I am interesting in the turbulent plane Couette flow.
I would like to model the mean velocity profile, which only exists in streamwise direction and is time independent as $(U(x_2),0,0)$.
Using the Navier Stokes equation as:
$\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial x_1}+V\frac{\partial U}{\partial x_2}+W \frac{\partial U}{\partial x_3}+\frac{\partial p}{\partial x_1} =\nu(\frac{\partial^2 U}{\partial x_1^2}+\frac{\partial^2 U}{\partial x_2^2}+\frac{\partial^2 U}{\partial x_3^2})$
$\frac{\partial V}{\partial t}+U\frac{\partial V}{\partial x_1}+V\frac{\partial V}{\partial x_2}+W \frac{\partial V}{\partial x_3}+\frac{\partial p}{\partial x_2} =\nu(\frac{\partial^2 V}{\partial x_1^2}+\frac{\partial^2 V}{\partial x_2^2}+\frac{\partial^2 V}{\partial x_3^2})$
$\frac{\partial W}{\partial t}+U\frac{\partial W}{\partial x_1}+V\frac{\partial W}{\partial x_2}+W \frac{\partial W}{\partial x_3}+\frac{\partial p}{\partial x_3} =\nu(\frac{\partial^2 W}{\partial x_1^2}+\frac{\partial^2 W}{\partial x_2^2}+\frac{\partial^2 W}{\partial x_3^2})$
$\frac{\partial U}{\partial x_1}+\frac{\partial V}{\partial x_2}+\frac{\partial W}{\partial x_3}=0$
Applying the above mentioned mean velocity profile simplifies the system of Navier Stokes equation into:
$0=\nu \frac{\partial^2 U}{\partial x_2^2}$
Thus the mean velocity is abtained as $U=A x_2$, leading to a liear profile.
However in reality the mean velocity is seen to be more of a "S-type" of a profile, resembling a cubic profile. 
However I do not understad how such a profile can fulfill the Navier Stokes equation.
Can someone help me understand how to model the mean velocity profile for a turbulent plane Couette flow?
I would highly appreciate any help.
For a complete turbulent flow solution, you cannot ignore the unsteady terms (the time derivatives), so your simplified equation is incorrect. The most common simplification in turbulent flow analyses is expressing all field variables as being the sum of mean and fluctuating/unsteady/turbulent components, after which the N-S equations are "Reynolds-averaged" to produce the famous Reynolds-averaged-Navier-Stokes (RANS) equations for the time-independent mean flow. These equations contain extra terms, especially the so-called turbulent shear stress terms that are missing in your simplified equation.
Please look up any graduate-level fluid mechanics text for more details on turbulent flow in general and the RANS equations in particular.