  In this paper, we establish the existence and the regularity of a unique weak
solution to turbulent flows in a bounded domain ${\Omega}\subset \mathbb R^3$
governed by the so-called Leray-{\alpha} model. We consider the Navier slip
boundary conditions for the velocity. Furthermore, we show that, when the
filter coefficient {\alpha} tends to zero, the weak solution constructed
converges to a suitable weak solution to the incompressible Navier Stokes
equations subject to the Navier boundary condition. Similarly, if {\lambda}
tends to 1- we recover a solution to the Leray-{\alpha} model with the
homogeneous Dirichlet boundary conditions.
