Multiscaling in the randomly forced and conventional Navier-Stokes equations

We present an overview of some results we have obtained recently (A. Sain, Manu and R. Pandit, Phys. Rev. Lett. 81 (1998) 4377) from a pseudospectral study of the randomly forced Navier–Stokes equation (RFNSE) stirred by a stochastic force with zero mean and a variance ~k4−d−y, with k the wavevector and the dimension d=3. These include the multiscaling of velocity structure functions for y>4 and a demonstration that the multiscaling exponent ratios p/2 for y=4 are in agreement with those obtained for the Navier–Stokes equation forced at large spatial scales (3dNSE). We also study a coarse-graining procedure for the 3dNSE and examine why it does not lead to the RFNSE