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

摘要

We study the fundamental physics of cascades and spectra in 2DCahn-Hilliard-Navier-Stokes (CHNS) turbulence, and compare and contrast thissystem with 2D MagnetoHydroDynamic (MHD) turbulence. The important similaritiesinclude basic equations, ideal quadratic invariants, cascades and the role oflinear elastic waves. Surface tension induces elasticity, and the balancebetween surface tension energy and turbulent kinetic energy determines a lengthscale (Hinze scale) of the system. The Hinze scale may be thought of as thescale of emergent critical balance between fluid straining and elasticrestoring forces. The scales between the Hinze scale and dissipation scaleconstitute the elastic range of the 2D CHNS system. By direct numericalsimulation, we find that in the elastic range, the mean square concentrationspectrum $H^\psi_k$ 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 cascaderegime 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 systemis $E^K_k\sim k^{-3}$ if forced at large scale, suggestive of the directenstrophy cascade power law of 2D Navier-Stokes (NS) turbulence. The differencefrom the energy spectra of 2D MHD turbulence implies that the back reaction ofthe concentration field to fluid motion is limited. We suggest this is becausethe surface tension back reaction is significant only in the interfacialregions. The interfacial regions fill only a small portion of the 2D CHNSsystem, and their interface packing fraction is much smaller than that for 2DMHD.