Multiple eigenmodes of the Rayleigh-Taylor instability observed for a fluid interface with smoothly varying density. II. Asymptotic solution and its interpretation

It was observed in the first part of this work [C. X. Yu et al., Phys. Rev. E 97, 013102 (2018)] that a Rayleigh-Taylor flow with a smoothly varying density at the interface permits a multiplicity of solutions for instability modes. Based on numerical solutions of the eigenvalue problem, a fitting expression for the multiple eigenmodes of the Rayleigh-Taylor instability was provided. However, the fitted curves showed poor agreement with the numerical solutions when the Atwood number was relatively high. This paper develops an asymptotic solution based on the Wentzel-Kramers-Brillouin approximation for high wave numbers in the direction tangential to the interface. The asymptotic solution of the eigenmode of each order can provide a fine prediction for moderate and high wave numbers as confirmed by a comparison with numerical solutions and, more importantly, the physical interpretation of the multiple-mode phenomenon is exhibited. We also show simpler expressions of the growth rates when the Atwood number approaches 0 or 1.

Figure 1

Sketch of the physical problem.

Figure 2

Distribution of $\mathrm{\Pi }\left(z\right)$ and locations of $z_A$ and $z_B$.

Figure 3

Five-layer structure of the WBK solution of $\widehat{u}$.

Figure 4

RTI linear growth rates $\gamma$ vs wave number $k$ for (a) $A=0.2$, (b) $A=0.4$, (c) $A=0.6$, and (d) $A=0.8$.

Figure 5

Eigenfunctions of $\widehat{u}$ for $A=0.8$ and $k=10$, where curves A1–A4 are the four modes from the asymptotic solution and curves N1–N4 are those from the numerical solution.

Figure 6

Comparison between (12) and (20) for $A\to 1$, where $k=50$.

Figure 7

Comparison between (12) and (26) for $A\to 0$, where $k=20$.

Figure 8

Composite solution ${\widehat{u}}_{\mathrm{com}}$ and the asymptotic solutions ${\widehat{u}}^i$ ($i=\text{I}–\text{V}$) for mode 3 with $A=0.8$ and $k=10$.