Three-dimensional viscous Rayleigh-Taylor instability at the cylindrical interface

In this paper, the rotational part of the disturbance flow field caused by viscous Rayleigh-Taylor instability (RTI) at the cylindrical interface is considered, and the most unstable mode is revealed to be three-dimensional for interfaces of small radii $R$. With an increase in $R$, the azimuthal wave number of the most unstable mode increases step by step, and the corresponding axial wave number increases as well at each step of the azimuthal wave number. When the amplitude of the wave-number vector is much larger or much smaller than $1/R$, the cylindrical RTI is close to the semi-infinite planar viscous RTI limit or the finite-thickness creeping-flow RTI limit, respectively. The effect of the viscosity ratio is double-edged; it may enhance or suppress the cylindrical RTI, depending on $R$ and the amplitude range of the wave-number vector.

Figure 1

Schematic of Rayleigh-Taylor instability. The cylindrical interface is subjected to an inward deceleration.

Figure 2

The isocontours of the growth rate $\tilde{\sigma }$ in the wave-number $n\text{−}\tilde{k}$ space. $A_t=0.5$ and $S=1$. $(n,\tilde{k})$ of the most unstable modes are (1, 0.134), (1, 0.206), (1, 0.226), (1, 0.251), (2, 0.249), (2, 0.311), and (3, 0.344) for (a)–(g), respectively. The growth rate is almost axisymmetric for case (h).

Figure 3

The perturbed interface of the most unstable mode, $(n,\tilde{k})=(2,0.311)$, at $(R,A_t,S)=(8,0.5,1)$ is shown in (a), and the corresponding isosurfaces of the positive (red areas) and negative (yellow areas) axial components of vorticity are shown in (c). As references, the corresponding interface and vorticity isosurfaces of the superposed modes with $(n,\tilde{k})=(2,0.311)$ and $(2,-0.311)$ are shown in (b) and (d), respectively.

Figure 4

(a) The azimuthal wave number $n$, (b) the axial wave number $\tilde{k}$, (c) the total wave number $\widetilde{k_t}$, and (d) the angle of the wave-number vector of the most unstable mode as functions of $R$ at $A_t=0.5$, $S=1$.

Figure 5

The growth rate $\tilde{\sigma }$ as a function of $\widetilde{k_t}$ at different values of $R$. $A_t=0.5$, $S=1$, and $n=0$. The result of semi-infinite planar viscous RTI [7] is shown as the dot-dashed line, and the growth rates for planar RTI with different $h_1$ values are shown as gray curves for reference.

Figure 6

The growth rate $\tilde{\sigma }$ as a function of $\widetilde{k_t}$ at different values of $n$. The solutions for the planar interface between semi-infinite viscous fluids [7] are shown as solid red lines as references. $A_t=0.5$, $S=1$. Inset in (a): Illustration that the maximum $\tilde{\sigma }$ corresponds to a three-dimensional mode of $n=1$ and $\tilde{k}$ is about 0.13.

Figure 7

The growth rate ${\tilde{\sigma }}_c$ as a function of ${\widetilde{k_t}}_c$ at different $A_t$ and $S$ values. Inset in (a): Illustration that the ${\tilde{\sigma }}_c$ curve for $A_t=0.9$ and $n=1$ has a maximum value when ${\tilde{k}}_c$ is about 0.15. The growth rates for planar RTI with $h_{1c}=10$ are shown as dotted curves in (d) for reference.