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

TL;DR

This study uses direct numerical simulations to show how coupling in 2D binary-fluid turbulence arrests phase separation at a specific length scale, affecting the inverse energy cascade.

Contribution

It demonstrates that coupling between Cahn-Hilliard and Navier-Stokes equations leads to phase separation arrest at the Hinze scale, modifying the inverse energy cascade in 2D turbulence.

Findings

Phase separation is arrested at the Hinze scale $L_c$.

The length scale $L_c$ is independent of diffusivity $D$.

Inverse energy cascade is blocked at wavenumber $k_c \,\simeq \\ 2\\pi / L_c$.

Abstract

We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numerical simulation (DNS) of the forced, statistically steady turbulence in the coupled Cahn-Hilliard and Navier-Stokes equations. In the absence of any coupling, we choose parameters that lead (a) to spinodal decomposition and domain growth, which is characterized by the spatiotemporal evolution of the Cahn-Hilliard order parameter $\phi$, and (b) the formation of an inverse-energy-cascade regime in the energy spectrum $E(k)$, in which energy cascades towards wave numbers $k$ that are smaller than the energy-injection scale $k_{inj}$ in the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale $L_c$, which we evaluate from $S(k)$, the spectrum of the fluctuations of $\phi$. We demonstrate that (a) $L_c \sim L_H$, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b) $L_c$ is independent, within error bars, of the diffusivity $D$. We elucidate how this coupling modifies $E(k)$ by blocking the inverse energy cascade at a wavenumber $k_c$, which we show is $\simeq 2\pi/L_c$. We compare our work with earlier studies of this problem.