  In this paper, we continue the discussion as done in \cite{CTZ15} on
turbulent channel flow described by the Navier-Stokes model and the
Navier-Stokes-alpha model. We study the non-stationary solutions for the
Navier-Stokes equations and Navier-Stokes-$\alpha$ model having particular
function forms. In particular, the term of sum of pressure and potential can be
shown to be harmonic in the space variable.
