Reynolds number

The Reynolds number (Re) is the primary parameter used to define the transition of fluid motion between laminar and turbulent flow patterns ¹. The Reynolds number represents the ratio of inertia forces to viscous forces, and as such has no units (i.e. is a dimensionless quantity) ¹. 

Calculation

In a straight pipe of constant diameter ²:

Re = (ρvD) / μ

Where: ρ = fluid density; v = fluid velocity; D = diameter of pipe; μ = dynamic viscosity of fluid.

Laminar flow is characterised by a parabolic flow profile comprised of concentric fluid laminae, with each layer increasing in velocity towards the vessel centre ². At low Reynolds numbers (<2000), viscous forces sufficiently outweigh inertia forces, and laminar flow predominates ^1,2. If this balance shifts in favour of inertia (e.g. by increasing fluid velocity or vessel diameter), the Reynolds number will increase. Past a critical Reynolds number (generally >2000), flow becomes chaotic, generating vortices and eddies characteristic of a turbulent pattern ^1,2. When turbulence reigns, a greater driving pressure is required to generate an equivalent degree of flow in the same vessel.

The critical Reynolds number is highly variable depending on pipe geometry, which in the human body varies considerably due to vessel curvature, branching and viscoelastic properties ². These factors, combined with the pulsatile nature of the cardiac cycle, lead to the approximation of laminar flow only along short distances in small abdominal vessels ². Despite this, the Reynolds number retains its use as a guiding relationship for predicting likely blood flow patterns.

Clinical use

Immediately distal to a vessel stenosis, blood velocity is increased such that the Reynolds number exceeds critical. Post-stenotic turbulence and flow separation leads to an increased range of blood velocities which manifests as spectral broadening on Doppler ultrasound ².

Turbulent flow in a vessel region increases endothelial shear stress and promotes thrombosis (see Virchow’s triad) ¹.

History and etymology

In 1851, Irish physicist George Stokes first documented the concept behind the Reynolds number, which in 1883, Osborne Reynolds (1842-1912) demonstrated in experiment ³. The governing parameter he described was named after him in 1908 by German physicist, Arnold Sommerfeld.