Turbulence and Multiscaling in the Randomly Forced Navier-Stokes Equation

We present a pseudospectral study of the randomly forced Navier-Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance $\ensuremath{\sim}{k}^{4\ensuremath{-}d\ensuremath{-}y}$, with $k$ the wave vector and the dimension $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$. We provide the first evidence for multiscaling of velocity structure functions for $y\ensuremath{\ge}4$. We extract the multiscaling exponent ratios ${\ensuremath{\zeta}}_{p}/{\ensuremath{\zeta}}_{2}$ by using extended self-similarity, examine their dependence on $y$, and show that, if $y\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4$, they are in agreement with those obtained for the Navier-Stokes equation forced at large spatial scales (3DNSE). Also well-defined vortex filaments, which appear clearly in studies of the 3DNSE, are absent in the RFNSE.