The Reynolds number can be obtained when one uses the nondimensional form of the incompressible Navier–Stokes equations for a newtonian fluid expressed in terms of the Lagrangian derivative:
Each term in the above equation has the units of a "body force" (force per unit volume) with the same dimensions of a density times an acceleration. Each term is thus dependent on the exact measurements of a flow. When one renders the equation nondimensional, that is when we multiply it by a factor with inverse units of the base equation, we obtain a form that does not depend directly on the physical sizes. One possible way to obtain a nondimensional equation is to multiply the whole equation by the factor
where
*  is the mean velocity,  or , relative to the fluid (m/s),
*  is the characteristic length (m),
*  is the fluid density (kg/m).
If we now set
we can rewrite the Navier–Stokes equation without dimensions:
where the term .
Finally, dropping the primes for ease of reading:
This is why mathematically all Newtonian, incompressible flows with the same Reynolds number are comparable. Notice also that in the above equation, the viscous terms vanish for . Thus flows with high Reynolds numbers are approximately inviscid in the free stream.