Flow regime in the tube

Is the medium flowing smoothly through the tube, or turbulently — and how fast?

Re = \frac{ρ \cdot v \cdot D}{η}, \qquad v = \frac{4 \cdot Q}{\pi \cdot D^2}

Method

1. Convert everything to SI: diameter to metres, flow rate to cubic metres per second, viscosity to pascal-seconds.
2. Find the mean velocity: flow rate divided by the tube's cross-section, π D² ÷ 4. This is the average across the bore — the centre of a laminar flow moves twice as fast as this, the edge not at all.
3. Multiply density by velocity by diameter, and divide by viscosity. The result has no units: it is the ratio of the flow's inertia to the viscous forces holding it in order.
4. Read the regime off the number. Under about 2300 the flow is laminar; over about 4000 it is turbulent; in between it is transitional and can flip either way with a disturbance.

Assumptions

• A straight, smooth, circular tube of uniform bore, full of fluid. Bends, fittings, a rough wall or an obstruction — a spiral divider, for one — all trip a flow into turbulence well below the 2300 figure.
• A Newtonian fluid. A heavily loaded slurry is not one, and neither is a magnetised one: its effective viscosity depends on how hard it is sheared and on the field, so its Reynolds number is not a single figure.
• Steady flow. A pulsed drive, or a pump that delivers in strokes, gives a flow whose regime changes through the cycle.
• The transition thresholds are the textbook ones for pipe flow. They are not sharp — the 2300–4000 band is exactly the range where a careful experiment can produce either regime.