Dear Suleyman Cengizci,

To my understanding, the dimensionless analysis of N-S equations is helpful to compare the main mechanisms governing the viscous fluid flow. For this it may be easier to neglect some mechanisms that not dominant for some special flow. For example, for the low Renold number flow (this may include friction flow,  drag force around fine particles, etc.), with the aid of dimensionless analysis, we can easily to neglect the inertial force and pay more attention on the viscous force, and thus the N-S equation is simplified and easy to be solved, with results in good agreement with experiments. Actually such treatment is also used for other cases but not only N-S equations. For example, for heat and mass transfer, the dimensionless statements for the governing equations are also widely used. 

In another word, the dimensionless statements are just like a prism which guide us to see the N-S equations from another view, which featured the main mechanism governing the flow.

Hope above discussions are helpful. 

Thank you so much Dear Prof. Zhao