  We solve the advection-diffusion equation for a stochastically stationary
passive scalar $\theta$, in conjunction with forced 3D Navier-Stokes equations,
using direct numerical simulations in periodic domains of various sizes, the
largest being $8192^3$. The Taylor-scale Reynolds number varies in the range
$140-650$ and the Schmidt number $Sc \equiv \nu/D$ in the range $1-512$, where
$\nu$ is the kinematic viscosity of the fluid and $D$ is the molecular
diffusivity of $\theta$. Our results show that turbulence becomes an
ineffective mixer when $Sc$ is large. First, the mean scalar dissipation rate
$\langle \chi \rangle = 2D \langle |\nabla \theta|^2\rangle$, when suitably
non-dimensionalized, decreases as $1/\log Sc$. Second, 1D cuts through the
scalar field indicate increasing density of sharp fronts on larger scales,
oscillating with large excursions leading to reduced mixing, and additionally
suggesting weakening of scalar variance flux across the scales. The scaling
exponents of the scalar structure functions in the inertial-convective range
appear to saturate with respect to the moment order and the saturation exponent
approaches unity as $Sc$ increases, qualitatively consistent with 1D cuts of
the scalar.
