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

Abstract

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.

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

“…For the binary mixture plots we also overlay the iso-contour of the order parameter φ for φ = 0 to highlight the interface. size of binary-fluid domains lie within the inertial range (Aronovitz and Nelson 1984;Perlekar et al 2014;Fan et al 2016).…”

Forces driving cell sorting inHydra

“…The definition of the typical blob size varies from one study to the next. Since this measure is also commonly used in the study of phase separation through spinodal decomposition (Fan et al, 2016), we chose here to use the same definition (see Methods). This allowed us to directly compare our dynamics to a purely physical situation.…”

Formation and evolution of target patterns in Cahn-Hilliard flows

“…In our previous study [9], we stressed the analogy between 2D CHNS and 2D MHD. The energy in MHD consists of two parts: kinetic energy and magnetic energy.…”

“…The two systems have identical ideal quadratic conserved quantities, and they both exhibit dual cascades. Our previous work [9] showed that the mean square concentration spectrum for the 2D CHNS system in the elastic range is ∼ k −7/3 , and it is associated with an inverse cascade of mean square concentration. Note that the power −7/3 is the same as the power for the mean square magnetic potential spectrum in 2D MHD.…”