There are different methods to approximate naiver stokes equations, actually all the solutions available now for various flow, heat transfer problems are solutions of these reduced form.

For example, for turbulence modeling a time averaged naiver stokes equations are used. Here velocity terms are divided into two, a constant term and a variable. Thus resulting equations are called Reynolds averaged naiver stokes equations.

You can also use similarity solution to approximate the solutions of partial differential equations. Please search for similarity solution for PDE's.

Hope this help you.

Regards.

If you are solving the Navier-Stokes equation with the periodic boundary conditions I would recommend you to use the spectral method, see Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, Thomas A. Zang "Spectral Methods: Fundamentals in Single Domains", Springer, 2007.

Hammad.

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Since the early twentieth century began to use numerical methods for solving the Navier-Stokes equations, so it is necessary to know the problem to suggest something.

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A very general question.

Of course, there are a lot of methods of NS numerical approximation that can be found in any CFD book. If you have simple geometry then spectral methods are the best.

Please, make concrete your question, what you mean under your question

I have been suggesting exact solutions in 1D, 2D, 3D. These solutions may be approximated by dropping some terms. I refer interested researchers to my published  papers in Results in Physics (2011), Physics Letters A (2014) and my recent post of a technical report, submitted for publication in Physica A, all are available from ResearchGate.  The learning curve will be steep in the beginning, but the interested reader will find many applications. Good luck.

There are many methods, approximating through numerical (FEM, FDM, FVM, etc) or analytical (ADM, HPM, etc) paths, satisfying all the tastes.

It's depends upon the accuracy, sometimes one can start from simplest damped oscillatory model. I have listened somewhere that infinitesimal Lie transform reduces ideal fluid to just a linear pendulum.

There are simple and taugh answers for this question. 

The simple is : Yes, there are several methods, that includes finite differences, finite volumes, finite elements and spectral methods as well.

The taugh answer needs more space to answer it , but basiclly :

1) Navier Stokes equations contains material derivatives and temporal derivatives.

2)  you have to choose a numerical method or an aproach to approximate the material derivatives ; 

3) then, you have to choose other numerical  (usually an iterative)  method to solve the temporal derivative. 

4) During these processes in 2 and 3, you realize (or you need a mentor to explain to you) that you need to do solve a "coupling" situation, that the velocity depends on the pressure. You have look for a method to solve this . For FiniteVolume there is the SIMPLE method (and similars) to solve it. .

I really suggest you to read and study some books : 

For the basics :  Computational Fluid Dynamics by John Anderson

For methods :  

An Introduction to Computational Fluid Dynamics: The Finite Volume Method (2nd Edition) by H. Versteeg and W. Malalasekera;

Essential Computational Fluid Dynamics by Oleg Zikanov

There's an online book, I don't have the opportunity to verify all the book, but it seems very introductory and I think it could help you : http://www.cfdbooks.com/cfdbooks.html 

These are only 3 of several. 

I do not know if you have studied CFD (or Computational Fluid Dynamics ) previously. If not so, I would suggest you start by the basics problems and try to follow an introductory course. 

Mrs Lorena Barba has a very very good course in YouTube that I really recommend. 

Navier Stokes equations request you some understanding about the physics and also about the methods you want (or the method you feel confortable) to use.

I hope my answer help you.

There are lots of simplified solutions for N-S equations, but each one has its own assumptions for a particular problem (for example shallow water, jets, Boundary layer,…). Therefore first of all you have to define the problem.

Few problems have analytical solutions and most problems don't, so the next step is choosing a numerical method to solve the N-S equations.