Two-dimensional Turbulence in Symmetric Binary-Fluid Mixtures: Coarsening Arrest by the Inverse Cascade

We study the fundamental physics of cascades and spectra in two-dimensional (2D) Cahn-Hilliard-Navier-Stokes (CHNS) turbulence, and compare and contrast this system with 2D magnetohydrodynamic (MHD) turbulence. The important similarities include basic equations, ideal quadratic invariants, cascades, and the role of linear elastic waves. Surface tension induces elasticity, and the balance between surface tension energy and turbulent kinetic energy determines a length scale (Hinze scale) of the system. The Hinze scale may be thought of as the scale of emergent critical balance between fluid straining and elastic restoring forces. The scales between the Hinze scale and dissipation scale constitute the elastic range of the 2D CHNS system. By direct numerical simulation, we find that in the elastic range, the mean square concentration spectrum H ψ k of the 2D CHNS system exhibits the same power law (−7/3) as the mean square magnetic potential spectrum H A k in the inverse cascade regime of 2D MHD. This power law is consistent with an inverse cascade of H ψ , which is observed. The kinetic energy spectrum of the 2D CHNS system is E K k ∼ k −3 if forced at large scale, suggestive of the direct enstrophy cascade power law of 2D Navier-Stokes turbulence. The difference from the energy spectra of 2D MHD turbulence implies that the back reaction of the concentration field to fluid motion is limited. We suggest this is because the surface tension back reaction is significant only in the interfacial regions. The interfacial regions fill only a small portion of the 2D CHNS system, and their interface packing fraction is much smaller than that for 2D MHD.

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“…In the inverse energy cascade regime of the 2D CHNS system, the characteristic length scale is also consistent with the Hinze scale [25].…”

Section: -2supporting

confidence: 49%

“…In addition, the external forcing properties f 0A,φ , and k f A,φ are also adjustable. Important dimensionless numbers here are as follows [25,31]: (4) Ch = ξ/L 0 , the Cahn number, which is the ratio of the interfacial thickness to the system size.…”

Section: A Basic Setupmentioning

confidence: 99%

Cascades and spectra of a turbulent spinodal decomposition in two-dimensional symmetric binary liquid mixtures

et al. 2016

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“…This was theoretically understood by using eddy diffusivity arguments by Aronovitz and Nelson (1984). However, only recent numerical investigations using a multicomponent Lattice-Boltzmann method in three-dimensions (Perlekar et al 2014) and direct numerical simulations (DNSs) of Cahn-Hilliard-Navier-Stokes equations in two-dimensions (Berti et al 2005;Fan et al 2016;Perlekar et al 2017;Fan et al 2018) have been able to study emulsification by turbulence in symmetric binary-fluid mixtures. Surprisingly, unlike shear flows, here numerically calculated domain size is in excellent agreement with Hinze's prediction in both two-and three-dimensions.…”

Section: Introductionmentioning

confidence: 99%

Kinetic energy spectra and flux in turbulent phase-separating symmetric binary-fluid mixtures

Perlekar¹

2019

J. Fluid Mech.

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We conduct direct numerical simulation (DNS) of the Cahn-Hilliard-Navier-Stokes (CHNS) equations to investigate the statistical properties of the turbulent phaseseparating symmetric binary-fluid mixture. Turbulence causes an arrest of the phaseseparation which leads to the formation of a statistically steady emulsion. We characterize turbulent velocity fluctuations in an emulsion for different values of the Reynolds number and the Weber number. Our scale-by-scale kinetic energy budget analysis shows that the interfacial terms in the CHNS provide an alternate route for the kinetic energy transfer. By studying the probability distribution function (PDF) of the energy dissipation rate, the vorticity magnitude, and the joint-PDF of the velocity-gradient invariants we show that the statistics of the turbulent fluctuations do not change with the Weber number.

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“…For the opposite scenario (i.e. when the length scale of the arrested domains exceeds the energy-injection scale), the arrest scale has been shown by other researchers to be comparable to the Hinze length et al [8]; in other words, the arrest scale in this regime is governed by a balance between inertia and surface tension (effectively, the backreaction or 'active' term in the momentum equation). Our recent work [9] for the passive case in two and three dimensions has further highlighted the key role played by the Péclet number (measuring the strength advection term relative to the Cahn-Hilliard antidiffusion term) in the outcome of the phase separation: changing the Péclet number by orders of magnitude involves dramatic changes in the outcome of the phase separation.…”

Section: Introductionmentioning

confidence: 90%

A flow-pattern map for phase separation using the Navier–Stokes–Cahn–Hilliard model

Naso

Náraigh

2018

European Journal of Mechanics - B/Fluids

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We use the Navier-Stokes-Cahn-Hilliard model equations to simulate phase separation with flow. We study coarsening -the growth of extended domains wherein the binary mixture phase separates into its component parts. The coarsening is characterized by two competing effects: flow, and the Cahn-Hilliard diffusion term, which drives the phase separation. Based on extensive two-dimensional direct numerical simulations, we construct a flow-pattern map outlining the relative strength of these effects in different parts of the parameter space. The map reveals large regions of parameter space where a standard theory applies, and where the domains grow algebraically in time. However, there are significant parts of the parameter space where the standard theory does not apply. In one region, corresponding to low values of viscosity and diffusion, the coarsening is accelerated compared to the standard theory. Previous studies involving Stokes flow report on this phenomenon; we complete the picture by demonstrating that this anomalous regime occurs not only for Stokes flow, but also, for flows dominated by inertia. In a second region, corresponding to arbitrary viscosities and high Cahn-Hilliard diffusion, the diffusion overwhelms the hydrodynamics altogether, and the latter can effectively be ignored, in contrast to the prediction of the standard scaling theory. Based on further highresolution simulations in three dimensions, we find that broadly speaking, the above description holds there also, although the formation of the anomalous domains in the low-viscosity-low-diffusion part of the parameter space is delayed in three dimensions compared to two. * onaraigh@maths.ucd.ie arXiv:1707.06067v1 [physics.flu-dyn]

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Two-dimensional Turbulence in Symmetric Binary-Fluid Mixtures: Coarsening Arrest by the Inverse Cascade

Cited by 65 publications

References 61 publications

Cascades and spectra of a turbulent spinodal decomposition in two-dimensional symmetric binary liquid mixtures

Cascades and spectra of a turbulent spinodal decomposition in two-dimensional symmetric binary liquid mixtures

Kinetic energy spectra and flux in turbulent phase-separating symmetric binary-fluid mixtures

A flow-pattern map for phase separation using the Navier–Stokes–Cahn–Hilliard model

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