Fluid Mechanics — Flow & Pressure

The Reynolds number — smooth or turbulent?

continues from lesson 4 — values defined earlier in the course stay live here

ELI5: one dimensionless number, Re = ρ·v·D / μ, predicts whether flow is smooth (laminar) or chaotic (turbulent). Below ~2300 it's orderly; above ~4000 it churns. Because it carries no units, the same threshold works for a capillary and a pipeline.

\mu _{water} := 1e-3 Pa*s

= 0.001 Pa s water's dynamic viscosity

Re = \frac{\rho_{water} \cdot v_{1} \cdot d_{1}}{\mu_{water}}

400000

Reynolds number = 400000 (no units)

✓ pass

Re > 4000

firmly turbulent — expect mixing and pressure loss

1 - x^2

laminar velocity profile across a pipe: fastest at the centre (x = 0), zero at the walls (x = ±1)

Real-world hook: Re tells an engineer whether a pipeline runs smooth or rough, why golf balls have dimples (tripping turbulence cuts drag), and how wind-tunnel models scale to full size.