§ 3.1  Module 3 — Head Loss: Friction

Laminar, Turbulent, and Reynolds

One dimensionless number decides which physics you are allowed to use. For everything a municipal station pumps, it is not a close call.

Module 2 left you with total dynamic head as an accounting problem: static lift plus every loss between the two water surfaces. This module prices the largest single entry in that ledger for a long pipeline. But no friction equation is usable until you know which of two entirely different regimes the pipe is in — they have different resistance laws, different velocity profiles, and different lists of what matters, and in one of them the pipe's roughness does not appear at all. One dimensionless number decides. For a force main the honest answer is not "probably turbulent" but "turbulent by a factor of eleven, at the slowest velocity any standard permits, in the smallest pipe any standard permits".

3.1.1The ratio that decides which physics applies

Water driven along a pipe is resisted by two mechanisms that scale differently. Momentum carried bodily by the moving fluid — inertia — grows with ρV²; momentum handed sideways between layers by molecular friction — viscosity — grows with μV/D. Take the ratio and the units cancel completely:

Re = ρVD/μ = VD/ν              V  mean velocity, m/s
                               D  inside diameter, m
                               ν  kinematic viscosity, m²/s  (= μ/ρ)

and because you are always given a FLOW, not a velocity,
substitute V = 4Q/πD² :

Re = 4Q/(πDν)                  Q  flow, m³/s

The two forms you will actually need

That the result is dimensionless is the whole point. One number describes every pipe: a 6 mm laboratory tube and a 900 mm transmission main at the same Reynolds number have geometrically similar flow. That is why a friction chart drawn in the 1940s still sizes pipe today.

• Q, the flow — the only term an operator moves hour to hour; Re is linear in it.
• D, the inside diameter — a nominal 250 mm pipe is not 250 mm bore, and that error lands in Re, in velocity and in the friction factor at once.
• ν, the kinematic viscosity — a property of the fluid and its temperature, not of the pipe. 1.1395 × 10^-6 m²/s at 15 °C, 1.5694 × 10^-6 at 4 °C, 8.9273 × 10^-7 at 25 °C: a factor of 1.76 across the range a buried main sees.
• Nothing else. No roughness, no length, no pressure, no pump. Re describes the flow, not the pipeline.

Q = 0.040 m³/s     D = 0.250 m     T = 15 °C

  A  = πD²/4  = π × 0.250² / 4            = 0.049087 m²
  V  = Q/A    = 0.040 / 0.049087          = 0.8149 m/s
  ν  at 15 °C                             = 1.1395 × 10⁻⁶ m²/s
  Re = VD/ν   = 0.8149 × 0.250 / 1.1395e-6 = 178,780

  in one step, from the flow:
  Re = 4Q/(πDν) = 4(0.040) / (π × 0.250 × 1.1395e-6)
                                          = 178,780   turbulent

Worked: 40 L/s in a 250 mm force main at 15 °C

The flow form is the one worth memorising, for the sign it exposes: at fixed flow the Reynolds number falls as the pipe gets larger. Velocity drops as D², the explicit D in the numerator recovers only one power of it, and the net dependence is 1/D. The instinct that a bigger pipe is "more turbulent" is exactly backwards — and it is the diameter ceiling of §1.3 wearing different clothes: oversize a force main and one decision lowers its velocity, lowers its Reynolds number and lengthens its detention.

3.1.2Two laws, and what changes when you cross between them

Below Re of about 2,000 the fluid moves in sliding layers that do not trade parcels. All the resistance is viscous shear, and the problem has an exact solution — found forty years before anyone knew why it stopped working: h_f = 32μLV/(ρgD²), which in Darcy-Weisbach's bookkeeping is f = 64/Re, exactly. Notice three features before they disappear: head loss is linear in velocity, there is nothing to look up, and the wall's roughness does not appear anywhere. A laminar pipe does not care what it is made of.

Above Re of about 4,000 the layers break down. Eddies carry parcels bodily across the flow, so momentum moves by bulk motion rather than by molecules and the resistance becomes mostly the cost of stirring. Every item above inverts: loss grows with roughly the square of velocity, the wall's texture matters because the eddies reach it, and there is no closed form — f comes from a correlation, which is what §3.2 is for.

The velocity profile is where the difference stops being notational. Laminar flow is a parabola — centreline exactly twice the mean, shear spread across the whole radius. Turbulent flow is far blunter: Prandtl's one-seventh power law puts the centreline at 60/49 = 1.2245 times the mean, with almost the entire velocity gradient compressed into a thin layer against the wall. That steep wall gradient is the friction.

3.1.3Why the transition is a band and not a number

Between the two figures the pipe is not "somewhat laminar". It is intermittently both: turbulent puffs are born, travel with the bulk flow and die, separated by stretches of undisturbed laminar flow, so a pressure tapping at a fixed station sees resistance flickering between two laws rather than settling on one.

How far apart are those two laws? Take Re = 4,000, where both are still written down. The laminar law gives f = 64/4000 = 0.0160, no iteration and no roughness term. Colebrook-White for 250 mm unlined ductile iron — e/D = 0.26 mm / 250 mm = 1.040 × 10^-3 — gives 0.04095: a factor of 2.559. The band is not a region of a few percent uncertainty. It contains two different answers and no basis for choosing.

Two working consequences. First, 2,000 and 4,000 are not the same kind of number: the lower is close to a physical constant, the upper a convention placed anywhere from 3,000 to 5,000. Second, the design rule — if a duty point lands inside the band, do not report a friction factor for it. Bound it with both laws and see whether the decision changes. src/core/hydraulics.js interpolates across the band to keep its curves continuous; every instrument here labels the band rather than quoting that interpolation as a result.

3.1.4Why a force main is never anything but turbulent

Two requirements already in force push the Reynolds number up from below, and they are requirements rather than tendencies. Ten State Standards asks for at least 0.6 m/s (2 ft/s) in a force main and forbids piping smaller than 100 mm (4 in) on raw wastewater. Put the two minima together — slowest permitted velocity, smallest permitted pipe — at a cold 10 °C, where ν is 1.3079 × 10^-6 m²/s:

Re = VD/ν = 0.6 × 0.100 / 1.3079e-6 = 45,900      11.5 × the 4000 floor

Going the other way — the flow at which a pipe LEAVES turbulence,
from the closed-form inverse  Q = πDνRe/4  at Re = 4000, 15 °C:

  100 mm main   0.358 L/s
  250 mm main   0.895 L/s   =  3.22 m³/h, about a garden tap
  600 mm main   2.148 L/s

Note the direction: at fixed Re, Q ∝ D. The bigger pipe needs MORE
flow to be laminar, because Re falls as 1/D at fixed flow.

The floor under Reynolds number, and the flow needed to escape it

Real duties sit far above that floor: the 40 L/s force main above at Re = 1.79 × 10⁵, a 300 mm potable main at 1.2 m/s at 3.16 × 10⁵, a 900 mm transmission main at 1.5 m/s at 1.18 × 10⁶. Even the gravity sewer arriving at the station is turbulent: a 300 mm sewer half full at 0.4% grade with Manning n = 0.013 flows at 0.865 m/s, and on a hydraulic diameter of 0.300 m that is Re = 2.28 × 10⁵. Reaching the band in the 250 mm main means throttling below 0.9 L/s — a flow at which self-cleansing and detention have been violated for hours.

Three things follow from being firmly turbulent, and they are the working content of this module.

1. Every correlation ahead may assume turbulence. That licenses Hazen-Williams (§3.4), which carries no viscosity term and is invalid in laminar flow. Measured against iterated Colebrook-White by tools/verify-hydraulics.mjs, the explicit Swamee-Jain fit's worst disagreement stays under 1% for Re ≥ 10⁵ over e/D 10^-5 to 10^-2 — 0.80% at the grid points that file samples, and 0.90% at the true worst point of the range, Re = 10⁵ with e/D ≈ 2.3 × 10^-3, which sits between two of its roughness columns. It rises to 2.75% only at Re = 5 × 10³ with very rough pipe — worst exactly where municipal flow never is.
2. Roughness becomes a first-order design variable. At 40 L/s in a 250 mm main the loss per 100 m is 0.217 m in PVC and 0.441 m in aged cast iron: a factor of 2.03 for the same bore and duty. It is also the one input that worsens every year the main is buried (§3.5).
3. Loss scales as roughly Q^1.9, not Q. Over 600 m of 250 mm unlined ductile iron, 40 L/s costs 1.733 m and 80 L/s costs 6.703 m: a factor of 3.868, exponent 1.9514. Under four because f itself eases from 0.02133 to 0.02063 as Re rises.

That last exponent is why a main sized for average flow is punished at peak, and why diameter is the decision that governs. The module epigraph says friction is levied on velocity squared; the exact local exponent is 1.9374 at this duty on unlined ductile iron and 1.8044 on smooth PVC, reaching 2 only in the fully-rough limit §3.2 arrives at. Squared is the right instinct and the wrong number.

3.1.5Where the regime question does bite

None of this makes the Reynolds number a formality: a station contains more than its force main. Where the question stays live is small bore, low flow, or not water.

• Chemical metering and seal-water lines. Here the flows are small enough that the regime is genuinely in play — but check it, because the tubing sizes invite exactly the error §3.1.1 warned about. At 15 °C a 6 mm tube is laminar only below 0.64 L/min and a 12 mm tube only below 1.29 L/min, and note the direction: Re = 4Q/(πDν), so at the same flow the narrower tube has the higher Reynolds number. It is the 12 mm line that stays laminar longest, not the 6 mm one. A few litres a minute is already turbulent in either — 3 L/min gives Re 9,310 in 6 mm and Re 4,660 in 12 mm. Even 0.045 L/s in a 25 mm seal-water line is Re = 2,010 — not laminar but just inside the band, and it takes 0.0447 L/s to sit on 2,000. Where the line really is laminar, a turbulent correlation overstates the loss.
• Sludge, polymer and dosed chemicals. Thickened sludge is not Newtonian, so ν is not a number. Reynolds still decides the regime; you cannot look up the input.
• Start-up, shut-down and slow filling. Every main crosses the band going up and again coming down. What matters there is not the friction estimate but what else is true at those velocities: nothing scours, no air is swept.
• Anything but water near 15 °C. Glycol, fuel oil, digester recirculation at 35 °C. Only ν moves — a long way.

What this lesson has deliberately not done is name the friction factor for turbulent flow. We used the fact that one exists and quoted values from src/core/hydraulics.js; where 0.02133 comes from, and what the Moody chart's flat right-hand side means, is §3.2. Competing correlations and their error bars are §3.3 and §3.4. Fittings and valves are Module 4, and inside a station they are not minor. The pump is still a black box supplying whatever head the system asks for; what it can actually deliver is Module 6.