Cascades and Spectra of a Turbulent Spinodal Decomposition in 2D Symmetric Binary Liquid Mixture

TL;DR

This study investigates the cascades and spectra in 2D Cahn-Hilliard-Navier-Stokes turbulence, revealing similarities to 2D MHD turbulence and identifying key spectral power laws and the role of elastic and interfacial effects.

Contribution

It provides the first detailed comparison of 2D CHNS turbulence with 2D MHD, highlighting spectral similarities and differences, and introduces the concept of the Hinze scale as a critical balance point.

Findings

Mean square concentration spectrum follows a -7/3 power law in the elastic range.

Kinetic energy spectrum exhibits a -3 power law, indicating a direct enstrophy cascade.

Back reaction of concentration field to flow is limited due to small interfacial regions.

Abstract

We study the fundamental physics of cascades and spectra in 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 $H^\psi_k$ 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 $H^\psi$, which is observed. The kinetic energy spectrum of the 2D CHNS system is $E^K_k\sim k^{-3}$ if forced at large scale, suggestive of the direct enstrophy cascade power law of 2D Navier-Stokes (NS) 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.