What is the Reynolds number?
Re = ρvD/μ — density times velocity times diameter, divided by dynamic viscosity. It is the ratio of inertial forces to viscous ones, and it is dimensionless. That is the whole value of it: the same Reynolds number means the same flow behaviour whether you are pushing water through a garden hose or crude oil through a pipeline, which is what lets a scale model in a wind tunnel tell you anything about the real aircraft.
What Reynolds number is laminar, and what is turbulent?
For flow in a round pipe the usual figures are below 2,300 for laminar and above 4,000 for turbulent, with a transitional band between. Treat those as conventions rather than physics: they mark where disturbances reliably die out and where they reliably grow. A very carefully prepared, vibration-free flow has been held laminar past Re = 100,000, and a rough inlet can trip turbulence early.
What is the difference between laminar and turbulent flow?
Laminar flow moves in parallel layers that slide over one another without mixing, so a dye streak injected at the centre stays a thin line the whole length of the pipe. Turbulent flow is chaotic and well mixed, with eddies at every scale, and the dye spreads across the pipe within a short distance. The practical consequences are large: turbulent flow mixes and transfers heat far better, and costs far more to pump.
Why is the pressure drop so much higher in turbulent flow?
In laminar flow, pressure drop rises in direct proportion to velocity. In turbulent flow it rises with roughly the square of velocity, because energy goes into generating and dissipating eddies as well as into shearing the fluid. Doubling the flow rate therefore roughly quadruples the pumping cost, which is why oversizing a pipe is usually cheaper over its life than fitting a bigger pump.
Does pipe roughness matter in laminar flow?
No — and this surprises people. In laminar flow the friction factor is exactly 64/Re, with no roughness term at all. The fluid touching the wall is stationary and the layers above it slide smoothly, so they never interact with the bumps. Roughness only starts to matter once turbulent eddies reach down into the wall layer, which is why polishing a pipe is worth nothing at low Reynolds numbers.
What is the Darcy friction factor and how is it found?
It is the f in the Darcy–Weisbach equation, h = f·(L/D)·v²/2g, which converts a flow into a head loss. Laminar flow has the exact answer f = 64/Re. Turbulent flow does not: the Colebrook–White equation relates f to both the Reynolds number and the relative roughness, and it is implicit — f appears on both sides — so it has to be solved by iteration. This calculator iterates it rather than using an explicit approximation.
Why is the velocity at the centre of the pipe faster than the average?
The fluid at the wall is stationary, so the flow has to speed up towards the middle. In laminar flow the profile is an exact parabola and the centreline moves at precisely twice the mean. In turbulent flow, mixing flattens the profile into a blunt plug, and the centre is typically only about 20% above the mean — with almost the whole velocity change squeezed into a very thin layer at the wall.
What is entrance length?
The distance a flow needs to travel from the inlet before its velocity profile settles into the shape the standard formulas assume. It is roughly 0.05·Re·D for laminar flow, which for a slow-moving viscous flow can be many metres, and much shorter for turbulent flow. In a pipe shorter than its entrance length the friction is higher than calculated, which is a common reason a laboratory measurement disagrees with a textbook answer.
How do I make a flow laminar?
The three levers are visible in ρvD/μ: slow the flow down, use a narrower pipe, or use a thicker fluid. Velocity and diameter are usually the ones you control. One counter-intuitive point: heating a liquid thins it and therefore pushes it towards turbulence, while heating a gas thickens it and pushes it the other way.
Does the Reynolds number apply to things other than pipes?
Yes — it applies to any flow, using a relevant length rather than pipe diameter: an aerofoil chord, a sphere's diameter, a ship's length. The transition values differ for each geometry, so the 2,300 and 4,000 here are specific to round pipes. Its power is that it makes flows comparable across scales, which is the foundation of model testing in wind tunnels and towing tanks.