Multiple eigenmodes of the Rayleigh-Taylor instability observed for a fluid interface with smoothly varying density. III. Excitation and nonlinear evolution

A compressible Rayleigh-Taylor (RT) unstable flow with a diffuse interface where the density varies smoothly supports a multiplicity of unstable eigenmodes that grow exponentially in time. This paper studies two relevant problems in a two-dimensional (2-D) RT flow by use of multimode decomposition. The first one is the excitation of these modes by arbitrarily introduced infinitesimal perturbations. The initial amplitude of each mode is calculated by projecting the introduced perturbation into the eigenvector space, and the obtained initial amplitude can be used to predict the linear evolution of the perturbation accurately, as is confirmed by direct numerical simulations. If the initial perturbation includes pressure or velocity components, then in addition to the growth of the RT mode, a strong acoustic wave package would be radiated to the heavy fluid, whereas a mild acoustic wave package would be radiated to the light fluid. As the second problem, in order to reveal the interaction feature of the unstable normal modes in the nonlinear phase, we perform the multimode decomposition based on the instantaneous perturbation field under the Fourier transformation in the direction tangential to the interface. For the current case studies, the emergence of the bubble-spike structure is mainly attributed to the nonlinear amplification of the odd modes decomposed from the fundamental disturbances. As time advances to the nonlinear phase, the projection to the low-order normal modes possesses less portion of the total energy due to the complexity of the multipeak perturbation profiles.

Figure 1

Sketch of the physical problem.

Figure 2

Complex growth rate $\gamma$ for $(A,k)=(0.4,4.672)$.

Figure 3

Eigenfunctions $\widehat{\rho }$ (a) and $\widehat{u}$ (b), $-i\widehat{v}$ (c), and $\widehat{p}$ (d) of the first four unstable modes, where modes 2, 3, and 4 are shifted for demonstration convenience.

Figure 4

Amplitude evolution for cases 1(a), 2(b), 3(c), and 4(d).

Figure 5

Contours of the density at representative instants of the DNS result for case 1, where only a part of the computational domain is shown.

Figure 6

Evolution of the normalized perturbation profiles $\widehat{u}$ for initial perturbation $f_2$.

Figure 7

Contours of the density at representative instants of the DNS result for case 4.

Figure 8

Temporal evolution of the projected normal modes from the fundamental disturbance ${\tilde{\phi }}_1$ for cases 1 (a), 2 (b), and 3 (c).

Figure 9

Contours of the density at representative instants of the DNS result for case 3.

Figure 10

Velocity profile $u$ at $y=0$ (a) and $y=y_l/2$ (b) for case 3.

Figure 11

Evolution of the lengths of spike and bubble for case 3.

Figure 12

Amplitude of the Fourier spectra $A_N$ and its projection coefficients to the normal modes of the first four orders $a_n$ ($n=1$ to 4).

Figure 13

Eigenfunctions of the acoustic-branch continuous mode, where the profiles are normalized by their maxima. (a) $\omega =50$; (b) $\omega =80$.

Figure 14

Eigenfunctions of ${\mathrm{g}}^{+}$ mode forced from the heavy-fluid side with $\omega =10$ and 40.

Figure 15

Eigenfunctions of ${\mathrm{g}}^{-}$ mode for $\gamma =0.1$ (a) and 0.05 (b).

Figure 16

Amplitude evolution for initial perturbation (D1) (a), (D2) (b), and (D3) (c).

Figure 17

Contours of the perturbation pressure $p^{\prime }(x,0,t)$ in $x-t$ plane for initial perturbation (D1) (a), (D2) (b), and (D3) (c).