  Fluid motion driven by thermal effects, such as that due to buoyancy in
differentially heated three-dimensional (3D) enclosures, arise in several
natural settings and engineering applications. It is represented by the
solutions of the Navier-Stokes equations (NSE) in conjunction with the thermal
energy transport equation represented as a convection-diffusion equation (CDE)
for the temperature field. In this study, we develop new 3D lattice Boltzmann
(LB) methods based on central moments and using multiple relaxation times for
the three-dimensional, fifteen velocity (D3Q15) lattice, as well as it subset,
i.e. the three-dimensional, seven velocity (D3Q7) lattice to solve the 3D CDE
for the temperature field in a double distribution function framework. Their
collision operators lead to a cascaded structure involving higher order terms
resulting in improved stability. In this approach, the fluid motion is solved
by another 3D cascaded LB model from prior work. Owing to the differences in
the number of collision invariants to represent the dynamics of flow and the
transport of the temperature field, the structure of the collision operator for
the 3D cascaded LB formulation for the CDE is found to be markedly different
from that for the NSE. The new 3D cascaded (LB) models for thermal convective
flows are validated for natural convection of air driven thermally on two
vertically opposite faces in a cubic cavity enclosure at different Rayleigh
numbers against prior numerical benchmark solutions. Results show good
quantitative agreement of the profiles of the flow and thermal fields, and the
magnitudes of the peak convection velocities as well as the heat transfer rates
given in terms of the Nusselt number.
