Fluid-flow topology optimization (FTO) allows the generation of innovative flow-channel layouts with minimal pressure drop (power dissipation) between inlet and outlet ports in a given design domain. FTO was first explored using Stokes flow with the material in the design domain modeled as a porous medium governed by Darcy’s law. More recently, Navier-Stokes flow has been implemented to consider higher Reynolds numbers. The objective of this work is to demonstrate the effect of the Reynolds number on the FTO results and generate a set of design rules. To this end, a density-based FTO algorithm and an in-house finite element analysis code for incompressible Navier-Stokes flow are developed. The optimization process is updated using the method of moving asymptotes so that the flow’s potential power is maximized. The nonlinear Navier-Stokes equations are solved using a trust region Newton’s method. Sensitivity analysis is carried out using the adjoint method. A parametric study of the underlying parameters of the Reynolds number in two numerical examples shows the effect of the fluid’s dynamic viscosity and velocity on the optimized flow channels. The results show that fluids with the same Reynolds number, but with different dynamic viscosity or velocity values, can generate significantly different flow channels.