  The gas-solid particle two-phase flow in a fluidized bed shows complex
physics. Following our previous work, the multi-scale framework based on
gas-kinetic scheme (GKS) and unified gas-kinetic wave-particle method (UGKWP)
for the gas-particle system is firstly extended to the three-dimensional
simulation of the fluidized bed. For the solid particle evolution, different
from the widely-used Eulerian and Lagrangian approaches, the UGKWP unifies the
wave (dense particle region) and discrete particle (dilute particle region)
formulation seamlessly according to a continuous variation of particle cell's
Kundsen number ($Kn$). The GKS-UGKWP for the coupled gas-particle evolution
system can automatically become an Eulerian-Eulerian (EE) method in the high
particle collision regime and Eulerian-Lagrangian (EL) formulation in the
collisionless particle regime. In the transition regime, the UGKWP can achieve
a smooth transition between the Eulerian and Lagrangian limiting formulation.
More importantly, the weights of mass distributions from analytical wave and
discrete particle are related to the local $Kn$ by $\exp(-1/Kn)$ for wave and
$(1-\exp(-1/Kn))$ for discrete particle. As a result, the UGKWP provides an
optimal modeling for capturing the particle phase in terms of physical accuracy
and numerical efficiency. In the numerical simulation, the UGKWP does not need
any prior division of dilute/dense regions, which makes it suitable for the
fluidized bed problem, where the dilute/transition/dense regions
instantaneously coexist and are dynamically interconvertible. In this paper,
based on the GKS-UGKWP formulation two lab-scale fluidization cases are
simulated in 3D and the simulation results are compared with the experimental
measurements. The typical heterogeneous flow features of the fluidized bed are
well captured and the statistics are in good agreement with experiment data.
