They are serveral types of bounary value problems, and the numerical methods maybe differents, you should read some books concerning numerical method of PDE or Computational Fluid Dynamics

I am trying to solve one research problem but at the end I am stuck this PDE navier Stokes Equation at the end

May be this article can help you.

Pakdemirli, Mehmet, and Muhammet Yurusoy. "Similarity transformations for partial differential equations." SIAM review 40.1 (1998): 96-101.

There are many methods to solve N-S equations depends on the flow is compressible or incompressible, even for compressible flow, the method chosen for solving also depends on the initial flow conditions (subsonic or supersonic etc.), because these kinds of conditions determine the mathematical characteristics of the N-S equations.

I advice you to read about some CFD books.

For nonliner PDE 2nd order you can apply Homotopy analysis method Generlise seperable method and successesive linearization mrthods to solve

Numerical methods like finitr differece Crenck Nicolsion schem and finit element method in some cases applicable.