  In this article, we consider a special class of initial data to the 3D
Navier-Stokes equations on the torus, in which there is a certain degree of
orthogonality in the components of the initial data. We showed that, under such
conditions, the Navier-Stokes equations are globally wellposed. We also showed
that there exists large initial data, in the sense of the critical norm
$B^{-1}_{\infty,\infty}$ that satisfies the conditions that we considered.
