Direct numerical simulation ͑DNS͒ data at grid resolution up to 2048 3 in isotropic turbulence are used to investigate the statistics of acceleration in a Eulerian frame. A major emphasis is on the use of conditional averaging to relate the intermittency of acceleration to fluctuations of dissipation, enstrophy, and pseudodissipation representing local relative motion in the flow. Pseudodissipation ͑the second invariant of the velocity gradient tensor͒ has the same intermittency exponent as dissipation and is closest to log-normal. Conditional acceleration variances increase with each conditioning variable, consistent with the scenario of rapid changes in velocity for fluid particles moving in local regions of large velocity gradient, but in a manner departing from Kolmogorov's refined similarity theory. Acceleration conditioned on the pseudodissipation is closest to Gaussian, and well represented by a novel "cubic Gaussian" distribution. Overall the simulation data suggest that, with the aid of appropriate parameterizations, Lagrangian stochastic modeling with pseudodissipation as the conditioning variable is likely to produce superior results. Reduced intermittency of conditional acceleration also makes the present results less sensitive to resolution concerns in DNS.

We perform direct numerical simulations (DNS) of the hyperviscous Navier-Stokes equations in a periodic box. We consider values of the hyperviscosity index h = 1, 2, 8, and vary the hyperviscosity to obtain the largest range of lengthscale ratios possible for well-resolved pseudo-spectral DNS. It is found that the spectral bump, or bottleneck, in the energy spectrum observed at the start of the dissipation range becomes more pronounced as the hyperviscosity index is increased. The calculated energy spectra are used to develop an empirical model for the dissipation range which accurately represents the bottleneck. This model is used to predict the approach of the turbulent kinetic energy k to its asymptotic value, k ϱ , as the hyperviscosity tends to zero.

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals' theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick's law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation

Lagrangian statistics of fluid-particle velocity and acceleration conditioned on fluctuations of dissipation, enstrophy and pseudo-dissipation representing different characteristics of local relative motion are extracted from a direct numerical simulation database of stationary (forced) homogeneous isotropic turbulence. The grid resolution in the simulations is up to 20483, and the Taylor-scale Reynolds number ranges from about 40 to 650, where characteristics of small-scale intermittency in the Eulerian flow field are well developed. A key joint statistic of the conditioning variables is the dissipation-enstrophy cross-correlation, which is asymmetric, but becomes less so at high Reynolds number. Conditional velocity autocorrelations are consistent with rapid changes in the velocity of fluid particles moving in regions of large velocity gradients. Examination of statistics conditioned upon enstrophy, especially in a local coordinate frame moving with the vorticity vector, and of the centripetal acceleration suggests the presence of vortex-trapping effects which persist for several Kolmogorov time scales. Further results on acceleration statistics and joint velocity-acceleration autocorrelations are also presented to help characterize in detail the properties of a joint stochastic process of velocity, acceleration and the pseudo-dissipation. Together with recent work on Eulerian conditional acceleration and Reynolds-number dependence of basic Lagrangian quantities, the present results are directly useful for the development of a new stochastic model formulated to account for intermittency and Reynolds-number effects as described in detail in a companion paper. © Cambridge University Press 2007

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