  Rayleigh-Taylor instability (RTI) as a multi-scale, strongly nonlinear
physical phenomenon which plays an important role in the engineering
applications and scientific research. In this paper, the mesoscopic lattice
Boltzmann method is used to numerically study the late-time evolutional
mechanism of three-dimensional (3D) single-mode RTI and the influences of
extensive dimensionless Reynolds number and Atwood number on phase interfacial
dynamics, spike and bubble growth are investigated in details. For a high
Reynolds number, it is reported that the development of 3D single-mode RTI
would undergo four different stages: linear growth stage, saturated velocity
growth stage, reacceleration stage and turbulent mixing stage. A series of
complex interfacial structures with large topological changes can be observed
at the turbulent mixing stage, which always preserve the symmetries with
respect to the middle axis at a low Atwood number, and the lines of symmetry
within spike and bubble are broken as the Atwood number is increased. Five
statistical methods for computing the spike and bubble growth rates are then
analyzed to reveal the growth law of 3D single-mode RTI in turbulent mixing
stage. It is found that the spike late-time growth rate shows an overall
increase with the Atwood number, while the bubble growth rate seems to be
independence of the Atwood number, approaching a constant of around 0.1. When
the Reynolds number decreases, the later stages cannot be reached gradually and
the evolution of phase interface presents a laminar flow state.
