Spinodal Decomposition in Homogeneous and Isotropic Turbulence

Prasad Perlekar, Roberto Benzi, Herman J. H. Clercx, David R. Nelson, and Federico Toschi

Phys. Rev. Lett. 112, 014502 – Published 8 January 2014

Abstract

We study the competition between domain coarsening in a symmetric binary mixture below critical temperature and turbulent fluctuations. We find that the coarsening process is arrested in the presence of turbulence. The physics of the process shares remarkable similarities with the behavior of diluted turbulent emulsions and the arrest length scale can be estimated with an argument similar to the one proposed by Kolmogorov and Hinze for the maximal stability diameter of droplets in turbulence. Although, in the absence of flow, the microscopic diffusion constant is negative, turbulence does effectively arrest the inverse cascade of concentration fluctuations by making the low wavelength diffusion constant positive for scales above the Hinze length.

Received 27 June 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.014502

Authors & Affiliations

Prasad Perlekar^1,2, Roberto Benzi³, Herman J. H. Clercx¹, David R. Nelson⁴, and Federico Toschi^1,5,6

¹Department of Physics and J.M. Burgerscentrum, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands; and International Collaboration for Turbulence Research
²TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India
³Dipartimento di Fisica and INFN, Università “Tor Vergata”, Via della Ricerca Scientifica 1, I-00133 Roma, Italy
⁴Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
⁵Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
⁶CNR, Istituto per le Applicazioni del Calcolo, Via dei Taurini 19, 00185 Rome, Italy

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Vol. 112, Iss. 1 — 10 January 2014

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Images

Figure 1
Pseudocolor plots of the concentration fields, with the two symmetric fluids indicated in red and blue. (Top panel, left–right) Time evolution of the concentration field undergoing coarsening process from an initially well-mixed state. Notice the formation of ever-larger concentration patches as the time evolves. (Middle panel, left–right) Time evolution of the concentration field undergoing a coarsening process from a well-mixed state in the presence of turbulence generated by external driving. The coarsening process goes on uninhibited until arrested by the turbulence at later times. (Bottom panel, left–right) Time evolution of a very coarse phase-separated mixture in the presence of turbulence with the same intensity as the middle panel. In this case, the domains are broken up until the mixture attains a steady state domain size that is the same as the one in the middle panel. This behavior indicates a positive renormalized eddy diffusivity at large length scales even though the microscopic diffusion constant is negative. The Taylor-microscale Reynolds number for the middle and bottom panels is ${Re}_{λ} = 103$ (run ST2, Table 1). From the plots, it is clear that, in case of turbulence, the coarsening of concentration gets arrested whereas coarsening length attains domain size in the absence of turbulence. In all the panels, the snapshots are taken at times $t = 5.0 \times 1 0^{3}$ , $1.0 \times 1 0^{4}$ , $2.5 \times 1 0^{4}$ , and $1.0 \times 1 0^{5}$ .
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Figure 2
Coarsening arrest for phase separating binary mixtures in the presence of turbulence. In the absence of an external turbulent forcing (red circles), the coarsening length keeps on growing as $L (t) \sim t$ (black line). Switching on turbulence, the coarsening length initially grows undisturbed, but then, it arrests as the system attains a steady state. The time and length scale are nondimensionalized by the corresponding characteristic length $L_{0} = ν^{2} / (ρ σ)$ and time $T_{0} = ν^{3} / (ρ σ^{2})$ .
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Figure 3
Growth of the coarsening length scale $L (t)$ in the arrested state normalized by the Hinze length $L_{H}$ for ${Re}_{λ} = 35$ , 49 (blue three-quarter-filled circle and green half-filled circle) [run ST1], ${Re}_{λ} = 72$ , 103 (purple square and brown filled square) [run ST2], and ${Re}_{λ} = 86$ (blue filled circle) [run ST4]. In the inset, we plot the average value of $L (t) / L_{H}$ , calculated over the time window $t = 5 \times 1 0^{4}$ to $2 \times 1 0^{5}$ , for different Reynolds numbers ${Re}_{λ}$ . Within error bars, $L (t) / L_{H} \approx 4.4 \pm 0.5$ is found to be a good indicator for the arrested length scale. We believe that the smaller value of $L (t) / L_{H}$ (although within our error bars) for the ${Re}_{λ} = 35$ , 49 arises because of the lower grid resolution.
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Figure 4
(Top panel) The inverse cascade of concentration spectrum $k^{2} S (k)$ for spinodal decomposition at times $t = 1 0^{3} - 5 \times 1 0^{5}$ ( $25 6^{3}$ , $ρ = 2.4$ [run S2]) without turbulence. The Fourier mode associated with the peak of the spectrum gives an estimate of the instantaneous coarsening length. On the other hand, in the presence of turbulence (bottom panel, ${25 6^{3}, ρ = 2.4, and {Re}_{λ} = 103 [run ST 2]}$ ), we do observe an initial inverse cascade of concentration that saturates around $t = 5 \times 1 0^{4}$ indicating a blockage of the inverse cascade.
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Figure 5
Comparison of the energy spectrum for the spinodal decomposition in the presence of turbulence (triangle, run ST3 [ ${Re}_{λ} = 185$ ]) with the pure fluid case (circle, run NS2). The black line indicates the Kolmogorov scaling $k^{- 5 / 3}$ . We find that the large- $k$ crossover takes place roughly around the inverse Hinze scale $k_{H} \equiv 1 / L_{H}$ . This crossover was also confirmed by comparing runs NS1 and ST2 (not shown here).
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