We aim at studying an incompressible fluid  model used to simulate the long time average of the velocity and pressure of a turbulent flow. The model is a steady like Navier-Stokes Equations system, with an additional eddy viscosity term and a wall law as boundary conditions.  We first detail the mixed variational formulation of this system expressed not only in terms of the velocity, but also in terms of the pressure. We show how to approximate it by a sequence of variational problems, derived from  singular perturbations of the incompressibility constrain with an additional pressure term, that allow to get stable estimates for the pressure. We construct a solutions to those systems by the Galerkin method, and then we take the limit in the equations, which yields solutions to the initial variational problem. Uniqueness is obtained subject to restrictive hypothesis about the data. Next, we consider the linearized variational problem derived from the initial one, which is shown to have a unique solution. This leads to construc a further solution to the model by the Leray-Hopf fixed point theorem, after having inferred some compactness properties from the energy method.