Before you can calculate friction in a pipe you have to know which KIND of flow you are looking at, because the two kinds obey different arithmetic. The number that decides it is $Re = \dfrac{\rho v D}{\mu}$ — read aloud R-e equals rho v D over mu, with $\mu$ the Greek letter mu, said “mew”.

$\rho$ is the fluid density in kg/m³, $v$ is the average velocity in m/s, $D$ is the characteristic length — for a full round pipe, its inside diameter — in metres, and $\mu$ is the dynamic viscosity in pascal-seconds, the fluid's resistance to being sheared. Water at room temperature is about 0.001 Pa·s; a warm hydraulic oil is twenty times that or more. Push the units through and every one of them cancels: $Re$ is dimensionless, a pure ratio of inertial forces to viscous ones. The top of the fraction is the fluid's willingness to keep going; the bottom is the fluid's willingness to be restrained.

Below about 2300 viscosity wins and the flow is laminar: the fluid moves in smooth layers that slide past one another without mixing, and the velocity profile is a tidy parabola. Above about 4000 inertia wins and the flow is turbulent: the layers break down, eddies mix everything, and the profile flattens out. Between the two lies a transitional band nobody wants to design in. Notice that $D$ appears to the first power only — double the diameter at the same velocity and Re doubles, which is why a big main is turbulent at speeds where a capillary is still perfectly laminar.

In the laminar world the friction factor is exact and beautifully simple: $f = \dfrac{64}{Re}$, where $f$ is the Darcy friction factor, dimensionless, and the 64 comes straight out of solving the parabolic profile. Roughness does not appear anywhere — the viscous layer blankets the wall's texture completely, so a laminar flow cannot tell a rough pipe from a smooth one.

And here is the trap this lesson exists to set. $f = 64/Re$ is a permission, not a formula you may reach for at will. Used above 2300 it does not give a rough answer — it gives a number several times too SMALL, which under-sizes the pump and under-states the head loss. Wrong in the direction that hurts. Always check the regime before you pick the relation.