Engineer's Notebook
Reynolds number
The Reynolds number compares a flow's inertia with its viscosity. In a pipe it predicts whether the flow is smooth and layered or chaotic:
Re =ρ·v·Dμ
- Re
- Reynolds number
- —
- ρ
- density
- kg/m³
- v
- mean flow speed
- m/s
- D
- pipe diameter
- m
- μ
- dynamic viscosity
- Pa·s
- ν
- kinematic viscosity, μ/ρ
- m²/s
- x
- distance from a plate's leading edge
- m
- Dh
- hydraulic diameter
- m
- P
- wetted perimeter
- m
| ν = μ/ρ | 1 × 10⁻⁶ m²/s |
| Regime | laminar |
| Re = ρvD/μ | 1,497 |
Worked example
Uses the values in Fig. 1.3-1. Change them and the working updates.
Given
- ρwater at 20 °C998 kg/m³
- μdynamic viscosity1 × 10⁻³ Pa·s
- vmean speed0.03 m/s
- Dpipe diameter50 mm
Find
The Reynolds number, and whether the flow is laminar or turbulent.
Solution
Re = ρvD/μ
= 998 kg/m³ × 0.03 m/s × 0.05 m / 1 × 10⁻³ Pa·s
= 1,497
Answer
Re = 1,497
Pipe flow is laminar below about 2,300 and turbulent above about 4,000, so this flow is laminar.
Related forms
On a flat plate, transition to turbulence starts near Rex ≈ 5 × 10⁵.
-
With kinematic viscosity
Re = vDν
-
Flat plate, distance x along it
Rex = ρvxμ
-
Non-circular duct
Dh = 4AP
Assumptions
- The thresholds, laminar below about 2,300 and turbulent above about 4,000, are for fully developed flow in smooth, straight, round pipes.
- Transition is not sharp. Very smooth, undisturbed pipes can stay laminar well above 2,300.
- Fluid properties are typical values at 20 °C. Viscosity changes strongly with temperature, especially for oils and glycerol.
Derivation
- Inertial force on a parcel of fluid scales as dynamic pressure times area:
ρv² · D². - Viscous force scales as shear stress times area:
(μv/D) · D² = μvD. - Their ratio is
ρv²D² / μvD = ρvD/μ.
When Re is large, inertia wins and small disturbances grow into turbulence. When it is small, viscosity damps them out.