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

The late-stage demixing following spinodal decomposition of a three-dimensi onal symmetric binary fluid mixture is studied numerically, using a thermod ynamically consistent lattice Boltzmann method. We combine results from sim ulations with different numerical parameters to obtain an unprecedented ran ge of length and time scales when expressed in reduced physical units. (The se are the length and time units derived from fluid density, viscosity, and interfacial tension.) Using eight large (256(3)) runs, the resulting compo site graph of reduced domain size I against reduced time t covers 1 less th an or similar to l less than or similar to 10(5), 10(5), 10 less than or si milar to t less than or similar to 10(8). Our data are consistent with the dynamical scaling hypothesis that 1(t) is a universal scaling curve. We giv e 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 gradie nt fields. Using the conventional definition of Reynolds number for this pr oblem, Re-phi = l dl/dt, we attain values approaching 350. At Re-phi greate r than or equal to 100 (which requires t greater than or equal to 10(6)) we find clear evidence of Furukawa's inertial scaling (l similar to t(2/3)), although the crossover from the viscous regime (l similar to t) is both bro ad and late (10(2) less than or similar to t less than or similar to 10(6)) . Though it cannot be ruled out, we find no indication that Reo is self-lim iting (l less than or similar to t(1/2)) at late times, as recently propose d by Grant & Elder. Detailed study of the velocity fields confirms that, fo r our most inertial runs, the RMS ratio of nonlinear to viscous terms in th e Navier-Stokes equation, R-2, is of order 10, with the fluid mixture showi ng incipient turbulent characteristics. However, we cannot go far enough in to the inertial regime to obtain a clear length separation of domain size, Taylor microscale, and Kolmogorov scale, as would be needed to test a recen t 'extended' scaling theory of Kendon (in which R2 is self-limiting but Re- phi not). Obtaining our results has required careful steering of several nu merical 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 repor t l similar to t(2/3) for t less than or similar to 10(4).) 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 requi re much larger computational resources and/or a breakthrough in algorithm d esign.