Laminar & Turbulent Flow Notes

Last reviewed 16 Sept 2026 · 6 min read

Reynolds experiment

Osborne Reynolds injected dye into water flowing through a glass tube:

At low velocity, the dye moved as a straight thread — laminar flow.
At intermediate velocity, the thread wavered — transition.
At high velocity, the dye mixed across the tube — turbulent flow.

FormulaReynolds number

R e = \frac{ρ V D}{μ} = \frac{V D}{ν} = \frac{inertia force}{viscous force}

Pipe flow (typical limits): laminar $R e < 2000$ ; transition 2000–4000; turbulent $R e > 4000$ . Open channels (using hydraulic radius $R$ ): laminar below about 500.

The lower critical Reynolds number (≈ 2000) is the more meaningful limit — below it, disturbances die out.

Laminar flow through a circular pipe (Hagen–Poiseuille)

For steady, fully developed laminar flow in a pipe of radius $R$ :

FormulaHagen–Poiseuille flow

Shear stress: $τ = (- \frac{d p}{d x}) \frac{r}{2}$ — zero at the centre, maximum at the wall (linear)

Velocity: $u = \frac{1}{4 μ} (- \frac{d p}{d x}) (R^{2} - r^{2})$ — parabolic

Maximum velocity (centre) $= 2 \times$ mean velocity: $u_{ma x} = 2 \overset{u}{ˉ}$

Pressure drop: $p_{1} - p_{2} = \frac{32 μ u ˉ L}{D ^{2}}$ → $h_{f} = \frac{32 μ u ˉ L}{ρ g D ^{2}}$

Friction factor (Darcy): $f = \frac{64}{R e}$

Wall shear stress: $τ_{0} = \frac{8 μ u ˉ}{D}$ ; correction factors $α = 2$ , $β = 4/3$

Head loss is proportional to velocity ( $h_{f} \propto V$ ) in laminar flow; in fully turbulent rough flow it is proportional to $V^{2}$ .
The velocity equals the mean velocity at $r = R / 2 = 0.707 R$ .

Laminar flow between parallel plates

Both plates fixed (plane Poiseuille flow)

Gap $B$ , $y$ measured from one plate:

u = \frac{1}{2 μ} (- \frac{d p}{d x}) (B y - y^{2}), u_{ma x} = 1.5 \overset{u}{ˉ}, p_{1} - p_{2} = \frac{12 μ u ˉ L}{B ^{2}}

Shear stress varies linearly, zero at mid-plane.

One plate moving (Couette flow)

Upper plate moving at $U$ , no pressure gradient: linear velocity $u = U y / B$ , uniform shear $τ = μU / B$ . With a pressure gradient, the parabolic and linear profiles add.

Viscous resistance of bearings

Journal bearing (shaft diameter $D$ , bearing length $L$ , oil film thickness $t$ , speed $N$ rpm):

τ = μ \frac{π D N}{60 t}, T = \frac{μ π ^{2} N D ^{3} L}{120 t}, P = \frac{μ π ^{3} N ^{2} D ^{3} L}{3600 t}

Measurement of viscosity

Method	Principle / formula
Capillary tube viscometer	Hagen–Poiseuille: $μ = \frac{π ρ g h _{f} D ^{4}}{128 Q L}$
Falling sphere viscometer	Stokes' law: drag $F_{D} = 3 π μ D V$ ; terminal velocity $V = \frac{D ^{2} ( ρ _{s} - ρ ) g}{18 μ}$ (valid for very small $R e$ , below about 0.2)
Rotating cylinder viscometer	Torque on the inner cylinder due to the shear of fluid in a narrow annulus
Orifice (efflux) viscometers	Time for a fixed volume to flow through an orifice — Saybolt, Redwood, Engler

Turbulent flow

In turbulent flow the velocity at a point fluctuates: $u = \overset{u}{ˉ} + u^{'}$ . The fluctuations transfer momentum and produce additional (apparent) shear stresses.

FormulaTurbulent shear stress

Reynolds stress: $τ = - ρ \overline{u^{'} v^{'}}$
Boussinesq eddy viscosity: $τ = η \frac{d u ˉ}{d y}$ ( $η$ depends on the flow, not a fluid property)
Prandtl mixing length: $τ = ρ l^{2} (\frac{d u ˉ}{d y})^{2}$ , with $l = κ y$ near the wall, $κ \approx 0.4$ (Kármán constant)

Total shear = viscous + turbulent: $τ = μ \frac{d u}{d y} + η \frac{d u}{d y}$

Shear velocity and laminar sub-layer

Shear (friction) velocity: $u_{*} = τ_{0} / ρ$ . Using $τ_{0} = f ρ V^{2} /8$ : $u_{*} = V f /8$ .
Next to the wall, a very thin laminar sub-layer exists where viscous shear dominates: $δ^{'} = \frac{11.6 ν}{u _{*}}$ .

Hydrodynamically smooth and rough boundaries

Compare average roughness height $k$ with $δ^{'}$ :

$k / δ^{'}$	Boundary	Friction factor depends on
< 0.25	Smooth — roughness buried in the sub-layer	$R e$ only
0.25 – 6	Transition	$R e$ and $k / D$
> 6	Rough — roughness projects through	$k / D$ only

A pipe can behave smooth at low $R e$ and rough at high $R e$ (sub-layer thins as $u_{*}$ increases).

Velocity distribution in turbulent pipe flow

FormulaLogarithmic velocity profiles (Prandtl–Kármán, Nikuradse)

Smooth pipes: $\frac{u}{u _{*}} = 5.75 lo g_{10} \frac{u _{*} y}{ν} + 5.5$
Rough pipes: $\frac{u}{u _{*}} = 5.75 lo g_{10} \frac{y}{k} + 8.5$
Velocity defect law (both): $\frac{u _{ma x} - u}{u _{*}} = 5.75 lo g_{10} \frac{R}{y}$
Mean velocity: $\frac{u _{ma x} - u ˉ}{u _{*}} = 3.75$

Power law (smooth pipes, moderate $R e$ ): $\frac{u}{u _{ma x}} = (\frac{y}{R})^{1/7}$

The turbulent profile is much flatter than the laminar parabola ( $u_{ma x} / \overset{u}{ˉ}$ ≈ 1.2 or less).

Friction factor for turbulent flow

Regime	Relation
Smooth, $4000 < R e < 1 0^{5}$	Blasius: $f = \frac{0.316}{R e ^{1/4}}$
Smooth, higher $R e$	$\frac{1}{f} = 2 lo g_{10} (R e f) - 0.8$
Fully rough	$\frac{1}{f} = 2 lo g_{10} \frac{R}{k} + 1.74$
Transition (commercial pipes)	Colebrook–White: $\frac{1}{f} = - 2 lo g_{10} (\frac{k / D}{3.7} + \frac{2.51}{R e f})$

The Moody diagram plots $f$ against $R e$ for various relative roughness $k / D$ ; explicit approximations (e.g. Swamee–Jain) avoid iteration.

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Laminar & Turbulent Flow