy the no-slip boundary condition on the particle surface by a localized force-density distribution in an otherwise force-free suspending fluid. The technique is used to model the sedimentation of prolate spheroids of aspect ratio b/a=5 at Reynolds number 0⋅3. For a periodic lattice of single spheroids, the ideas of Hasimoto are extended to obtain an estimate for the finite-size correction to the sedimentation velocity. For a system of several spheroids in periodic arrangement, a maximum of the settling speed is found at the effective volume fraction φ(b/a)2≈0⋅4, where φ is the solid-volume fraction. The occurence of a maximum of the settling speed is partially explained by the competition of two effects: (i) a change in the orientation distribution of the prolate spheroids whose major axes shift from a mostly horizontal orientation (corresponding to small sedimentation speeds) at small φ to a more uniform orientation at larger φ, and (ii) a monotonic decrease of the the settling speed with increasing solid-volume fraction similar to that predicted by the Richardson–Zaki law ∝(1−φ)5⋅5 for suspensions of spheres.

Interacting Particle-Liquid Systems

We present two Euler-Lagrangian simulation methods for particles immersed in fluids described by the Navier-Stokes equation. These implement the coupling between particle and fluid phase by (i) direct integration of the stress tensor on the particle surface discretized according to the grid topology and (ii) by a tracer particle method, which employs the volume force term in the Navier-Stokes equation to emulate “rigid” body motion. Both methods have been parallelized and applied to bulk sedimentation of about 65 000 particles (in one simulation 106 particles have been simulated). We also report results for the rheology of shear-thinning suspensions, modelled by hydrodynamically interacting particles in shear flows. Aggregation occurs due to attractive, short range forces between the particles. We also address a deficiency of the MPI communication library on the CRAY T3E which had to be resolved to improve the performance of our algorithm.

Simulation of hindered settling in bidisperse suspensions of rigid spheres

Abstract We use a recently developed algorithm to simulate and study the physics of bidisperse suspensions of rigid spheres. We compute the settling velocity as a function of the ratio λ ≡ a L / a S ≥1 of the radius of the large a L and small a S particles at a solid volume fraction of φ =0.14 and compare our results to predictions of the theory by Batchelor and a semi-analytical formula by Davis and Gecol. None of these predictions agree well with our findings, but the Davis and Gecol form seems to capture some aspects of the physics in the semi-dilute regime more accurately.

Design and Application of Object Oriented Parallel Data Structures in Particle and Continuous Systems

The complexity of parallel computing hardware calls for software that eases the development of numerical models for researchers that do not have a deep knowledge of the platforms’ communication strategies and that do not have the time to write efficient code themselves. We show in three examples how two “container” data structures, one for particles, the other an array for arbitrary data types, can address the needs of many potential users of parallel machines that have so far been deterred by the complexity of parallelizing code. These examples comprise (i) parallel solutions of partial differential equations, (ii) a molecular dynamics simulation of more than 109 particles and (iii) a simulation of particles sedimenting in a viscous fluid.

Non-Brownian suspensions: simulation and linear response

We have developed and applied numerical techniques to study the dynamics of non-Brownian suspensions of spheres in viscous fluids. The numerical approaches reproduce experimental results like the mean settling velocity and pressure drops in particle arrays for solid volume fractions up to about 0.30 in 3D. In two dimensions we study the correlations between the velocity and the density distribution at small particle Reynolds numbers Re≈1 under the influence of gravity both by full hydrodynamic simulations and by linear analysis of corresponding continuum equations. For the case of the stratification of a dense homogeneous fluids on top of a less dense one, classical Rayleigh–Taylor theory predicts exponential growth of the arising initial velocity fluctuations. We find, however, that the system is driven by the initial density inhomogeneities that necessarily exist in a particulate suspension. The corresponding velocity modes saturate exponentially. The spatial correlation length of the emerging fingers grows in time until it reaches a limit, which, in our simulations, depends on the system size.

Container Size Dependence of the Velocity Fluctuations in Suspensions of Monodisperse Spheres

We show that a constraint force method can be used in an advantageous fashion to replace the no-slip boundary conditions in a Navier-Stokes simulation of rigid particles suspended in Newtonian fluids. The constraint forces mimic the presence of particles in the fluid. We use the method to study the container-size dependence of the velocity fluctuations in batch sedimentation under periodic boundary conditions and in quadrilateral containers with fixed walls.