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Reynolds experiment and flow regimes

Short answer

Reynolds injected a dye filament into pipe flow; at low velocity it remained a straight line (laminar), at high velocity it diffused (turbulent). Reynolds number Re = ρVD/μ = VD/ν. Critical values: Re < 2000 laminar, 2000–4000 transition, >4000 turbulent. Laminar flow is orderly with parabolic velocity profile; turbulent flow is chaotic with flatter profile and higher mixing.

Describe Reynolds' experiment with a neat sketch. Define Reynolds number and state its critical values for pipe flow. Differentiate between laminar and turbulent flow, and sketch their velocity profiles.

Reynolds' experiment

Osborne Reynolds (1883) studied the nature of flow in a glass tube by injecting a thin filament of dye into flowing water.

  water tank
 +-----------+      dye
 |           |       |
 |           |   ____v___
 |           |==|  tube  |=====> to valve
 +-----------+  |_________|

Observations

  1. At low velocity, the dye moves as a straight, thin line parallel to the tube axis: laminar flow (layers slide over each other).
  2. When the velocity is increased, the dye line begins to wave: transition.
  3. At high velocity, the dye mixes completely with the water, showing eddies and random motion: turbulent flow.

Reynolds found that the change depends on the dimensionless group:

Re=ρVDμ=VDν=inertia forceviscous forceRe = \frac{\rho V D}{\mu} = \frac{V D}{\nu} = \frac{\text{inertia force}}{\text{viscous force}}

Critical values (pipe flow)

  • Re<2000Re < 2000: laminar
  • 2000<Re<40002000 < Re < 4000: transition (unstable)
  • Re>4000Re > 4000: turbulent

The exact values depend on the disturbances present; the lower critical value of about 2000 is the commonly used limit.

Laminar vs turbulent flow

BasisLaminarTurbulent
MotionOrderly layers, no mixingRandom, eddying, mixing
Reynolds numberRe<2000Re < 2000Re>4000Re > 4000
Shear stressDue to viscosity: τ=μ du/dy\tau = \mu\,du/dyViscous plus turbulent (Reynolds) stress
Velocity profileParabolic, Vavg=0.5 umaxV_{avg} = 0.5\,u_{max}Flatter, Vavg≈0.8 umaxV_{avg} \approx 0.8\,u_{max}
Head losshf∝Vh_f \propto Vhf∝V1.75 to 2h_f \propto V^{1.75\text{ to }2}
Friction factorf=64/Ref = 64/ReDepends on ReRe and roughness
  Laminar          Turbulent
  |   .--          |  .----
  |  /             | |
  | |              | |
  |  \             | |
  |   '--          |  '----

✓ Answer checked by Claude Opus 5.5 (high effort)

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