Many models include both nonlinear and non-analytical equations. There are indeed many nonlinear models with exact solutions. There are also many nonlinear models with no exact solution. Turbulence is one of these due to its highly random nature. The NS equations can be applied to fluid flow in general, but at high Reynold's numbers we cannot make the assumptions (steady, incompressible for air) required to simplify the equations into a solution which can be solved by hand. Depending on the Reynold's number, this solution can be highly nonlinear to the extent that it passes current computer's computational ability for analytical or numerical methods.Don't confuse nonlinearity to mean non-analytical. There are many nonlinear models that have exact solutions. As with your example with turbulence, we do have a a model to describe it (The Navier-Stokes eqs), but we don't have and exact solution. For now, numerical methods work, but we are always trying to find ways to simplify enormously complex situations to solves things quickly and on much greater scales.
I was wrong here as well with what I wrote, bad example.And in most cases, linearity is a simplification of a nonlinear problem. As an example, F=ma is the sum of the forces acting on an object, and doesn't model the individual components of the system.