If we know that the relevant physical quantities in a physical system are only , then the Reynolds number is essentially fixed by the Buckingham π theorem.
In detail, since there are 4 quantities , but they have only 3 dimensions (length, time, mass), we can consider , where  are real numbers. Setting the three dimensions of  to zero, we obtain 3 independent linear constraints, so the solution space has 1 dimension, and it is spanned by the vector . 
Thus, any dimensionless quantity constructed out of  is a function of , the Reynolds number.
Alternatively, we can take the incompressible Navier–Stokes equations (convective form):Remove the gravity term , then the left side consists of inertial force , and viscous force . Their ratio has the order of , the Reynolds number. This argument is written out in detail on the Scallop theorem page.