Numerical computation of the Rayleigh-Taylor instability for a viscous fluid with regularized interface properties

In this article, the computation of the linear growth rates and eigenfunctions of the viscous version of the Rayleigh-Taylor instability by numerically solving the corresponding eigenvalue problem in the case of one-dimensional (1D) and two-dimensional (2D) geometries is studied. The 1D version is first validated in the particular inviscid case to be compared to the previous literature. The most unstable mode, also known as the first mode, which has the maximal linear growth rate has been extensively studied in previous literature. Higher modes have smaller eigenvalues, but the corresponding eigenfunctions present a more complex structure that contains multipeak shapes. In the extension to the 2D geometry, the length of the domain limits the wave number of the eigenvectors computed. In the extension to the 2D geometry the length of the domain limits the wave number of the eigenvectors computed. The importance of extending the results to the two-dimensional case is twofold. First, it opens up the possibility of generalizing the computation to more complex geometries that could contain fixed or floating objects and, second, allows the computation of flow instabilities in nonzero basic flows that could come from the steady Navier-Stokes solutions.

Figure 1

Scheme of the geometry used for the 2D viscous simulation. A dashed line indicates the $y=0$ coordinate. The boundary conditions are also written at each boundary.

Figure 2

Growth rate $\sigma$ of the most unstable eigenvalue in the inviscid case versus the wave number $k$ for different values of the Atwood number $A_{\rho }=0.2,0.4,0.6,0.8$ for $H=1$ and $L_s=0.01$. The expression (43) is added in black lines for comparison.

Figure 3

Growth rate $\sigma$ of the most unstable and three less stable internal modes versus the wave number $k$ for $A_{\rho }=0.2$, $L_s=0.203$, and $H=1$. The results from the analytic expression (44) when $H=1$ have been added for comparison.

Figure 4

Eigenvectors of the vertical velocity component $\tilde{v}$ corresponding to the four most unstable modes $m=1$ (a), $m=2$ (b), $m=3$ (c), and $m=4$ (d) for the wave numbers $k=4$ and $k=6$ for $A_{\rho }=0.2$. Other parameters are $L_s=0.01$ and $H=1$.

Figure 5

Eigenvectors of the vertical velocity component $\tilde{v}$ corresponding to the four most unstable modes $m=1$ (a), $m=2$ (b), $m=3$ (c), and $m=4$ (d) for the wave numbers $k=1$ and $k=4$ for $A_{\rho }=0.8$. Other parameters are $L_s=0.203$ and $H=6$.

Figure 6

Dependence of the growth rate $\sigma$ on $k$ for $A_{\rho }=A_{\mu }=0.1$ (a), $A_{\rho }=A_{\mu }=0.5$ (b), and $A_{\rho }=A_{\mu }=0.9$ (c). The upper and the lower curves correspond to $H=4$ and $H=2$, respectively. Previous numerical calculations [8] and spectral method ($L_s=0.01$ and $N=256$) are plotted.

Figure 7

Two most unstable vertical velocity, density and pressure eigenvectors $m=1$ (a) and $m=2$ (b) for the 1D domain in the viscous case, where $k=\pi A_{\rho }=A_{\mu }=0.9$, $L_s=0.5$, and $H=1$.

Figure 8

Dependence of the viscous growth rate $\sigma$ on $k$ for $A_{\rho }=A_{\mu }=0.9$, $L_s=0.5$, and $H=1$ for the four least unstable modes $m=1,2,3,4$.

Figure 9

Mesh used for the 2D viscous simulation.

Figure 10

Dependence of the viscous growth rate $\sigma$ on $k$ for $A_{\rho }=A_{\mu }=0.8,0.6,0.4,0.2$ and $H=1$. Referenced numerical calculations [8] and FEM in 2D ($L_s=0.01$) are plotted.

Figure 11

Most unstable perturbation for the case $k_x=\pi$, $H=1$, $L=4$, and $A_{\mu }=A_{\rho }=0.9$. (a) Density, (b) pressure, (c) horizontal velocity, and (d) vertical velocity. The computation was performed with the interface parameter $L_s=0.5$.

Figure 12

Second-most unstable perturbation for the case $k_x=\pi$, $H=1$, and $A_{\mu }=A_{\rho }=0.9$. (a) Density, (b) pressure, (c) horizontal velocity, and (d) vertical velocity. The computation was performed with the interface parameter $L_s=0.5$.