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, Reϕ = ldl/dt,
we attain values approaching 350. At Reϕ [gsim ] 100 (which requires
t [gsim ] 106) we find clear evidence of Furukawa's inertial
scaling (l ∼ t2/3), 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 Reϕ is self-limiting
(l ∼ t1/2) 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
Reϕ 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 ∼ t2/3 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.
Constitutive relations for compressible granular flow in the inertial regime
