Yes, Navier-Stokes equation would help us understand non-laminar flow. My work involves a dumbed down version of this equation to a simple system of odes with a penalty function on the boundary conditions. I have had no luck in solving these equations, I am going to build a super computer using several raspberry pi computers to try to solve them numerically.
Terry Tao has written a series of papers trying to understand the blow up behavior of the Navier-Stokes problem. His latest is here https://terrytao.wordpress.com/2016/06/29/finite-time-blowup-for-lagrangian-modifications-of-the-three-dimensional-euler-equation/