Colebrook–White — the equation behind every curve
The Darcy friction factor is implicit — λ appears on both sides — so every point on this chart is obtained by fixed-point iteration on \(x = 1/\sqrt{\lambda}\): evaluate the right-hand side, invert, square, repeat until the value stops moving (here to 1e-13, typically 3–6 iterations). The two terms inside the logarithm are the physics: 2.51/(Re√λ) is the viscous contribution, (k/d)/3.7 the roughness contribution. Whichever dominates writes the law.
The three families on the chart
Below Re ≈ 2300 viscosity orders the flow and λ depends on Re alone. On the smooth line the viscous term dominates everywhere; Blasius (dashed) reproduces it only up to Re ≈ 1e5, which is why the solid smooth curve continues with the full Colebrook form — at Re = 1e8 Blasius would claim λ ≈ 0.0032 while the true smooth value is ≈ 0.0059, nearly a factor of two apart. The six roughness curves are the same Colebrook equation at k/d = 1e-5, 1e-4, 1e-3, 5e-3, 1e-2 and 5e-2; each one peels away from the smooth curve as Re grows and flattens into a vertical asymptote.
Regime criteria — the numbers behind the verdict card
| Criterion | Range | Verdict | Meaning |
|---|---|---|---|
| Re | < 2300 | 🟢 Laminar | λ = 64/Re exactly; roughness has no effect. Viscosity dominates the whole boundary layer. |
| Re | 2300 – 4000 | 🟡 Critical zone | Flow flips between laminar and turbulent states; λ is genuinely uncertain (the blend drawn here is a schematic). Avoid steady operation in this band. |
| Re·k/d | < 65 | 🟢 Hydraulically smooth | Roughness tips are buried inside the viscous sublayer; the k/d term in Colebrook is negligible and λ follows the smooth law. |
| Re·k/d | 65 – 1300 | 🟡 Transition | Both terms act: λ falls with Re and rises with k/d simultaneously — the true Moody chart region. |
| Re·k/d | > 1300 | 🟡 Fully rough | Viscous sublayer destroyed; λ = 1/[2·lg(3.7·d/k)]² is independent of Re — the curves flatten into vertical lines. |
Symbol table
| Symbol | Meaning | Units |
|---|---|---|
| λ | Darcy friction factor | dimensionless |
| Re | Reynolds number, Re = ρVD/μ | dimensionless |
| k/d | relative roughness (equivalent sand-grain roughness over inner diameter) | dimensionless |
| k | absolute roughness | mm / in |
| D | inner pipe diameter | mm / in |
Worked anchor (self-check)
Re = 1e5 and k/d = 0.001: three iterations of the Colebrook fixed point converge to \(1/\sqrt{\lambda} \approx 6.716\), i.e. λ ≈ 0.0222 — the marker lands exactly there on the k/d = 1e-3 curve. Two more checkpoints you can verify by eye: at Re = 2000 the point sits on the laminar line (λ = 64/2000 = 0.032 regardless of k/d), and at Re = 1e6 with k/d = 0.05 the product Re·k/d = 50,000 ≫ 1300 puts the point on the flat fully-rough asymptote λ = 1/[2·lg(3.7/0.05)]² ≈ 0.0716.
Engineering criteria applied
- Laminar bound — 2300; the 2300–4000 critical band is shaded and its λ is marked as a schematic blend, not a solved value.
- Smooth / rough split — Re·k/d = 65 and 1300 decide which Colebrook term dominates; the verdict card restates them with the live product.
- Curve generation — every roughness curve is Colebrook evaluated point-by-point in log-Re (no pre-tabulated chart data), so the marker and the curves can never disagree.