Flow

Flow is smooth (laminar) or chaotic (turbulent) depending on the Reynolds number. Learn Re, transition, velocity profiles, boundary layers and why turbulence matters for mixing, heat transfer and pressure loss.

Watch smoke rising from a candle: it starts as a smooth thread (laminar) and then breaks into swirls (turbulent). The same transition happens in pipes, over wings and around cars, and it changes friction, heat transfer and noise dramatically.

One dimensionless number, the Reynolds number, predicts which regime you're in.

An analogy

Laminar flow is a disciplined marching band in neat lanes; turbulent flow is a crowd rushing out of a stadium — the same average direction but constant mixing and jostling.

Inertia vs viscosity

Re compares inertial forces (which amplify disturbances) with viscous forces (which damp them). Low Re (slow, small, viscous) = laminar; high Re = turbulent. Most engineering water and air flows are turbulent.

Velocity profiles

Laminar pipe flow has a parabolic profile; turbulent flow is much flatter in the core with steep gradients near the wall — which is why turbulent flow has higher wall friction and heat transfer.

Boundary layers

Near any surface a thin layer slows to zero velocity at the wall (no-slip). On a flat plate it transitions at Re_x ≈ 5 × 10⁵. Adverse pressure gradients can make it separate, causing stall on wings and drag behind bluff bodies.

Modelling turbulence

Turbulence contains eddies from the pipe size down to the Kolmogorov scale. Direct simulation (DNS) is extremely costly, so engineering CFD uses RANS models (k-ε, k-ω SST) or LES. Dimples on golf balls deliberately trip turbulence to delay separation and cut drag.

Formulas

• Reynolds number: Re = ρVD/μ = VD/ν
• Pipe regimes: Re < 2300 laminar, 2300–4000 transitional, > 4000 turbulent
• Laminar profile: u/u_max = 1 − (r/R)², V_avg = u_max/2
• Turbulent profile (approx.): u/u_max = (1 − r/R)^(1/7)

Worked example: Water in a pipe

Given: V = 1 m/s, D = 50 mm; ρ = 998, μ = 1.0 × 10⁻³ Pa·s

Find: Reynolds number and regime.

• Re = 998 × 1 × 0.05/0.001 = 49 900
• > 4000 → turbulent

Answer: Re ≈ 5 × 10⁴ (turbulent)

Worked example: Oil in a hydraulic line

Given: V = 0.5 m/s, D = 25 mm; ν = 1 × 10⁻⁴ m²/s

Find: Regime.

• Re = 0.5 × 0.025/10⁻⁴ = 125

Answer: Laminar

Common mistakes

• Wrong: Mixing mm and m in Re. — Right: Re is dimensionless — use consistent SI units.
• Wrong: Using laminar formulas for turbulent pipe flow. — Right: Check Re first; use Moody/Colebrook when turbulent.

Applications

• Heat exchangers: Turbulent flow is desired for high heat-transfer coefficients.
• Lubrication & microfluidics: Laminar flow dominates thin films, blood capillaries and lab-on-chip devices.
• Aerodynamics: Laminar-flow wings and riblets reduce drag; vortex generators delay separation.

Key takeaways

• Re = ρVD/μ decides the regime.
• Laminar < 2300 < transition < 4000 < turbulent.
• Turbulence: flatter profile, more friction and mixing.
• Boundary layers and separation govern drag.

Going further

• Law of the wall: Near-wall turbulent velocity scales as u⁺ = y⁺ (viscous sublayer) and u⁺ = (1/κ)ln y⁺ + B (log layer); CFD meshes are built around y⁺.
• Transition control: Suction, smooth surfaces and favourable pressure gradients extend laminar flow on aircraft wings.
• Non-dimensional similarity: Matching Re (and Mach, Froude) lets scale models in wind tunnels predict full-size behaviour.