What is the Reynolds number formula?

Reynolds number is defined as Re = (ρ · v · L) / μ = (v · L) / ν, where ρ is fluid density, v is velocity, L is characteristic length (such as pipe diameter), μ is dynamic viscosity, and ν is kinematic viscosity.

What are the critical Reynolds number limits for pipe flow?

In circular pipes, flow is typically laminar when Re 4,000.

What physical ratio does the Reynolds number represent?

It quantifies the ratio of inertial forces to viscous forces within a moving fluid.

Fluid Mechanics Calculator

Reynolds Number Calculator

Calculate the dimensionless Reynolds number (Re) to evaluate the ratio of inertial forces to viscous forces and predict laminar, transitional, or turbulent flow regimes.

Flow Regime Profile

Internal velocity streamlines across laminar and turbulent regimes

Flow & Fluid Inputs

Fluid kinematics and physical properties

Flow Velocity (v)

m/s

Mean relative fluid flow velocity.

Characteristic Length / Diameter (L, D)

Pipe internal diameter or body dimension.

Fluid Properties

Fluid Density (ρ)

kg/m³

Fluid mass density.

Dynamic Viscosity (μ)

Pa·s

Fluid dynamic shear viscosity.

Regime Bounds

For circular pipe conduits:
Laminar: Re < 2,000
Transitional: 2,000 ≤ Re ≤ 4,000
Turbulent: Re > 4,000

Reynolds Number (Re)

74,700.6

Scientific Notation: 7.4701e+4

Pipe Flow Regime

Turbulent

Kinematic Viscosity (ν)

1.0040e-6 m²/s

External Flat Plate Boundary Layer: Laminar (Transition at Re ≈ 5×10⁵)

Governing Formula

Re = (ρ · v · L) / μ = (v · L) / ν

Re—

Dimensionless Reynolds number

vm/s

Fluid velocity

Lm / mm

Characteristic length (diameter)

ρkg/m³

Fluid density

μPa·s

Dynamic shear viscosity

νm²/s

Kinematic viscosity (μ / ρ)

Calculation Assumptions

Newtonian fluid behavior[cite: 4]
Homogeneous continuum flow
Constant fluid temperature
Uniform 1D bulk velocity
No-slip wall boundary condition
Standard critical threshold limits

Engineering Code

Reuse the calculation in your own engineering workflow.

Python

def reynolds_number(velocity, length_mm, density, dynamic_viscosity):
    """
    Calculate dimensionless Reynolds number.
    velocity: Flow velocity (m/s)
    length_mm: Characteristic dimension (mm)
    density: Fluid density (kg/m³)
    dynamic_viscosity: Dynamic shear viscosity (Pa·s)
    """
    L_m = length_mm / 1000.0
    Re = (density * velocity * L_m) / dynamic_viscosity
    
    if Re < 2000:
        regime = "Laminar"
    elif 2000 <= Re <= 4000:
        regime = "Transitional"
    else:
        regime = "Turbulent"
        
    return Re, regime

# Example
v = 1.5
L = 50
rho = 998
mu = 0.001002

Re, regime = reynolds_number(v, L, rho, mu)
print(f"Reynolds Number: {Re:.1f} ({regime})")

MATLAB

function [Re, regime] = reynolds_number(v, L_mm, rho, mu)
    L_m = L_mm / 1000;
    Re = (rho * v * L_m) / mu;
    if Re < 2000
        regime = 'Laminar';
    elseif Re <= 4000
        regime = 'Transitional';
    else
        regime = 'Turbulent';
    end
end

% Example
[Re, regime] = reynolds_number(1.5, 50, 998, 0.001002);
fprintf('Reynolds Number: %.1f (%s)\n', Re, regime);

Excel Formula

=(rho * v * (L_mm / 1000)) / mu

Example Calculation

For water (density ρ = 998 kg/m³, dynamic viscosity μ = 0.001002 Pa·s) flowing at 1.5 m/s through a 50 mm (0.050 m) diameter pipe:

Re = (998 × 1.5 × 0.050) / 0.001002

Re = 74,700.6 (Turbulent Flow Regime)

Technical Explanation: Understanding the Reynolds Number

Named after Osborne Reynolds, the Reynolds number ($Re$) is the most fundamental dimensionless quantity in fluid mechanics. It predicts whether fluid flow along a surface or conduit will be smooth and organized (laminar) or chaotic and eddy-dominated (turbulent).

Physical Meaning

The Reynolds number represents the ratio of dynamic pressure forces (momentum/inertia) resisting disturbances to internal viscous forces dampening instabilities:

Re = (Inertial Forces) / (Viscous Forces) = (ρ · v · L) / μ = (v · L) / ν

Flow Regimes in Enclosed Pipes

Laminar Flow (Re < 2,000): Viscous forces dominate. Fluid layers slide past one another smoothly with parabolic velocity profiles and minimal mixing.
Transitional Flow (2,000 ≤ Re ≤ 4,000): Unstable intermediate zone where flow randomly oscillates between laminar streams and turbulent bursts.
Turbulent Flow (Re > 4,000): Inertial forces overwhelm fluid shear damping. Rapid cross-stream momentum exchange produces flat velocity profiles and higher wall friction.

Continue Calculating

built

Ohms Law

Quickly calculate and analyze parameters for your ohms law requirements.

Open

built

Pcb Trace Calculator

Quickly calculate and analyze parameters for your pcb trace calculator requirements.

Open

built

Ugv 4Wd Platform

Quickly calculate and analyze parameters for your ugv 4wd platform requirements.

Open

Real-World Engineering Cases

Flow Meter Calibration Inaccuracies Across Viscosity Changes

A turbine flow meter calibrated for light diesel (low viscosity, Re > 50,000) was used to meter heavy fuel oil during cold start conditions (Re ~ 1,500). The shift from fully turbulent to laminar velocity profiling introduced a 14% metering volume error.

Engineering Lesson

Differential pressure and mechanical velocity meters are highly Reynolds-number dependent. Metering calibrations must account for kinematic viscosity shifts across varying fluid temperatures.

Inadequate Heat Transfer in Laminar Tube Heat Exchangers

In a viscous chemical reactor cooling jacket, fluid flow remained deep in the laminar regime (Re ~ 450). Because laminar flow lacks turbulent eddies to mix core fluid with the tube wall, the thermal boundary layer acted as an insulator, causing reactor overheating.

Engineering Lesson

High heat transfer coefficients generally require turbulent mixing. Where viscosity forces laminar conditions, designers must incorporate static turbulators or rifled tube profiles to break thermal boundary layers.

Frequently Asked Questions

What is the characteristic length (L) for non-circular ducts?

For non-circular conduits or rectangular ducts, use the hydraulic diameter $D_h = 4A / P$, where $A$ is the cross-sectional flow area and $P$ is the wetted perimeter.

What is the difference between dynamic and kinematic viscosity?

Dynamic viscosity (μ, in Pa·s or cP) measures internal shear resistance under applied force. Kinematic viscosity (ν, in m²/s or cSt) is the ratio of dynamic viscosity to fluid density (ν = μ / ρ), measuring resistance to flow under gravity.

Does temperature affect the Reynolds number?

Significantly. Liquid viscosity drops sharply as temperature rises (increasing the Reynolds number), whereas gas viscosity increases with temperature (decreasing the Reynolds number).