On the solutions of Navier-Stokes equations for turbulent channel flows in a particular function class

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.