Get Re, flow regime, and full step-by-step working — plus 3 solved practice problems
kg/m³
m/s
m
Pa·s (kg/m·s)
Reynolds Number:
--
--
What is the Reynolds Number?
The Reynolds number (Re) is a dimensionless quantity used to predict whether flow in a pipe will be smooth (laminar) or chaotic (turbulent). It is calculated as:
Re = (ρ × v × D) / μ
where ρ is fluid density, v is flow velocity, D is the pipe's internal diameter, and μ is the fluid's dynamic viscosity.
Flow Regimes
Reynolds Number
Regime
Description
Re < 2300
Laminar
Smooth, orderly flow in parallel layers
2300 ≤ Re ≤ 4000
Transitional
Unstable, switches between laminar and turbulent
Re > 4000
Turbulent
Chaotic, mixing flow with eddies
Why It Matters
Engineers use the Reynolds number to select the right friction factor correlation, predict pressure drop, and size pumps correctly. Using a laminar-flow formula on turbulent flow (or vice versa) gives badly wrong pressure drop estimates.
Solved Examples (Practice Problems)
Click "Try This Example" to auto-fill the calculator above and see the full step-by-step working.
Example 1 — Water in a pipe (turbulent)
Water (ρ = 998 kg/m³, μ = 0.001 Pa·s) flows at 2.5 m/s through a 50 mm diameter pipe. Find the Reynolds number and flow regime.
Example 2 — Lubricating oil in a pipe (laminar)
A lubricating oil (ρ = 900 kg/m³, μ = 0.29 Pa·s) flows at 0.6 m/s through a 25 mm diameter pipe. Find the Reynolds number and flow regime.
Example 3 — Air in a duct (transitional zone)
Air (ρ = 1.2 kg/m³, μ = 0.000018 Pa·s) flows at 2 m/s through a 25 mm diameter duct. Find the Reynolds number and flow regime.
Worked Solutions in Full
These are the same three problems as above, written out line by line so you can follow every step of the arithmetic without the calculator.
Example 1 — Water in a pipe
Given: ρ = 998 kg/m³, v = 2.5 m/s, D = 0.05 m, μ = 0.001 Pa·s
Step 1 — Write the formula: Re = ρ × v × D / μ
Step 2 — Substitute the values: Re = (998 × 2.5 × 0.05) / 0.001
Step 3 — Multiply the top line: 998 × 2.5 = 2495, then 2495 × 0.05 = 124.75
Step 4 — Divide by the viscosity: 124.75 / 0.001 = 124,750
Step 5 — Classify the flow: Re is far above 4,000, so the flow is turbulent. Pressure drop for this pipe must be calculated with a turbulent friction factor (the Friction Factor Calculator does this next).
Answer: Re ≈ 124,750
Example 2 — Lubricating oil in a pipe
Given: ρ = 900 kg/m³, v = 0.6 m/s, D = 0.025 m, μ = 0.29 Pa·s
Step 1 — Write the formula: Re = ρ × v × D / μ
Step 2 — Substitute the values: Re = (900 × 0.6 × 0.025) / 0.29
Step 3 — Multiply the top line: 900 × 0.6 = 540, then 540 × 0.025 = 13.5
Step 4 — Divide by the viscosity: 13.5 / 0.29 = 46.55
Step 5 — Classify the flow: Re is far below 2,300, so the flow is laminar. A thick oil moving slowly in a narrow pipe is the classic laminar case, and the friction factor is simply 64/Re.
Answer: Re ≈ 46.55
Example 3 — Air in a duct
Given: ρ = 1.2 kg/m³, v = 2 m/s, D = 0.025 m, μ = 0.000018 Pa·s
Step 1 — Write the formula: Re = ρ × v × D / μ
Step 2 — Substitute the values: Re = (1.2 × 2 × 0.025) / 0.000018
Step 3 — Multiply the top line: 1.2 × 2 = 2.4, then 2.4 × 0.025 = 0.06
Step 4 — Divide by the viscosity: 0.06 / 0.000018 = 3,333.33
Step 5 — Classify the flow: Re lies between 2,300 and 4,000, so the flow is in the transitional zone. It can flip between laminar and turbulent, so designers avoid depending on a single friction-factor value here.
Answer: Re ≈ 3,333.33
Getting velocity when you only know the flow rate
Often a problem gives volumetric flow rate instead of velocity. Use v = Q / A, where A = π D² / 4. For water at 10 m³/h in a 50 mm pipe:
Q = 10 / 3600 = 0.002778 m³/s. A = π × 0.05² / 4 = 0.001963 m². v = 0.002778 / 0.001963 = 1.415 m/s.
Then Re = (998 × 1.415 × 0.05) / 0.001 = 70,594, which is turbulent.
Practical Notes and Common Mistakes
The Reynolds number decides which friction-factor correlation you may use, which heat-transfer correlation applies, and how a mixer or pipe will behave. It appears at the start of almost every fluid-flow and heat-transfer calculation in a chemical process design, so an error here carries into everything after it.
Mistakes that give wrong answers
Using the outside or nominal pipe diameter. Always use the inside diameter, because that is the flow area the fluid actually sees.
Mixing units. Diameter in millimetres with velocity in m/s gives a result 1,000 times too large. Convert to metres first.
Viscosity in centipoise. 1 cP = 0.001 Pa·s. Water at room temperature is about 1 cP, which is 0.001 Pa·s.
Mixing up dynamic and kinematic viscosity. μ (Pa·s) goes with density; ν (m²/s) already includes density. Use one formula or the other, never both.
Treating 2,300 as a hard switch. Real pipes show transition over a range that depends on entrance conditions and disturbances.
Non-circular ducts. Replace D with the hydraulic diameter, Dh = 4 × flow area / wetted perimeter.
Quick unit reference
Quantity
Conversion
Viscosity
1 cP = 0.001 Pa·s = 1 mPa·s
Diameter
1 mm = 0.001 m; 1 inch = 0.0254 m
Flow rate
1 m³/h = 0.000278 m³/s; 1 L/s = 0.001 m³/s
Kinematic viscosity
1 cSt = 1 × 10⁻⁶ m²/s
Frequently Asked Questions
What is a good Reynolds number for turbulent flow?
For flow inside a circular pipe, Re above 4000 is generally treated as fully turbulent. Between 2300 and 4000 the flow is transitional and unstable — in practice, engineers usually design to avoid operating in this range because pressure drop becomes unpredictable.
Can I calculate Reynolds number without knowing density?
Yes — switch to "Kinematic Viscosity" mode above. Instead of Re = ρvD/μ, it uses Re = vD/ν, where ν (kinematic viscosity) already accounts for density. This is common when working from a fluid properties table that lists ν directly, such as for air or steam.
Does pipe diameter use inside diameter or outside diameter?
Always use the internal (inside) diameter — the actual cross-section the fluid flows through, not the pipe's nominal or outer diameter. For non-circular ducts, use the hydraulic diameter instead.
Why does Reynolds number matter for pump and pipe sizing?
The flow regime determines which friction factor correlation is valid. Laminar flow uses f = 64/Re; turbulent flow requires the Colebrook or Moody chart approach. Using the wrong one gives a badly wrong pressure drop, which leads to an under- or over-sized pump.
Related Tool
Already have your Reynolds number? Use it directly to find pressure drop with the Friction Factor & Head Loss Calculator — it picks up right where this one leaves off.