One dimensionless ratio that says whether the flow will glide or tumble. Reynolds compared the inertial force pushing a parcel of fluid forward against the viscous force smearing it out — and the whole regime hangs on that single number. Down the center, data flows: density, speed, size, viscosity go in, the engine forms Re = ρvL/μ, the regime comes out. The blue team builds and defends it; the red team tries to break it.
source Reynolds, O. (1883) An Experimental Investigation of the Circumstances Which Determine Whether the Motion of Water Shall Be Direct or Sinuous…, Phil. Trans. R. Soc. 174, 935–982 — doi.org/10.1098/rstl.1883.0029. Rendered, not quoted.
Re is not a fudge factor; it is the honest ratio of two stresses on the same parcel of fluid:
inertial stress ρv²/L (the momentum the flow carries) over viscous stress μv/L² (the friction resisting it). Divide, and length and speed collapse:
(ρv²/L) ÷ (μv/L²) = ρvL/μ = Re
For the current settings, the two stresses and their ratio, computed live:
| inertial ρv²/L | viscous μv/L² | ratio = Re |
|---|
Re is the switch between two neighbouring spheres. At low Re viscosity wins and drag is linear in speed — that is the-stokes-drag's world, 6πμrv.
At high Re inertia wins and drag turns quadratic — that is the-drag-equation's world, ½ρv²CdA. Reynolds 1883 built the one dial that tells you which regime you are standing in. Each sphere is the next one's boundary condition.
The blue team's live check: re-derive Re independently and confirm it is dimensionless (invariant under unit rescaling), linear in v, and inverse in μ. If red moves μ to the numerator, this badge is where it shows.
Re needs exactly four measured things and nothing else:
| symbol | quantity | SI unit |
|---|---|---|
| ρ | density | kg·m⁻³ |
| v | flow speed | m·s⁻¹ |
| L | characteristic length | m |
| μ | dynamic viscosity | Pa·s |
Their units cancel completely — kg·m⁻¹·s⁻¹ on top and bottom — so Re is a pure number. That cancellation is what the panel exploits, and what the witness re-checks.
Re is formed from ρ, v, L, μ on the spot — never looked up. Slide v or L and watch inertia overtake viscosity.
Threshold pipe values (AMBER ~2300 / ~4000) are conventional and disturbance-dependent — see the adversary.
What the machine produces, proven: a single dimensionless Re, its reading as a force ratio, and the pipe regime — laminar below Re ~ 2300, transitional between, turbulent above ~4000. Two different (v, L) pairs with the same Re are dynamically similar — the reason a scale model in a wind tunnel predicts the real thing.
The blue team's witness (left) confirms the algebra live; the red team (right) tries to make it lie.
And Re is only one group. Compressibility (Mach), free-surface waves (Froude), heat (Prandtl) each add their own dimensionless number. Matching Re alone does not guarantee similarity when those matter — which is why the panel is honest that ~2300/~4000 are conventions, not laws.
"Turbulence begins exactly at Re = 2300." Cut. 2300 is a practical lower bound for pipes; transition is a range (~2300–4000) and can be pushed far higher. The engine labels the middle band transitional, not a knife-edge.
"Reynolds number gives you the drag." Cut. It sets the regime; the drag coefficient is a function of Re, measured — not delivered by it. Re tells you which curve to read.
"The characteristic length is obvious." Kept, corrected. L is a convention — pipe diameter, plate length, airfoil chord. Re values only compare within the same choice of L.
The red team's move: put viscosity in the numerator — Re = ρvL·μ — so thicker fluid reads as more turbulent. Doubling μ would now double Re instead of halving it. The blue team's witness (window 7) is watching.
Flip μ upstairs and the inverse-in-μ law breaks: the witness recomputes, finds doubling μ no longer halves Re, and turns red. Nothing is faked; the attack is real and it is caught.