Scalar mixtures in porous media

Mihkel Kree and Emmanuel Villermaux

Phys. Rev. Fluids 2, 104502 – Published 23 October 2017

Abstract

Using a technique allowing for in situ measurements of concentrations fields, the evolution of scalar mixtures flowing within a porous medium made of a three-dimensional random stack of solid spheres, is addressed. Two distinct fluorescent dyes are injected from separate sources. Their evolution as they disperse and mix through the medium is directly observed and quantified, which is made possible by matching the refractive indices of the spheres and the flowing interstitial liquid. We decipher the nature of the interaction rule between the scalar sources, explaining the phenomenon that alters the concentration distribution of the overall mixture as it decays toward uniformity. Any residual correlation of the initially merged sources is progressively hidden, leading to an effective fully random interaction rule of the two distinct subfields.

Received 5 July 2017

DOI:https://doi.org/10.1103/PhysRevFluids.2.104502

Physics Subject Headings (PhySH)

Turbulent mixing

Porous media

Fluid Dynamics

Authors & Affiliations

Mihkel Kree^1,2,* and Emmanuel Villermaux^2,3,†

¹Institute of Cybernetics, Tallinn University of Technology, 12618 Tallinn, Estonia
²Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384 Marseille, France
³Institut Universitaire de France, 75005 Paris, France

^*mihkel.kree@gmail.com
^†emmanuel.villermaux@univ-amu.fr

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Vol. 2, Iss. 10 — October 2017

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Images

Figure 1
(a) A bead, made of polyacrylamide hydrogel, placed in fluorescent water. Green regions mark the propagation of vertical planar laser sheet directed from left to right. Three sketches: Simple geometrical optical ray tracing illustrates the emergence of shadows and caustics from the top and bottom regions of the sphere due to slight relative mismatch $δ = (n_{b} - n_{w}) / n_{w}$ of the refractive indices. (b) A sketch of the experimental setup: The direct imaging of the concentration fields inside the transparent porous medium is made possible by the fluorescence of the injected dyes at the plane of the laser sheet. (c) A raw, unfiltered snapshot of the tank filled with beads, cut by a planar laser sheet, showing how the two plumes of dyes are injected, disperse, mix, and interfere. The mean flow is from bottom to top.
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Figure 2
Direct visualization of the concentration fields of the injected dyes inside the transparent porous medium. The stationary water flow is directed upward, and the planar laser sheet exciting the fluorescent emission of the dyes is directed from right to left. (a) Single dye injected at the bottom of the tank. (b) Two distinct dyes are injected side by side. The three rectangles indicate regions used for calculating the two-dimensional joint distributions of the two colors shown in Fig. 4.
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Figure 3
(a) The concentration decays with distance $〈 C 〉 \sim x^{- 1}$ (solid line) as suggested by Eq. (14). Note that it is difficult to measure the true average concentration $〈 C 〉$ because the ever-increasing spacial domain needs to be properly taken into account in calculating the averages. Instead, we have plotted the average of top $1 %$ of the concentration content in observation windows of fixed width, denoted by ${〈 C 〉}_{max}$ , at various distances for a certain realization of the flow. (b) The variance $δ^{2} (x)$ of the spreading plume of dye increases linearly with downstream distance $x$ in accordance with Eq. (1). The data points correspond to two flow velocities, $u$ and $1.5 u$ . (c) The lines with open markers indicate three concentration $C$ (normalized by the injection concentration) distributions $P (C)$ at various downstream locations $x / d = 12$ (open squares), $x / d = 15$ (open triangles), and $x / d = 18$ (open circles) measured from an image such as the one shown on Fig. 2. The tails of these distributions are well fitted to weighted $Γ$ distributions $Γ_{n} (C)$ (solid lines) of various shape parameters: $n = 5, n = 11, n = 17$ (in the order of increasing distance $x)$ . Inset: The fitted shape parameters $n$ increase exponentially with downstream distance $x$ . The three sets of filled markers correspond to three random realizations of the porous medium.
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Figure 4
Top row: three regions of the field at various downstream distances $x$ , extracted from the image on Fig. 2 where the corresponding regions have been indicated by rectangular borders. Bottom row: the plots of joint distributions $Q (C_{1}, C_{2})$ are calculated from the corresponding regions of the fields shown on top. The transition from initially anticorrelated fields $C_{1}$ and $C_{2}$ to interacting fields is qualitatively evident. Note that the two axes $C_{1} = 0$ and $C_{2} = 0$ are not symmetrical as a result of different absorbance and emission behavior of rhodamine and fluorescein.
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Figure 5
(a) The three thick solid lines indicate the distributions of individual concentrations $P_{1} (C_{1})$ and $P_{2} (C_{2})$ and the total concentration $P (C = C_{1} + C_{2})$ extracted from the joint distribution $Q (C_{1}, C_{2})$ at location $x = 18 d$ shown on Fig. 4, right. The thick dashed line indicates the calculated convolution of $P_{1} (C_{1})$ and $P_{2} (C_{2})$ . The thin dotted line is a weighted $Γ$ distribution of shape parameter $n = 8$ . The thin solid line is self-convolution of the thin dotted line and thus a weighted $Γ$ distribution of shape parameter $n = 16$ . For comparison, the thin dashed line indicates the distribution of the sum $C_{1} + C_{2}$ in the case of two fully correlated $Γ$ distributions. (b) The two lines with markers correspond to conditional distributions $P (Z | C)$ of the mixing ratio $Z = C_{1} / C$ confined to range $0.2 < C < 0.4$ that are calculated from two of the three joint distributions $Q (C_{1}, C_{2})$ shown on Fig. 4. The thick solid line indicates a fitted $β$ distribution [Eq. (23)] of shape parameter $n = 8$ matching well with the measured $P (Z)$ at the late stage of the mixture. The fully correlated case would result in Dirac $δ$ in $Z = 0.5$ . The thick dashed line indicates a $β$ distribution of shape parameter less than unity $n < 1$ qualitatively matching the shape of measured $P (Z)$ at the segregated stage of the mixture.
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