  The late-time growth of single-mode immiscible Rayleigh-Taylor instability is
investigated over a comprehensive range of the Reynolds numbers ($1\leq Re \leq
10000$) and Atwood numbers $(0.05 \leq A \leq 0.7)$ using an improved lattice
Boltzmann multiphase method. We first reported that the instability with a
moderately high Atwood number of 0.7 undergoes a sequence of distinguishing
stages at high Reynolds numbers, named as the linear growth, saturated velocity
growth, reacceleration and chaotic development stages. The spike and bubble at
the secondary stage evolve with the constant velocities and their values agree
well with the potential flow theory. Owing to the increasing strengths of the
vortices, the spike and bubble are accelerated with velocities exceeding than
the asymptotic values and the evolution of the instability enters into the
reacceleration stage. Lastly, the curves for the spike and bubble velocities
have some fluctuations at the chaotic stage and also a complex interfacial
structure with large topological change is observed, while it still preserves
the symmetry property. To determine the nature of the late-time growth, we also
calculated the spike and bubble growth rates by using five popular statistical
methods and two comparative techniques are recommended, resulting in the spike
and bubble growth rates of about 0.13 and 0.022, respectively. When the
Reynolds number is gradually reduced, some later stages cannot be reached
successively and the structure of the evolutional interface becomes relatively
smooth. Moreover, we observe that the spike growth rate shows an overall
increase with the Atwood number, while it has little influence on the bubble
growth rate being approximately 0.0215.
