Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study

Abstract

The late-stage demixing following spinodal decomposition of a three-dimensional symmetric binary fluid mixture is studied numerically, using a thermodynamically consistent lattice Boltzmann method. We combine results from simulations with different numerical parameters to obtain an unprecedented range of length and time scales when expressed in reduced physical units. (These are the length and time units derived from fluid density, viscosity, and interfacial tension.) Using eight large (256³) runs, the resulting composite graph of reduced domain size l against reduced time t covers 1 [lsim ] l [lsim ] 10⁵, 10 [lsim ] t [lsim ] 10⁸. Our data are consistent with the dynamical scaling hypothesis that l(t) is a universal scaling curve. We give the first detailed statistical analysis of fluid motion, rather than just domain evolution, in simulations of this kind, and introduce scaling plots for several quantities derived from the fluid velocity and velocity gradient fields. Using the conventional definition of Reynolds number for this problem, Re_ϕ = ldl/dt, we attain values approaching 350. At Re_ϕ [gsim ] 100 (which requires t [gsim ] 10⁶) we find clear evidence of Furukawa's inertial scaling (l ∼ t^2/3), although the crossover from the viscous regime (l ∼ t) is both broad and late (10² [lsim ] t [lsim ] 10⁶). Though it cannot be ruled out, we find no indication that Re_ϕ is self-limiting (l ∼ t^1/2) at late times, as recently proposed by Grant & Elder. Detailed study of the velocity fields confirms that, for our most inertial runs, the RMS ratio of nonlinear to viscous terms in the Navier–Stokes equation, R₂, is of order 10, with the fluid mixture showing incipient turbulent characteristics. However, we cannot go far enough into the inertial regime to obtain a clear length separation of domain size, Taylor microscale, and Kolmogorov scale, as would be needed to test a recent ‘extended’ scaling theory of Kendon (in which R₂ is self-limiting but Re_ϕ not). Obtaining our results has required careful steering of several numerical control parameters so as to maintain adequate algorithmic stability, efficiency and isotropy, while eliminating unwanted residual diffusion. (We argue that the latter affects some studies in the literature which report l ∼ t^2/3 for t [lsim ] 10⁴.) We analyse the various sources of error and find them just within acceptable levels (a few percent each) in most of our datasets. To bring these under significantly better control, or to go much further into the inertial regime, would require much larger computational resources and/or a breakthrough in algorithm design.