Turbulent mixing of substances which are highly diffusive

How a substance gets mixed by a fluid, even when the motion is turbulent, depends to some extent on whether its diffusivity is small or large. The magnitude of the diffusivity is usually expressed by the Schmidt number ($Sc$, ratio of fluid viscosity to the diffusivity of the substance). The case of passive scalars (which have no back-reaction on the flow) with large $Sc$ (weak diffusivity) has received considerable attention, especially for its special features such as the -1 power roll-off of the spectrum of the fluctuations. Similar studies for passive scalars at low Schmidt numbers (or large diffusivity) do not yet exist, though the classical theory (Batchelor, Howells \& Townsend, J. Fluid Mech, {\bf 5}, 134 (1959)) is now more than fifty years old. In this talk we report direct numerical simulations for decaying scalar fields with $Sc$ as low as 1/2048, at grid resolution up to $4096^3$, in stationary isotropic turbulence with microscale Reynolds number in the range 140-390. We examine the validity of theoretical assumptions that lead to a spectral slope of -17/3 in the so-called inertial-diffusive range. Despite limitations on the range of scales in the simulations, the data support the theory as the Schmidt number decreases and the Reynolds number increases.