Reynolds number

Clinical relevance in anaesthesia

Predicts likelihood of transition from laminar to turbulent flow in tubes and around objects
- Turbulence increases resistance and work of breathing; affects gas delivery and airway pressures
Airway equipment: ETTs, LMAs, breathing circuits, filters, connectors
- Smaller diameter and higher flows increase Re → more turbulence → disproportionate rise in pressure drop
Clinical scenarios where Re rises
- High inspiratory flow rates (IPPV, high minute ventilation, recruitment manoeuvres)
- Low density/viscosity changes: density dominates in gases; heliox lowers density and reduces Re
- Obstruction/irregularity: secretions, kinks, narrowed ETT, bronchospasm → local high velocity and disturbed flow
Blood flow: usually laminar in large vessels; turbulence more likely with high velocity or stenosis
- Murmurs/bruits relate to turbulent flow; critical in valvular disease and arterial stenoses

Definition and formula

Reynolds number is a dimensionless ratio of inertial to viscous forces in a flowing fluid
Re = (ρ v D) / μ
- ρ = density (kg·m⁻³)
- v = mean velocity (m·s⁻¹)
- D = characteristic length (for a tube: internal diameter) (m)
- μ = dynamic viscosity (Pa·s)
Alternative form using kinematic viscosity (ν = μ/ρ): Re = (v D) / ν

Interpretation: laminar vs turbulent

In straight, smooth, rigid tubes: laminar flow usually when Re < ~2000
Transitional/unstable region: Re ~2000–4000 (disturbances can trigger turbulence)
Turbulent flow usually when Re > ~4000
Thresholds are not absolute: depend on tube roughness, pulsatility, bends, branching, and upstream disturbances

How changing variables affects Re (and turbulence risk)

Increase Re (more turbulence likely): ↑ρ, ↑v, ↑D, ↓μ
- In gases, density changes (e.g., heliox vs air) can be clinically significant
Decrease Re (more laminar likely): ↓ρ, ↓v, ↓D, ↑μ
- Reducing inspiratory flow (longer inspiratory time) can reduce turbulence and peak pressures

Link to resistance and pressure drop

Laminar flow: pressure drop is proportional to flow (ΔP ∝ Q) (Poiseuille relationship)
Turbulent flow: pressure drop is approximately proportional to flow squared (ΔP ∝ Q²)
- Explains large increases in airway pressure when flow is increased through narrow/irregular airways or ETTs
In practice, airway flow is often mixed: laminar in small peripheral airways; turbulent in trachea/large bronchi at high flows

Worked-style relationships commonly examined

If flow rate (Q) increases, velocity increases (v = Q/A); for a tube A ∝ D² so v ∝ Q/D²
Substitute into Re: Re ∝ ρ (Q/D²) D / μ = ρ Q / (μ D)
- At fixed volumetric flow Q, smaller diameter increases Re (counterintuitive if you only remember Re ∝ D)
At fixed velocity v (not fixed flow), increasing diameter increases Re (Re ∝ D)

Anaesthetic examples

ETT: narrowing (smaller ID, secretions) increases velocity for a given minute ventilation → increases Re and turbulence → higher peak pressures
Heliox (He/O2): lower density than air/O2 → reduces Re → promotes laminar flow and reduces pressure drop in obstructed airways
High flow oxygen devices and CPAP: high flow through connectors can be turbulent; pressure losses depend strongly on flow
Arterial stenosis: local velocity increases at narrowing → Re rises → turbulence distal to stenosis (bruit)

Define Reynolds number and state what it predicts.

Aim: definition + physical meaning + clinical use.

Reynolds number is a dimensionless ratio of inertial to viscous forces: Re = ρ v D / μ
It predicts the likelihood of laminar vs turbulent flow; higher Re means turbulence more likely

What are the typical Reynolds number thresholds for laminar and turbulent flow in a straight tube?

Laminar usually when Re < ~2000
Transitional ~2000–4000
Turbulent usually when Re > ~4000
Not absolute: depends on roughness, bends, pulsatility, upstream disturbances

List the variables in Reynolds number and describe how each affects turbulence risk.

Re = ρ v D / μ
Increase Re (more turbulence likely): ↑density, ↑velocity, ↑diameter (at fixed v), ↓viscosity
Decrease Re: ↓density, ↓velocity, ↓diameter (at fixed v), ↑viscosity

Why can a smaller endotracheal tube increase the likelihood of turbulent flow for a given minute ventilation?

Common FRCA trap: distinguish fixed flow vs fixed velocity.

For a given volumetric flow (Q), velocity v = Q/A and A ∝ D², so v ∝ Q/D²
Then Re = ρ v D / μ ∝ ρ (Q/D²) D / μ = ρ Q / (μ D)
So at fixed Q, decreasing D increases Re and promotes turbulence, increasing pressure drop and peak airway pressure

How does the pressure-flow relationship differ between laminar and turbulent flow, and why does this matter clinically?

Laminar: ΔP ∝ Q (linear); resistance is constant for a given tube and fluid properties
Turbulent: ΔP ∝ Q² (non-linear); small increases in flow can cause large increases in pressure required
Clinical impact: high inspiratory flows through narrow/obstructed ETTs or bronchospastic airways markedly increase peak pressure and work of breathing

Explain why heliox can improve airflow in upper airway obstruction using Reynolds number.

Helium has much lower density than nitrogen/air; Re ∝ ρ
Lower Re reduces turbulence and promotes more laminar flow through narrowed segments
This reduces pressure drop for a given flow and can reduce work of breathing in obstruction

A viva: compare the likelihood of turbulence in trachea vs small bronchi and explain using Reynolds number.

Re depends on velocity and diameter; in large airways, diameter is larger and flows/velocities can be high, especially during exercise/IPPV
In peripheral small airways, total cross-sectional area is very large so velocity is low despite small diameter → Re tends to be low → laminar predominates

How do bends, connectors, roughness, and pulsatile flow affect the usefulness of Reynolds number thresholds?

They introduce disturbances and energy losses, so transition to turbulence can occur at lower Re than in an ideal straight smooth tube
Therefore Re thresholds (2000/4000) are guides, not absolute cut-offs in clinical systems (airways, circuits, vessels)

Past FRCA-style calculation: A gas (ρ = 1.2 kg·m⁻³, μ = 1.8×10⁻⁵ Pa·s) flows in a 8 mm ID tube with mean velocity 2 m·s⁻¹. Estimate Re and comment on flow type.

Use Re = ρ v D / μ = (1.2 × 2 × 0.008) / (1.8×10⁻⁵)
Numerator = 0.0192; divide by 1.8×10⁻⁵ → Re ≈ 1067
Likely laminar in an ideal straight tube (Re < 2000), but fittings/bends could still generate disturbed flow

Past FRCA-style question: Show how Reynolds number changes with diameter at (i) fixed velocity and (ii) fixed volumetric flow.

(i) Fixed velocity v: Re = ρ v D / μ so Re ∝ D
(ii) Fixed volumetric flow Q: v = Q/A and A ∝ D² so v ∝ Q/D²
Then Re ∝ ρ (Q/D²) D / μ = ρ Q / (μ D), so Re ∝ 1/D

Past FRCA-style viva: A patient with severe bronchospasm is being ventilated. What ventilator adjustments reduce turbulence-related pressure losses?

Reduce inspiratory flow (e.g., lower peak flow, longer inspiratory time) to reduce velocity and Re
Accept permissive hypercapnia where appropriate to avoid high minute ventilation demands
Treat the cause: bronchodilators, suction secretions, ensure adequate ETT size/patency