Unstable yet static initial state: A universal method for studying Rayleigh-Taylor instability and lock exchange

Macroscopic flow induced by instability is a feature found in both natural phenomena and industrial processes; however, in situ observation of instabilities at an early stage in their formation is rare. In this article, we propose an experimental method for observing the early stages of a Rayleigh-Taylor instability and lock exchange. We successfully used physical gels to form a “static” initial state which is simultaneously “gravitationally” unstable. Fluid instabilities can be observed from the beginning by switching from a static state to a dynamic state by irradiating the system with light. We can also observe flow dynamics under arbitrary boundary conditions using this method. Here we present notable results in the early stages of a Rayleigh-Taylor instability and lock exchange. Our method promises to greatly advance experimental research into many types of instability phenomena, contributing to the elucidation of nonequilibrium phenomena in fluids.

Figure 1

A schematic of our experimental setup. We performed the experiment with glass sample chambers with internal dimensions (H, L, W) = (30 mm, 126 mm, 2.4 mm). The sample chamber is heated using light irradiation from the back. In order to improve the efficiency of heating, black paper is attached to the area to be heated; the rest is covered with insulating materials.

Figure 2

(a) An enlarged image near a boundary between a heating region and a heat-insulating region. Cross symbols are measured points of the temperature by a thermometer. (b) The spatial dependence of the temperature near the boundary. The length of the boundary effect was less than 5 mm from the boundary.

Figure 3

The time evolution of the Rayleigh-Taylor instability in the gelatin solution at (a) $t=0$ s, (b) $t=71\mathrm{s}$, (c) $t=138$ s, (d) $t=214$ s, and (e) $t=260$ s. The horizontal length of the images corresponds to 70 mm. The green region corresponds to the 5 wt% gelatin solution, and the black region corresponds to the 6 wt% gelatin solution. The orange region is background, unrelated to the flow. A stationary, flat interface is realized in a density inverted state (a). The instability starts by heating with light from the back (b). A fingering pattern grows over time (c–e).

Figure 4

The time evolutions of (a) the most unstable wavelength $\lambda$ and (b) the amplitude of the fluctuations $h_p$ at $h_b=5.9$ mm. $\lambda$ is constant, and $h_p$ exponentially grows. Those are typical behaviors of Rayleigh-Taylor instability in the early stage. The error bar in panel (a) comes from a distribution of the fluctuation. The solid line in panel (b) is a fitting line by the exponential curve.

Figure 5

(a) $\lambda$ as a function of $W$ at $h_b=10$ mm. $\lambda$ increases exponentially with $W$. (b) ${\lambda }_{\infty }$ as a function of $h_b$. ${\lambda }_{\infty }$ linearly increases with $h_b$.

Figure 6

The time evolution of the Rayleigh-Taylor instability with circular boundary conditions at (a) $t=0\mathrm{s}$, (b) $t=380$ s, (c) $t=435$ s, (d) $t=650$ s, and (e) $t=1028$ s. The diameter of the circle is 30 mm. The rest is covered with a heat insulator. The green and orange regions correspond to a 5 wt% gelatin solution. The black and white regions correspond to the 6 wt% gelatin solution. A ginkgo leaf pattern is formed inside the circle, while nothing changes outside the circle.

Figure 7

The time evolution of lock-exchange flow between two fluids, using 6 wt% gelatin solution (orange) and silicone oil (black) at (a) $t=0$ s, (b) $t=50$ s, (c) $t=121$ s, (d) $t=183$ s, and (e) $t=239$ s. We successfully form a stationary, vertical, flat interface without a traditional lock gate (a). A lock-exchange flow is subsequently driven by the density difference (b–e).

Figure 8

Distances between the front position of the fluid and the initial position at (a) $\delta \theta >0$ and (b) at $\delta \theta <$ 0. We find that the front of the fluid moves with constant velocity.

Figure 9

Time evolution of the lock-exchange flow in 6 wt% gelatin solution (orange) and silicone oil (black) with an inclined interface. (a) $t=0$ s, (b) $t=83$ s, (c) $t=112$ s, (d) $t=164$ s, and (e) $t=240$ s. We also realized the formation of an inclined stationary flat interface without a traditional lock gate (a). The gelatin solution surges over the silicone oil; this is significantly different behavior compared to flow with a vertical interface (b–c). The gelatin solution then slides into the silicon oil in a similar manner to the vertical interface system (d–e). The front velocity of the gelatin solution is much higher than with a vertical interface.

Figure 10

$V$ of the heavier fluid as a function of $\delta \theta$. A transition of $V$ can be observed at $\delta \theta =0$. It is also found for $V$ of the lighter fluid shown as an inset.

Figure 11

The front shapes of the heavier liquid extracted by the image analysis. The blue solid line and the blue dashed line correspond, respectively, to the front shape at $t=183$ s and 239 s at $\delta \theta >0$, while the red solid line and the red dashed line correspond, respectively, to the front shape at $t=164$ s and 240 s at $\delta \theta <0$. Tips of the heavier liquid are overlapped. The front shape is more streamlined for $\delta \theta <0$.