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 ϕ, 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 kin j in the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale Lc, which we evaluate from S(k), the spectrum of the fluctuations of ϕ. We demonstrate that (a) Lc ~ LH, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b) Lc 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 kc, which we show is ≃2π/Lc. We compare our work with earlier studies of this problem.
Highlights
Binary-fluid mixtures have played a pivotal role in the development of the understanding of (a) equilibrium critical phenomena at the consolute point, above which the two fluids mix[1,2,3], (b) nucleation[4], and (c) spinodal decomposition, the process by which a binary-fluid mixture, below the consolute point and below the spinodal curve, separates into the two, constituent liquid phases until, in equilibrium, a single interface separates the two coexisting phases[5,6]
2D, statistically steady, Navier-Stokes-fluid turbulence displays a forward cascade of enstrophy, from inj to smaller length scales, and an inverse cascade inverse-cascade regime, on which we concentrate here, of energy E(k) ~ k−5/3 to length and the energy
Our extensive study of two-dimensional (2D) binary-fluid turbulence shows how the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale Lc, which follows from S(k)
Summary
Binary-fluid mixtures (such as oil and water) have played a pivotal role in the development of the understanding of (a) equilibrium critical phenomena at the consolute point, above which the two fluids mix[1,2,3], (b) nucleation[4], and (c) spinodal decomposition, the process by which a binary-fluid mixture, below the consolute point and below the spinodal curve, separates into the two, constituent liquid phases until, in equilibrium, a single interface separates the two coexisting phases (this phase separation is known as coarsening)[5,6]. In the presence of flows, the demixing because of spinodal decomposition gets arrested and an emulsion is formed This process, known as coarsening arrest, is important in several three-dimensional (3D) and two-dimensional (2D) turbulent flows. The nature of coarsening arrest, for scales larger than inj, i.e., in the inverse-cascade regime, which is relevant for large-scale oceanic flows, still remains elusive. It is not clear what happens to the inverse energy transfer, in a 2D binary-fluid, turbulent mixture, in which the mean size of domains provides an additional, important length scale. We resolve these two issues in our study. We elucidate how these scaling forms for E(k) and S(k, t) are modified when we study forced 2D turbulence, in the inverse-cascade regime in the coupled Cahn-Hilliard-Navier-Stokes equations
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