Experimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability

We assess experimentally the scaling laws that characterize the mixing region produced by the Rayleigh-Taylor instability in a confined porous medium. In particular, we wish to assess experimentally the existence of a superlinear scaling for the growth of the mixing region, which was observed in recent two-dimensional simulations. To this purpose, we use a Hele-Shaw cell. The flow configuration consists of a heavy fluid layer overlying a lighter fluid layer, initially separated by a horizontal, flat interface. When small perturbations of concentration and velocity fields occur at the interface, convective mixing is eventually produced: Perturbations grow and evolve into large finger-like convective structures that control the transition from the initial diffusion-dominated phase of the flow to the subsequent convection-dominated phase. As the flow evolves, diffusion acts to reduce local concentration gradients across the interface of the fingers. When the gradients become sufficiently small, the system attains a stablystratified state and diffusion is again the dominant mixing mechanisms. We employ an optical method to obtain high-resolution measurements of the density fields, and we perform experiments for values of the Rayleigh-Darcy number (i.e., the ratio between convection and diffusion) sufficiently large to exhibit all the flow phases just described, which we characterize via the mixing length, a measure of the extension of the mixing region. We are able to confirm that the growth of the mixing length during the convection-dominated phase follows the superlinear scaling predicted by previous simulations.

Figure 1

(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left, and right, indicated as T, B, L, and R, respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sketched in Fig. 2) consisting of a pump, pipes, and valves. The cell is aligned in the vertical direction, namely parallel to the acceleration due to gravity, $\mathbf{g}$. (b) Sketch of the experimental domain with explicit indication of the boundary conditions (no flux of mass or solute through the walls). The reference frame ($x,z$) as well as the initial position of the interface (red dashed line) are indicated, with the heavy fluid (density ${\rho }_M$, concentration $C=C_M$) initially lying on top of the lighter fluid (${\rho }_0, C=0$). The background field consists of one of the images collected and provides a qualitative picture of the flow for the present physical configuration.

Figure 2

Circuital scheme of the experimental setup employed to obtain the desired initial conditions [33, 34, 35]. A pumping system (peristaltic pump, Watson-Marlow 502 S) is used to inject water (density ${\rho }_0$) and solution (${\rho }_M$) from two separate containers, through gates B and T, respectively. To set up the initial condition, the valves are placed in “loading mode” and the exceeding solution is collected from gates L and R into a residue container. The flow through the gates, which is used to modulate the shape of the interface between the two fluid layers, is controlled by a regulation valve. Finally, when the initial condition is achieved, the system is set in “experiment mode” and the cell is bypassed by the fluids, which are channeled to the residue container.

Figure 3

Reconstruction of the concentration field. (a) Raw image consists of a light intensity distribution, $I$. (b) Upon processing to remove light intensity noises, scratches and nonuniformities of backlight illumination (see the Supplemental Material [40] for further details), the dimensionless concentration field is obtained. To visualize the interface between the two fluids, corresponding to the fingers, three isocontours of normalized concentration are shown, corresponding to $C^{*}=C/C_M=0.3$ (blue), 0.5 (red), and 0.7 (green).

Figure 4

Snapshots of the dimensionless concentration field obtained experimentally (a)–(d) and numerically (e)–(h). The experiment (E4 in Table 1, $\mathrm{Ra}=19789$) is compared vis-à-vis against numerical simulations performed at $\mathrm{Ra}=19953$ by De Paoli et al. [4]. The dimensionless time instants ($t^{*}$) at which the fields refer are indicated on top of each panel. We remark that the experiments are realized in a square geometry, whereas the simulations are performed in wider domains (aspect ratio $\pi /2$). Therefore, for better comparison, only a portion of the numerical domain having unitary aspect ratio is reported. The desired initial position of the interface (located at $z=H/2$, i.e., $z^{*}=\mathrm{Ra}/2$) is also shown as a dashed black line. See also Movie S1 in the Supplemental Material [40] for a comparison of the time-dependent evolution of the concentration fields in the experiment and in the simulation.

Figure 5

Time evolution of the dimensionless mixing length, $h^{*}\left(t^{*}\right)$ (solid lines), for all the values of the global Rayleigh number considered in this study (and explicitly indicated next to each curve). The evolution of the system, initially controlled by diffusion, nicely follows the analytical solution, $h^{*}\approx (1-2\varepsilon )\sqrt{4\pi t^{*}}$ [Eq. (20), dashed blue line]. The intermediate phase, dominated by convection, exhibits the anomalous scaling $h^{*}\sim {\left(t^{*}\right)}^{1.2}$ first observed by De Paoli et al. [4].

Figure 6

Time taken by the fingers to reach the horizontal boundaries $\left(t_s^{*}\right)$ as a function of $\mathrm{Ra}$. Experimental (present work) and numerical [4] results are reported as open ($\circ$) and filled symbols ($\blacksquare$), respectively. The best-fitting power law for present results (red dashed line) is $9.64\times {\mathrm{Ra}}^{0.80}$, in excellent agreement with numerical findings ($14.98\times {\mathrm{Ra}}^{0.78}$, black dashed line). Experimental results correspond to the mean values obtained for several flow realizations at fixed $\mathrm{Ra}$.

Figure 7

Measurements of the number of fingers over time for $\mathrm{Ra}=5.43\times {10}^4$ (E6 in Table 1). Space-time map, consisting of the evolution of the concentration field along the centerline, $C^{*}(x^{*},z^{*}=\mathrm{Ra}/2,t^{*})$, is shown in (a). To compute the finger number, the space-time map is binarized, $\widehat{C}(x^{*},z^{*}=\mathrm{Ra}/2,t^{*})$, choosing $C^{*}=1/2$ as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at $t^{*}\approx 1.1\times {10}^5$ (indicated by the dashed blue line in panels a, b, and d) are also reported as a function of the horizontal coordinate $x^{*}$ (c). Finally (d), the time-dependent finger number is computed as from the binarized field $\widehat{C}$ (black line), from power-averaged mean wave number as defined in Eq. (21) ($\langle N\rangle =L\langle k\rangle /\pi$, red line) and from a $k_{\text{peak}}$ (green line), i.e., the wave number maximizing the spectrum amplitude of $C$. See also Movie S2 in the Supplemental Material [40] for the time-dependent evolution of the wave number and the corresponding concentration field.

Figure 8

(a) Experimental measurements. The evolution of the power-averaged mean wave number, $\langle k\rangle$ [defined in Eq. (21), main panel], and the corresponding finger number $\langle N\rangle$ (inset). (b) Comparison of experimental (solid lines) and numerical ([30], dashed lines) measurements. For ease of reading, three values of Rayleigh number are used to compare the behavior of $\langle k\rangle$.

Figure 9

Given the definition (A5) and the empirical correlations (A1) [37], the relative dependence of density of the solution, $\rho$, mass fraction, $\omega$, and solute concentration, $C$, is obtained. We report here $C\left(\omega \right)$ (a), $\rho \left(\omega \right)-{\rho }_0$ (b), and $\rho \left(C\right)-{\rho }_0$ (c), being ${\rho }_0$ the water density. The profiles shown here correspond to $\vartheta =25{}^{\circ }\text{C}$. We observe that the empirical values (symbols) are very well fitted by linear functions (solid lines). In the instance of density and concentration (c), the linear function corresponds to Eq. (5).

Figure 10

Example of calibration curve for a given illumination condition. A number of samples having different mass fraction ($\omega$) are used to uniformly fill the cell. The average value of light intensity, $\langle I^{\mathrm{cal}}\rangle$, is used to build the curve. Experimental measurements (symbols) are well fitted by a combination of exponential functions, $\omega \left(\langle I^{\mathrm{cal}}\rangle \right)=a_0+a_{1}exp\left(b_1\langle I^{\mathrm{cal}}\rangle \right)+a_{2}exp\left(b_2\langle I^{\mathrm{cal}}\rangle \right)$ (solid line). Uncertainties on the light intensity and mass fraction are also shown (see Sec. 2b for further details). Finally, the mass fraction-light intensity space considered in each experiment, labeled as in Table 1, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e., for low values of mass fraction.

Figure 11

Initial conditions for two different experiments, E1 and E4, characterized by $\text{Ra}=4391$ (a)–(c) and $\text{Ra}=19789$ (d)–(f), respectively. The domain centerline (red dashed line) is indicated on the unprocessed light intensity fields $I$ (a), (d) and on the normalized inferred concentration fields $C^{*}$ (b), (e). In panels (c) and (f) three isocontours of dimensionless concentration ($C^{*}=0.3$, blue, $C^{*}=0.5$, red, and $C^{*}=0.7$, green) are used to identify the interface shape in a close up view of the interfacial region. We observe that at both macroscopic (a), (b), (d), (e) and interfacial (c), (f) levels, the fluid-fluid separation region is well defined within a narrow area.