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Orlandi, P, Pirozzoli S, Bernardini M, Carnevale GF.  2014.  A minimal flow unit for the study of turbulence with passive scalars. Journal of Turbulence. 15:731-751.   10.1080/14685248.2014.927066   AbstractWebsite

The concept of a minimal flow unit (MFU) for the study of the basic physics of turbulent flows is introduced. The MFU is an initial vorticity configuration that consists of a few simple well-defined large-scale vortex structures. The form and position of these structures are chosen so that their interaction produces turbulence capturing many of the essential characteristics of isotropic homogeneous turbulence produced from random-phase initial conditions or that produced by continual random-phase forcing. The advantage of using the MFU is that the evolution of the vortex structures can be followed more clearly and the relationship between the evolving vortex structures and the various ranges in the energy spectrum can be more clearly defined. The addition of passive scalar fields to the MFU permits an investigation of passive scalar mixing that is relevant to the study of combustion. With a particular choice of the MFU, one that produces a trend to a finite-time singularity in the vorticity field, it is demonstrated that passive scalar distributed in the original large-scale vortices will develop intense gradients in the region where the vorticity is tending toward a singularity. In viscous flow, the evolution of the MFU clearly shows how the volume of the regions where originally well-separated passive scalars come into contact increases with increasing Reynolds number.

Orlandi, P, Pirozzoli S, Carnevale GF.  2012.  Vortex events in Euler and Navier-Stokes simulations with smooth initial conditions. Journal of Fluid Mechanics. 690:288-320.   10.1017/jfm.2011.430   AbstractWebsite

We present high-resolution numerical simulations of the Euler and Navier-Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier-Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (t(s)) with scaling vertical bar u vertical bar(infinity) similar to (t(s) - t)(-1/2), vertical bar omega vertical bar(infinity) similar to (t(s) - t)(-1). This blow-up is associated with the formation of a k(-3) spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward t(s), the total enstrophy is observed to increase at a slower rate, Omega similar to (t(s) - t)(-3/4), than would naively be expected given the behaviour of the maximum vorticity, omega(infinity) similar to (t(s) - t)(-1). This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various Re, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid t(s). In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching k(-513). The simulations show that the peak value of the enstrophy scales as Re-3/2, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of Re, supporting the validity of Kolmogorov's law of finite energy dissipation. At later times the kinetic energy shows a t(-1.2) decay for all Re, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to ts, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a k(-5/3) range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.