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We present a collocated-grid framework for Direct Numerical Simulations of polydisperse particles submerged in a viscous fluid. The fluid-particle forces are coupled with the Immersed Boundary Method (IBM) while the particle-particle forces are modeled with a combination of contact and lubrication models, adapted for collocated grids. Our method is modified from the staggered-grid IBM of previous authors to a collocated-grid IBM by adapting the fluid and particle solvers. The method scales well on high-performance parallel computing platforms. It has been validated against various cases and is able to reproduce experimental results. Tuning parameters have been thoroughly calibrated to ensure the accuracy of the method. Finally, we demonstrate the capability of the method to simulate both monodispersed and bidispersed fluidized beds and reproduce the power law relationship between the inflow velocity and the porosity., Comment: 32 pages, 17 figures

We present a method to simulate nonhydrostatic ocean flows on a horizontally-unstructured grid with a moving generalized vertical coordinate (GVC). The nonhydrostatic governing equations are transformed to a GVC system that can represent the well-known z-level, terrain-following, or isopycnal coordinates while also being able to employ a vertically-adaptive coordinate using r-adaptivity. Different vertical coordinates are accommodated with the arbitrary Lagrangian-Eulerian (ALE) approach in which the vertical coordinate lines translate vertically, and the layer heights are made consistent with the vertical grid velocities through a discrete layer-height equation. Vertical grid velocities are also accounted for in the discrete momentum and scalar trans-port equations. While momentum is approximately conserved, the mass, heat, and volume are conserved both locally and globally. The nonhydrostatic pressure is implemented using a pressure-correction method that enforces the transformed continuity equation. The proposed GVC framework is implemented in the SUNTANS (Fringer et al., 2006) ocean model. Non-hydrostatic internal solitary-like waves are simulated to demonstrate that isopycnal coordinates can represent similar dynamics as z-levels at a fraction of the computational cost. The nonhydrostatic lock-exchange is then simulated to demonstrate that adaptive vertical coordinates can improve the accuracy of the model by concentrating more grid layers in regions of higher vertical density gradients., Comment: 48 pages, 9 figures

We study the effects of fluid-particle and particle-particle interactions in a three-dimensional monodispersed reactor with unstable fluidization. Simulations were conducted using the Immersed Boundary Method (IBM) for particle Reynolds numbers of 20-70 with an Archimedes number of 23600. Two different flow regimes were identified as a function of the particle Reynolds number. For low particle Reynolds numbers ($20 < Re_p < 40$), the porosity is relatively low and the particle dynamics are dominated by interparticle collisions that produce anisotropic particle velocity fluctuations. The relative importance of hydrodynamic effects increases with increasing particle Reynolds number, leading to a minimized anisotropy in the particle velocity fluctuations at an intermediate particle Reynolds number. For high particle Reynolds numbers ($Re_p > 40$), the particle dynamics are dominated by hydrodynamic effects, leading to decreasing and more anisotropic particle velocity fluctuations. A sharp increase in the anisotropy occurs when the particle Reynolds number increases from 40 to 50, corresponding to a transition from a regime in which collision and hydrodynamic effects are equally important (Regime 1) to a hydrodynamic-dominated regime (Regime 2). The results imply an optimum particle Reynolds number of roughly 40 for the investigated Archimedes number of 23600 at which mixing in the reactor is expected to peak, which is consistent with reactor studies showing peak performance at a similar particle Reynolds number and with a similar Archimedes number. Results also show that maximum effective collisions are attained at intermediate particle Reynolds number. Future work is required to relate optimum particle Reynolds number to Archimedes number., Comment: 28 pages, 17 figures

We study confined solutions of certain evolutionary partial differential equations (pde) in 1+1 space-time. The pde we study are Lie-Poisson Hamiltonian systems for quadratic Hamiltonians defined on the dual of the Lie algebra of vector fields on the real line. These systems are also Euler-Poincare equations for geodesic motion on the diffeomorphism group in the sense of the Arnold program for ideal fluids, but where the kinetic energy metric is different from the L2 norm of the velocity. These pde possess a finite-dimensional invariant manifold of particle-like (measure-valued) solutions we call ``pulsons.'' We solve the particle dynamics of the two-pulson interaction analytically as a canonical Hamiltonian system for geodesic motion with two degrees of freedom and a conserved momentum. The result of this two-pulson interaction for rear-end collisions is elastic scattering with a phase shift, as occurs with solitons. In contrast, head-on antisymmetric collisons of pulsons tend to form singularities., Comment: 49 pages, 29 figures, animations at: http://rossby.stanford.edu/~fringer/pulsons